Ribbon categories from ind-exact algebras: simple current case

RIBBON CA TEGORIES FR OM IND-EXA CT ALGEBRAS: SIMPLE CURRENT CASE KENICHI SHIMIZU AND HARSHIT Y AD A V Abstract. W e giv e criteria for when ﬁnitely generated local modules o v er a comm utativ e algebra A in the ind-completion b C of a braided tensor category C inherit the structure of a (rigid, braided, ribb on) tensor category . W e then apply this to simple curren t algebras A = L g ∈ Γ E g , where Γ is a subgroup of inv ertible ob jects in C . Using a description of simple A -mo dules, we verify the required hypotheses for this class of algebras and deduce rigidity , braided, ribbon, and non- degeneracy prop erties for their ﬁnitely generated local mo dules. As applications, w e construct examples of ribb on tensor categories from quantum sup ergroup categories for unrolled gl (1 | 1). 1. Introduction Simple curren ts (i.e. in vertible ob jects) play a central role in constructions and classiﬁcation problems in braided tensor categories arising in representation theory and conformal ﬁeld the- ory . Giv en a braided tensor category C and a subgroup Γ of its group of isomorphism classes of in v ertible ob jects, the ob ject A = L g ∈ Γ E g (where E g ∈ C denotes an in v ertible ob ject corre- sp onding to g ∈ Γ) has a structure of an algebra in the ind-completion b C under an appropriate co cycle condition. W e call an algebra of this t yp e a simple curr ent algebr a (see Deﬁnition 2.8 for the precise deﬁnition). The goal of this pap er is to give work able criteria ensuring that suitable sub categories of mo dules ov er a comm utativ e algebra in b C inherit the structure of ribb on tensor category from C and to establish that comm utativ e simple current algebras satisfy these criteria under mild assumptions. 1.1. Main results. Throughout this paper, w e work o ver an algebraically closed ﬁeld k . The pap er has tw o parts. First, we set up a general framew ork for commutativ e algebras A ∈ b C , where C is a braided tensor category . A right A -mo dule is said to b e ﬁnitely gener ate d if it is a quotient of the free A -mo dule X ⊗ A for some ob ject X ∈ C . Based on this deﬁnition, one can deﬁne the category fg- b C A of ﬁnitely generated A -mo dules and its sub category fg- b C loc A of ﬁnitely generated lo cal mo dules. Theorem A. L et A ∈ b C b e a haploid c ommutative algebr a. (a) If A is Artinian, then fg - b C A and fg - b C loc A ar e ab elian c ate gories of ﬁnite length. (b) If mor e over A is ind-exact, then fg - b C A is tensor and fg - b C loc A is br aide d tensor. (c) If in addition C is ribb on and A is F r ob enius with θ A = id A , then fg - b C loc A is ribb on. See Deﬁnitions 3.2 and 3.5 for the deﬁnitions of Artinian and ind-exact algebras, resp ectiv ely . The term F rob enius algebra is used in an extended sense as deﬁned in Deﬁnition 3.9 , and in particular the algebra A in (c) may not b e a rigid ob ject in b C . Our discussion of Artinian and ind-exact algebras builds upon [CEO24] and our prior w ork [SY24]. P art (c) is the main new con tribution in this theorem: we in tro duce ‘inﬁnite’ F rob enius algebras and use this prop ert y to obtain the ribb on structure of fg - b C loc A . 1 2 KENICHI SHIMIZU AND HARSHIT Y AD A V Second, we discuss a practical wa y of c hec king ind-exactness of algebras. Etingof and Penneys pro v ed that in an ab elian braided monoidal category with right exact tensor pro duct, if all the simple ob jects are rigid, then all ﬁnite length ob jects are rigid [EP26, Lemma 4.2]. In App endix A , w e prov e a non-braided generalization of this result (Theorem A.2 ). Using this, we obtain the follo wing result (Theorem 3.6 ): Theorem B. L et C b e a tensor c ate gory and A an Artinian c entr al c ommutative algebr a in b C . Then, A is ind-exact if and only if every simple A -mo dule is rigid. Third, and this is the main contribution of the pap er: w e pro v e that simple curr ent algebr as (Deﬁnition 2.8 ) satisfy these hypotheses under explicit and c hec k able conditions, and we derive concrete consequences for their mo dule categories. F or a simple curren t algebra A = L g ∈ Γ E g , w e ﬁrst classify simple A -mo dules. W e then verify the hypotheses of Theorem A for this class: in particular, A is alwa ys Artinian (Prop osition 4.14 ) and F rob enius (Theorem 4.15 ). If A is central comm utativ e and char( k ) = 0, w e use the explicit description of simple A -mo dules to show that they are rigid. By Theorem B , this establishes that A is ind-exact. Section 4 also gives a general categorical non-degeneracy criterion for the category fg- b C loc A . Theorem C. Supp ose that char( k ) = 0 . L et Γ b e a sub gr oup of the gr oup of isomorphism classes of invertible obje cts of C such that c E g ,E g = id E g ⊗ E g for al l g ∈ Γ . Then the obje ct A = L g ∈ Γ E g has a unique (up to isomorphism) structur e of c ommutative simple curr ent algebr a. F or the c ommutative simple curr ent algebr a A , we have: (a) fg - b C A is a t ensor c ate gory and fg - b C loc A is a br aide d tensor c ate gory. (b) If C is ribb on and θ E g = id E g for al l g ∈ Γ , then fg - b C loc A is ribb on. Now supp ose that C is F r ob enius (i.e., it has enough pr oje ctives). Then, (c) fg - b C A and fg - b C loc A ar e F r ob enius. (d) fg - b C A is ﬁnite if and only if Irr( C ) / Γ is ﬁnite, and fg - b C loc A is br aide d ﬁnite if and only if Irr( C Γ ) / Γ is ﬁnite. Her e, C Γ is the ful l sub c ate gory C Γ := { X ∈ C | c E g ,X c X,E g = id X ⊗ E g ∀ g ∈ Γ } . (e) If the simple obje cts in C Γ that trivial ly double br aid with al l other simples in C Γ ar e pr e cisely E g for g ∈ Γ , then fg - b C loc A is non-de gener ate. One may call a ribb on tensor category with trivial M¨ uger center a mo dular tensor c ate gory . Then Theorem C giv es suﬃcient conditions that a comm utativ e simple curren t algebra A m ust satisfy to ensure that the category fg- b C loc A is mo dular. 1.2. V O A motiv ation and applications. The study of simple current extensions originated in conformal ﬁeld theory with the w ork of Sc hellekens and Y ankielowicz [SY89] on mo dular in v ariants generated by in teger-spin simple currents. In vertex op erator algebra theory , Dong, Li, and Mason [DLM96] gav e an early treatmen t of extensions by simple currents. In the rational tensor- categorical setting, the algebra-ob ject and lo cal-mo dule viewpoint was dev elop ed b y Kirillov– Ostrik [KJO02] and by F uchs, Runkel, and Sch weigert [FRS04]. Later, Huang, Kirillo v, and Lep o wsky [HKJL15] show ed that, under suitable hypotheses, extensions of a V O A V are equiv alen t to commutativ e asso ciative algebra ob jects with trivial t wist in the braided tensor category of V -mo dules. In this framew ork, the extended V O A W corresponds to an algebra ob ject A in C (or b C ), and the category of W -mo dules is described b y the category of lo cal A -modules; see RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 3 [HKJL15, CKM24]. If the underlying ob ject of the algebra is a direct sum of inv ertible ob jects, it is called a simple current extension (this is why we call an algebra of the form A = L g ∈ Γ E g a simple curren t algebra). Our motiv ation comes from studying simple curren t extensions of logarithmic VO As. The represen tation categories of such V OAs are t ypically not semisimple and not ﬁnite, and moreo v er, the simple curren t extensions are typically not ob jects of the original category , but rather lie in the ind-completion. There are many pap ers whic h discuss rigid, ribb on structures on categories of mo dules ov er simple current extensions in sp eciﬁc examples; see, for example [AR18, CMY22a, CR22, CMY22b]. Our results can b e view ed as a rigidity inheritance statemen t for simple curren t extensions of vertex op erator algebras. Let V b e a V O A suc h that a suitable category C of V -mo dules carries a braided tensor category structure. Assume moreo v er that C is rigid, and let V ⊂ W b e a simple current extension. In the tensor-categorical form ulation, the extension corresp onds to a commutativ e algebra ob ject A ∈ b C , and W -modules are describ ed b y lo cal A -mo dules in b C [HKJL15, CKM24]. Under the hypotheses of Theorem A , w e show that the category fg - b C loc A is rigid. Thus W also admits a rigid braided tensor category of represen tations. This is useful b ecause rigidit y is often one of the hardest prop erties to establish in v ertex tensor category theory . 1.3. Examples. W e also obtain examples of (braided, ribb on) tensor categories b y applying our criteria to comm utativ e simple curren t algebras in t w o families of non-semisimple ribb on cate- gories: the categories of integral weigh t mo dules for U H q ( g ) [Rup22] and U E q ( gl (1 | 1)) [GY25]. In these settings, one can sum inv ertibles o v er suitable subgroups to obtain commutativ e simple cur- ren t algebras, and our criteria pro duce new categories of lo cal mo dules. In the unrolled quantum group setting, this sp ecializes to the explicit lattice criterion of Creutzig–Rup ert [CR22]. Note that the weigh t mo dule categories of U H q ( g ) and U E q ( gl (1 | 1)) are relativ e mo dular cate- gories in the sense of [CGPM14]. It w ould b e interesting to examine whether the relativ e mo dular structure descends to the lo cal mo dule category obtained from our results. 1.4. Organization. The paper is organized as follows. Section 2 recalls basic terminology on tensor categories and centers. Section 3 develops a general framew ork for constructing (braided, ribb on) tensor categories from commutativ e algebras in ind-completions; the main criteria are summarized in Theorem 3.1 . Section 4 applies this framework to simple current algebras. Section 5 records concrete examples from the unrolled quantum groups of gl (1 | 1). App endix A contains a non-braided v ariant of the Etingof–Penneys [EP26] lemma used for rigidity argumen ts. 1.5. Ac kno wledgemen ts. The authors thank RIMS for hospitality during the completion of this w ork. W e thank P av el Etingof and Da v e P enneys for a helpful email exc hange regarding [EP26, Lemma 4.2]. HY is partially supp orted b y a start-up grant from the Universit y of Alb erta and an NSERC Disco v ery Grant. KS is supp orted by JSPS KAKENHI Gran t Num b er JP24K06676. 2. Back gr ound 2.1. Notations. Throughout the pap er, k will denote an algebraically closed ﬁeld. W e will write v ec to denote the category of ﬁnite-dimensional k -vector spaces and V ec to denote the category of all k -v ector spaces. 4 KENICHI SHIMIZU AND HARSHIT Y AD A V Giv en a category C , we write C op to denote the opp osite category of C . Unless otherwise noted, the monoidal pro duct and the unit ob ject of a monoidal category C are denoted b y ⊗ : C × C → C and 1 ∈ C , resp ectively . C rev will denote the category C with the opp osite monoidal pro duct. F or a braided category C with braiding c , we use the notation C to denote C rev with reversed braiding c X,Y = c − 1 Y ,X . Regarding the terminology related to duality , we follo w the con v entions used in [EGNO16]. An ob ject of a monoidal category C is left rigid and right rigid if it has a left and a right dual ob ject, resp ectiv ely . A rigid ob ject is a left and righ t rigid ob ject. W e sa y that C is rigid if ev ery ob ject of C is rigid. Giv en a left rigid ob ject X ∈ C , w e denote by ( X ∗ , ev X : X ∗ ⊗ X → 1 , co ev X : 1 → X ⊗ X ∗ ) the left dual ob ject of X . W e call ev X and co ev X the evaluation and the c o evaluation for X , resp ectiv ely . An (ob ject-part of ) a righ t dual ob ject of X is denoted by ∗ X . 2.2. T ensor categories. A lo c al ly ﬁnite ab elian c ate gory is an essentially small k -linear ab elian category where every ob ject is of ﬁnite length and every Hom space is ﬁnite-dimensional. A tensor c ate gory is a lo cally ﬁnite ab elian category endow ed with a structure of a rigid monoidal category suc h that the monoidal pro duct is k -bilinear and the unit is simple [EGNO16]. Let C b e a tensor category . W e record the following result for later use. Lemma 2.1 ([Del02, Prop osition 1.1]) . Supp ose that C is an ab elian k -line ar monoidal c ate gory with End C ( 1 ) ∼ = k whose obje cts have ﬁnite length. If C is rigid, then it is Hom-ﬁnite. A tensor category C is said to be F r ob enius if, in addition, ev ery simple ob ject of C has an injectiv e h ull [ACE15]. Sev eral equiv alen t conditions for C to b e F rob enius were given in [SS23]. 2.3. Braided tensor categories. 2.3.1. M¨ uger c enter and nonde gener acy. Giv en a braided tensor category C and a monoidal sub- category D , the c entr alizer of D in C is deﬁned as: Z 2 ( D ⊂ C ) = { X ∈ C | c Y ,X ◦ c X,Y = id X ⊗ Y , ∀ Y ∈ D } . The M¨ uger c enter of C is C ′ := Z 2 ( C ⊂ C ). W e call C non-de gener ate if C is a tensor category and its M ¨ uger cen ter is equiv alent to vec . W e will use the following criteria for nondegeneracy . Lemma 2.2. Supp ose that c har( k ) = 0 . Then a br aide d F r ob enius tensor c ate gory C is non- de gener ate if and only if the only simple obje ct of the M¨ uger c enter of C is 1 up to isomorphism. Pr o of. Since C is F rob enius, the pro jectiv e co ver of 1 exists in C . T o sho w the ‘if ’ part, w e note that Ext 1 C ( 1 , 1 ) = 0 under our assumption (the pro of in the ﬁnite case [EGNO16, Theorem 4.4.1] requires that a pro jective co ver of 1 exists in C but do es not use the ﬁniteness of C ). If 1 is the only simple ob ject of C ′ up to isomorphism, ev ery ob ject of C ′ is a ﬁnite direct sum of 1 since Ext 1 C ( 1 , 1 ) = 0, and therefore C ′ is equiv alent to vec . The conv erse is trivial. □ 2.3.2. R ibb on tensor c ate gories. A ribb on tensor c ate gory is a braided tensor category C equipp ed with a twist , i.e. a natural automorphism θ = { θ X : X ∼ − → X } X ∈C satisfying θ X ⊗ Y = c Y ,X ◦ c X,Y ◦ ( θ X ⊗ θ Y ) , θ X ∗ = ( θ X ) ∗ (2.1) for all X , Y ∈ C . One can deduce θ 1 = id 1 from the former equation. RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 5 2.4. Mo dules in a closed monoidal category. Given algebras A and B in a monoidal category C , we denote by A C , C B and A C B , the categories of left A -mo dules, righ t B -mo dules and A - B - bimo dules in C , resp ectively . A monoidal category C is said to b e close d if for ev ery X ∈ C the endofunctors X ⊗ − and − ⊗ X on C hav e right adjoints. Let C b e a closed monoidal category . Giv en X ∈ C , w e denote by [ X , − ] a righ t adjoin t of − ⊗ X . The assignment ( X , Y ) 7→ [ X , Y ] extends to a functor C op × C → C , whic h w e call the right internal Hom functor . By deﬁnition, there is a natural isomorphism Hom C ( W ⊗ X , Y ) ∼ = Hom C ( W , [ X, Y ]) for W, X , Y ∈ C . The left internal Hom functor is deﬁned in a similar manner. Since a functor admitting a righ t adjoin t preserv es colimits, the endofunctors X ⊗ ( − ) and ( − ) ⊗ X preserve colimits. W e assume that C admits equalizers and co equalizers. F or a right A -mo dule M and a left A -mo dule N in C , their tensor pro duct M ⊗ A N is deﬁned to b e the co equalizer of a r M ⊗ id N and id M ⊗ a l N , where a r M : M ⊗ A → M and a l N : A ⊗ N → N are the right and the left action of A on M and N , resp ectiv ely . Since C is closed, one can verify that the category A C A of A -bimo dules is a monoidal category with monoidal pro duct ⊗ A and unit A . Moreov er, A C A is a closed monoidal category . A construction of the in ternal Hom functor is found in [SY24, § 3.2]. 2.4.1. Mo dules over a c entr al algebr a. Let C be a closed monoidal category admitting equalizers and co equalizers. W e call an algebra A in C c entr al c ommutative if it is equipped with a half- braiding σ suc h that ( A, σ ) is a commutativ e algebra in the Drinfeld center Z ( C ). In this case, a righ t A -mo dule can b e viewed as a left A -mo dule by the action given b y σ , and the category C A is view ed as a full sub category of A C A . One can verify that C A is closed under ⊗ A , and therefore C A is a monoidal category . C A is also a closed monoidal category by the same intern al Hom functor as A C A [SY24, § 3.3]. 2.4.2. L o c al mo dules. Assume that the closed monoidal category C is braided and A is a comm u- tativ e algebra in C . Then, as noted ab o v e, the category C A is monoidal with the relative tensor pro duct ⊗ A . How ever, it is not braided in general. The full sub category of C A consisting of lo c al mo dules, that is, those A -modules ( M , ρ ) for which the action ρ : M ⊗ A → M satisﬁes ρ ◦ c A,M ◦ c M ,A = ρ , is braided with the braiding inherited from C [Par95]. When C admits equal- izers, the category C loc A is in fact a closed monoidal category b y the same internal Hom functor as A C A [SY24, § 3.3]. 2.5. Ind-completions and their sub categories. 2.5.1. Ind-c ompletions. Let C b e a tensor category . The ind-completion b C of C is the completion of C under ﬁltered colimits [KS06]. It is a Grothendiec k ab elian category , and the inclusion C  → b C is exact and fully faithful. Moreo v er, b C admits a monoidal structure extending that of C , and the inclusion is strong monoidal. While b C is not rigid, it is closed monoidal. When C is braided, the braiding of C naturally extends to b C . When C is moreov er a ribb on category , the t wist of C naturally extends to a natural isomorphism id b C → id b C satisfying the ﬁrst equation in ( 2.1 ). 2.5.2. Cate gory of mo dules over an algebr a in the ind-c ompletion. Let C b e a tensor category and A an algebra in the ind-completion b C . The category b C A is iden tiﬁed with the category of algebras o v er the monad T := − ⊗ A . Since b C is closed monoidal, the functor T admits a right adjoin t. Th us b C A is an ab elian category such that the forgetful functor b C A → b C is exact and faithful. 6 KENICHI SHIMIZU AND HARSHIT Y AD A V 2.5.3. V arious mo dule c ate gories. Let A b e an algebra in b C . W e recall the following three ﬁniteness conditions for a righ t A -mo dule. Deﬁnition 2.3. A righ t A -mo dule M in b C is said to b e ﬁnitely gener ate d if there exists an epimorphism X ⊗ A ↠ M in b C A for some X ∈ C . M is ﬁnitely pr esente d if there exists an exact sequence X ⊗ A → Y ⊗ A → M → 0 in b C A for some X , Y ∈ C . M is of ﬁnite length if it admits a ﬁnite comp osition series in b C A . W e denote by fg- b C A , fp- b C A and ﬂ- b C A the full sub categories of b C A consisting of ﬁnitely generated, ﬁnitely presented and ﬁnite length mo dules, resp ectively . It is obvious that the full sub category fg - b C A is closed under quotients. How ever, fg - b C A need not be closed under k ernels in b C A , so it ma y fail to b e ab elian. The full sub category ﬂ- b C A of ﬁnite-length ob jects is a Serre sub category of b C A ; in particular it is ab elian. W e recall that b C A is monoidal with respect to ⊗ A when A is cen tral commutativ e. Although w e do not know whether fp- b C A and ﬂ- b C A are closed under ⊗ A , one can pro ve: Lemma 2.4. If A is c entr al c ommutative, then fg - b C A is close d under ⊗ A . Pr o of. If M i ∈ fg - b C A ( i = 1 , 2) is a quotient of X i ⊗ A ( X i ∈ C ), then M 1 ⊗ A M 2 is a quotient of ( X 1 ⊗ A ) ⊗ A ( X 2 ⊗ A ) ∼ = ( X 1 ⊗ X 2 ) ⊗ A and th us it is ﬁnitely generated. □ F rom the deﬁnitions, we hav e fp- b C A ⊆ fg- b C A . As we show next, ﬂ- b C A ⊆ fg- b C A as w ell. Lemma 2.5. Every ﬁnite length A -mo dule is ﬁnitely gener ate d. Pr o of. Let M ∈ b C A b e of ﬁnite length. Let { F i } i ∈ I b e the family of all ﬁnitely generated sub- mo dules of M . W e ﬁrst claim that M = P i ∈ I F i . Since M is an ob ject of b C , we can write M = P λ ∈ Λ X λ for some family of sub ob jects X λ ⊂ M with X λ ∈ C . F or each λ ∈ Λ, let W λ b e the image of the sub ob ject X λ ⊗ A under the action M ⊗ A → M . Then W λ is a submo dule of M , and by construction it is a quotient of the free A -mo dule X λ ⊗ A . Hence W λ is ﬁnitely generated. Moreo v er, X λ ⊂ W λ b y the unit axiom. Therefore M = X λ ∈ Λ X λ ⊂ X λ ∈ Λ W λ ⊂ X i ∈ I F i ⊂ M , whic h pro ves the claim. Next, note that the family { F i } i ∈ I is directed under inclusion. Indeed, if F i and F j are ﬁnitely generated submo dules of M , then F i ⊕ F j is also ﬁnitely generated. Hence, its quotient F i + F j is a ﬁnitely generated submo dule of M . Thus there exists k ∈ I suc h that F i ⊂ F k and F j ⊂ F k . Assume for contradiction that M is not ﬁnitely generated. Cho ose j 1 ∈ I such that F j 1  = 0. Since F j 1  = M and M = P i ∈ I F i , there exists i 2 ∈ I such that F i 2 ⊂ F j 1 . By directedness, there exists j 2 ∈ I such that F j 1 + F i 2 ⊂ F j 2 . Then F j 1 ⊊ F j 2 . Rep eating this argument, we construct a strictly increasing chain F j 1 ⊊ F j 2 ⊊ F j 3 ⊊ · · · of submo dules of M , whic h is imp ossible b ecause M has ﬁnite length. Hence M is ﬁnitely generated. □ R emark 2.6 . As b C is a lo cally presen table category , so is b C A . In suc h a category , there is a general notion of ﬁnitely presen ted and ﬁnitely generated ob jects, deﬁned b y the condition that Hom b C A ( M , − ) commutes with ﬁltered colimits and ﬁltered colimits of monomorphisms, respec- tiv ely . Our deﬁnitions of ﬁnitely generated and ﬁnitely presen ted mo dules coincide with these general notions [SS23]. Thus, the lemma ab ov e can b e prov en b y app ealing to general results on lo cally presentable categories, but we hav e given direct pro ofs for the con v enience of readers. RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 7 2.5.4. L o c al mo dules in ind-c ompletion. Let C b e a braided tensor category , and let A b e a com- m utativ e algebra in b C . By [CKM24, Prop osition 2.56] (sp ecialized to the non-sup er case), b C loc A is closed under taking k ernels and cokernels. Hence it is an ab elian category and the inclusion functor i : b C loc A  → b C A is exact. In fact, b C loc A is a coreﬂective sub category of b C A [P ar95, Theorem 3.6]. That is, i admits a right adjoin t i R : b C A → b C loc A . Explicitly , for ( M , ρ ) ∈ b C A , i R ( M , ρ ) is the equalizer of the maps [ A, ρ ] ◦ η M , [ A, ρ ◦ c A,M ◦ c M ,A ] ◦ η M : M → [ A, M ] , (2.2) where η M : M → [ A, M ⊗ A ] is the comp onen t of the unit of the adjunction. In particular, b C loc A is closed under colimits in b C A and the inclusion i preserv es colimits. Deﬁnition 2.7. W e deﬁne the full sub categories ﬂ- b C loc A , fg- b C loc A and fp- b C loc A of b C loc A as the inter- sections of b C loc A with ﬂ- b C A , fg- b C A and fp- b C A , resp ectively . 2.6. In v ertibles in a tensor category. Let C b e a lo cally ﬁnite k -linear ab elian monoidal category with simple unit. W e recall some basic facts ab out inv ertible ob jects in C . An ob ject X ∈ C is called invertible if there exists an ob ject Y ∈ C suc h that X ⊗ Y ∼ = 1 ∼ = Y ⊗ X . The set of isomorphism classes of inv ertible ob jects in C forms a group under the tensor pro duct, denoted b y Inv ( C ). 2.6.1. Gr oup c ohomolo gy. W e ﬁx our conv en tion for group cohomology . Let G b e a group. F or an in teger n ≥ 1, we denote by C n ( G ) the set of all maps from G × · · · × G ( n times) to k × suc h that f ( x 1 , · · · , x n ) = 1 whenever one of x i is the identit y element. An element of C n ( G ) is called a normalize d n -c o chain . F or each n ≥ 1, there is the c ob oundary map ∂ n +1 : C n ( G ) → C n +1 ( G ). F or example, ∂ 2 ( f )( x, y ) = f ( x ) f ( xy ) − 1 f ( y ) , (2.3) ∂ 3 ( g )( x, y , z ) = g ( x, y ) g ( x, y z ) − 1 g ( xy , z ) g ( y , z ) − 1 (2.4) for f ∈ C 1 ( G ), g ∈ C 2 ( G ) and x, y , z ∈ G . An elemen t of Ker( ∂ n +1 ) and of Im( ∂ n ) is called an n -c o cycle and an n -c ob oundary , resp ectively . W e deﬁne the n -th c ohomolo gy gr oup H n ( G ) to be the quotien t group Ker( ∂ n +1 ) / Im( ∂ n ). 2.6.2. A 3-c o cycle arising fr om the asso ciator. F or eac h class g ∈ Inv ( C ), w e c ho ose an ob ject E g ∈ C representing g . W e assume that E 1 is the unit ob ject of C . Since E 1 is simple and the endofunctor E g ⊗ ( − ) on C is an equiv alence, E g ∼ = E g ⊗ E 1 is also a simple ob ject. Thus, by Sc h ur’s lemma and lo cal ﬁniteness, the space Hom C ( E g , E h ) for g , h ∈ Inv ( C ) is one-dimensional if g = h , and zero otherwise. W e further choose isomorphisms φ x,y : E x ⊗ E y → E xy ( x, y ∈ Inv ( C )) suc h that φ 1 ,x and φ x, 1 are unit isomorphisms for all x ∈ Inv ( C ). W e deﬁne the map ω : Inv ( C ) 3 → k × so that the follo wing diagram is commutativ e: ( E x ⊗ E y ) ⊗ E z E xy ⊗ E z E xy z E x ⊗ ( E y ⊗ E z ) E x ⊗ E y z E xy z , α E x ,E y ,E z ϕ x,y ⊗ id ϕ xy,z ω ( x,y ,z ) id id ⊗ ϕ y,z ϕ x,yz (2.5) 8 KENICHI SHIMIZU AND HARSHIT Y AD A V for all x, y , z ∈ Inv ( C ), where α is the asso ciator. As is w ell-kno wn, the p entagon axiom implies that ω is a 3-co cycle on the group Inv ( C ). The cocycle ω dep ends on the choice of the isomorphisms φ x,y ; ho w ever, its cohomology class do es not. 2.6.3. A n ab elian 3-c o cycle arising fr om the br aiding. W e assume that the monoidal category C has a braiding c . Then we also deﬁne the map β : Inv ( C ) 2 → k × b y β ( x, y )id E xy = φ y ,x ◦ c E x ,E y ◦ φ − 1 x,y (2.6) for x, y ∈ Inv ( C ). The hexagon axioms for the braiding imply that the pair ( ω , β ) forms an ab elian 3-c o cycle on Inv ( C ) in the sense of Eilen b erg–Mac Lane [EML54]; see also [JS93]. The function q : Inv ( C ) → k × deﬁned b y q ( g ) = β ( g , g ) is a quadratic form whose asso ciated symmetric bic haracter is the double braiding scalar b q ( g , h ) = β ( g , h ) β ( h, g ). This p ersp ective is developed in [JS93], [DGNO10, § 2.11, App endix D] and [EGNO16, § 8.4]. 2.6.4. (Commutative) algebr as fr om invertible obje cts. Let C b e a lo cally ﬁnite k -linear ab elian closed monoidal category , and deﬁne ω (and β ) as b efore (when C is braided). W e no w introduce the follo wing class of algebra in b C , whic h is the main ob ject of study in this pap er: Deﬁnition 2.8. Let Γ b e a subgroup of Inv ( C ). A simple curr ent algebr a ov er Γ is an algebra A such that A = L g ∈ Γ E g as an ob ject of b C , the m ultiplication A ⊗ A → A restricts to an isomorphism E x ⊗ E y → E xy for all elemen ts x, y ∈ Γ and the unit is given by the inclusion morphism 1 = E 1 → A . Lemma 2.9. We ﬁx a sub gr oup Γ of Inv ( C ) and set A = L g ∈ Γ E g . (a) The obje ct A b e c omes a simple curr ent algebr a over Γ if and only if the r estriction of the 3-c o cycle ω to Γ × Γ × Γ is a c ob oundary. If this is the c ase, for any c o chain η ∈ C 2 (Γ) such that ω | Γ × Γ × Γ = ∂ 3 ( η ) , the morphism µ : A ⊗ A → A induc e d by µ x,y := η ( x, y ) φ x,y : E x ⊗ E y → E xy ( x, y ∈ Γ) (2.7) makes A a simple curr ent algebr a over Γ . (b) We assume that C is br aide d. Then A is a c ommutative simple curr ent algebr a if and only if q | Γ ≡ 1 , i.e., β ( g , g ) = 1 for al l g ∈ Γ . If this is the c ase, then we have β ( x, y ) β ( y , x ) = 1 , for al l x, y ∈ Γ . (2.8) P arts (a) and (b) are well known; see [EGNO16, § 8.8] for the p oin ted fusion category case, [DGNO10, § 2.11] for the general statement, and [FRS04, § 3] for simple current extensions. Pr o of. (a) Giv en a normalized 2-co chain η : Γ × Γ → k × , it is straightforw ard to verify that the morphism µ : A ⊗ A → A induced by ( 2.7 ) mak es A a simple curren t algebra if and only if ω | Γ × Γ × Γ = ∂ 3 ( η ). In particular, we ha v e prov ed the ‘if ’ part. T o show the conv erse, w e assume that A is a simple current algebra with multiplication µ . By the deﬁnition of a simple current algebra, the multiplication must b e induced b y ( 2.7 ) for some map η : Γ × Γ → k × . The same computation as ‘if ’ part completes the pro of. (b) Suppose A is a simple curren t algebra with m ultiplication induced b y η . The comm utativity condition µ ◦ c A,A = µ on the ( x, y )-comp onent reads η ( y , x ) β ( x, y ) = η ( x, y ) ( x, y ∈ Γ) . (2.9) Setting x = y and using η ( x, x )  = 0 giv es β ( x, x ) = 1, i.e., q | Γ ≡ 1. Conv ersely , assume that q | Γ ≡ 1. Then the abelian 3-cocycle ( ω | Γ 3 , β | Γ 2 ) has trivial asso ciated quadratic form. By the RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 9 Eilen b erg–Mac Lane isomorphism b etw een ab elian cohomology and quadratic forms (see [Gal26] for a recen t reference), its class in ab elian cohomology is therefore trivial. Hence there exists a normalized 2-co chain η ∈ C 2 (Γ) suc h that ∂ 3 ( η ) = ω | Γ 3 and β ( x, y ) = η ( x, y ) η ( y , x ) − 1 ( x, y ∈ Γ) . The ﬁrst identit y gives an algebra structure on A by part (a), while the second identit y is exactly the comm utativit y condition ( 2.9 ). Th us A is a comm utativ e algebra. Equation ( 2.8 ) easily follo ws from ( 2.9 ). □ The c hoice of η in (a) is not unique. If a simple current algebra A i ( i = 1 , 2) is constructed from a co chain η i ∈ C 2 (Γ) satisfying ∂ 3 ( η i ) = ω | Γ × Γ × Γ , then A 1 ∼ = A 2 as algebras if and only if η 1 and η 2 are cohomologous. In other w ords, the set of the isomorphism classes of simple curren t algebras o v er Γ (if it is non-empty) is a torsor ov er H 2 (Γ). The c hoice of η in (b) is also not unique. Supp ose that cochains η 1 , η 2 ∈ C 2 (Γ) deﬁne tw o comm utativ e simple current algebra structures on the ob ject A = L g ∈ Γ E g . Then ρ := η 1 η − 1 2 is a symmetric 2-co cycle on the ab elian group Γ. Since the group k × is divisible, ev ery symmetric 2-co cycle Γ × Γ → k × is a cob oundary . Thus there exists λ ∈ C 1 (Γ) suc h that ρ = ∂ 2 ( λ ). Rescaling each summand E x b y the scalar λ ( x ) yields an algebra isomorphism b et ween the t w o structures. In particular, the cohomology class of η in H 2 (Γ), and hence the isomorphism class of the comm utativ e algebra A , is uniquely determined. 3. A construction of ribbon braided tensor ca tegories In this section, we present a framew ork for constructing new ribb on braided tensor categories . W e begin with a braided tensor category C and consider comm utativ e algebras within its ind- completion b C . The full category of mo dules ov er such an algebra is typically to o large to form a tensor category of the type w e study , as it will include non-rigid and inﬁnite length ob jects. Therefore, we m ust identify sp eciﬁc sub categories that are closed under tensor pro ducts and satisfy necessary ﬁniteness conditions. W e establish that for Artinian algebr as the three ﬁniteness conditions of ﬁnitely generated, ﬁnitely presented, and ﬁnite length mo dules coincide (Theorem 3.4 ), yielding a k -linear abelian monoidal category where ev ery ob ject is of ﬁnite length. In Section 3.2 , we in tro duce ind-exact algebras (following [CEO24]) to ensure rigidit y of the category of ﬁnitely generated mo dules. In Section 3.3 , w e introduce the notion of a F rob enius algebra in a closed monoidal category b y extending the commonly used deﬁnition in the rigid case (see Deﬁnition 3.9 ), and sho w that the category of ﬁnitely generated lo cal mo dules is ribb on if the algebra A is Artinian, ind-exact and F rob enius in our sense and the t wist is the identit y on A . The main results are summarized in Theorem 3.1 b elow. Con v en tion. Throughout this section, C denotes a tensor category (with additional structure imp osed when needed), b C its ind-completion, and A an algebra in b C . Theorem 3.1. Supp ose that C is a br aide d tensor c ate gory and A ∈ b C is a c ommutative algebr a. If A is A rtinian, haploid and ind-exact, then fg - b C A is a tensor c ate gory and fg - b C loc A is a br aide d tensor c ate gory. If in addition, C is ribb on, A is F r ob enius and θ A = id A , then fg - b C loc A is a ribb on tensor c ate gory. Pr o of. This theorem is a summary of results pro v ed later in this section. The ﬁrst claim follows from Theorem 3.13 , and the second claim follows from Theorem 3.15 . □ 10 KENICHI SHIMIZU AND HARSHIT Y AD A V 3.1. Artinian algebras. Recall the sub categories ﬂ- b C A , fg - b C A , and fp- b C A of ﬁnite length, ﬁnitely generated, and ﬁnitely presen ted A -mo dules, resp ectively , from Deﬁnition 2.3 . Ideally , one would like to work inside the full sub category of b C A consisting of ob jects that are sim ultaneously ﬁnitely generated, ﬁnitely presented, and of ﬁnite length. This leads to the guiding problem: ﬁnd algebras A for which these three ﬁniteness conditions coincide. Deﬁnition 3.2. W e call an algebra A ∈ b C Artinian [CEO24] if ev ery ﬁnitely generated right A -mo dule is of ﬁnite length. W e ﬁrst record the key prop erty of Artinian algebras. Lemma 3.3. If A is A rtinian, every ﬁnitely gener ate d right A -mo dule is ﬁnitely pr esente d. Pr o of. Let M b e a ﬁnitely generated A -mo dule. By deﬁnition there exists an epimorphism f : X ⊗ A ↠ M in b C A for some X ∈ C . Let K := ker( f ) and let k : K  → X ⊗ A b e the k ernel inclusion, so w e hav e an exact sequence 0 → K k − → X ⊗ A f − → M → 0 . Since X ⊗ A is ﬁnitely generated and A is Artinian, X ⊗ A has ﬁnite length. Hence K has ﬁnite length as a subob ject of X ⊗ A in the ab elian category b C A . By Lemma 2.5 , K is ﬁnitely generated. Thus there exists X ′ ∈ C and an epimorphism g : X ′ ⊗ A ↠ K . Then k ◦ g : X ′ ⊗ A → X ⊗ A has image K = k er( f ), so f is the cokernel of k ◦ g . Equiv alently , X ′ ⊗ A k ◦ g − − → X ⊗ A f − → M → 0 is exact, pro ving that M is ﬁnitely presented. □ Our in terest in Artinian algebras stems from the following theorem, which sho ws that being Artinian forces these ﬁniteness notions to coincide. Theorem 3.4. L et A ∈ b C b e an A rtinian algebr a. Then we have ﬂ - b C A = fg - b C A = fp - b C A and these c ate gories ar e ab elian k -line ar c ate gories wher e obje cts have ﬁnite length. If C has enough pr oje ctive obje cts, then so do these c ate gories. Pr o of. By Lemma 3.3 , it is immediate that fg - b C A = fp- b C A . Next, using Lemma 2.5 together with the Artinian prop ert y , we see that an ob ject of b C A is ﬁnitely generated if and only if it is of ﬁnite length, i.e. ﬂ- b C A = fg- b C A . Thus, the three categories are equal. In particular, fg- b C A is abelian, k -linear and ob jects ha v e ﬁnite length. Assume that C has enough pro jective ob jects. Let M ∈ fg- b C A , and c hoose an epimorphism q : X ⊗ A ↠ M with X ∈ C . Cho ose an epimorphism π : P ↠ X in C with P pro jective. Then π ⊗ id A : P ⊗ A ↠ X ⊗ A is an epimorphism in b C A , and hence so is q ◦ ( π ⊗ id A ) : P ⊗ A ↠ M . Moreo v er P ⊗ A is pro jectiv e in b C A : indeed, for any A -mo dule N we hav e an adjunction Hom b C A ( P ⊗ A, N ) ∼ = Hom b C ( P , U ( N )) , where U is the forgetful functor, whic h is exact b ecause k ernels and cokernels in b C A are computed in b C . Since Hom b C ( P , − ) is exact, it follo ws that Hom b C A ( P ⊗ A, − ) is exact. Th us fg- b C A has enough pro jectiv es. □ 3.2. Ind-exact algebras. W e consider the following notion from [CEO24, § 6]. RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 11 Deﬁnition 3.5. W e call an algebra A in b C ind-exact if the functor − ⊗ A − : fg- b C A × fg - A b C → b C is bi-exact. Here, fg- A b C is the category of ﬁnitely generated left A -mo dules, whic h is deﬁned in the same manner as fg- b C A . Theorem 3.6. L et A b e an Artinian c entr al c ommutative algebr a in b C . The fol lowing ar e e quiv- alent: (a) A is ind-exact. (b) The c ate gory fg - b C A is rigid. (c) Every simple A -mo dule is rigid. Pr o of. ( a ) ⇒ ( b ): The Artinian assumption implies that fp- b C A = fg- b C A . Thus, every ob ject in fg- b C A is ﬁnitely presented, that is, it is the cokernel of a morphism X ⊗ A → Y ⊗ A for some X , Y ∈ C . Here X ⊗ A and Y ⊗ A are rigid. So, the claim follo ws because rigid ob jects in an ab elian monoidal category with biexact tensor pro duct are closed under taking cok ernels [Wig18]. ( b ) ⇒ ( a ): Rigidit y implies that the functor − ⊗ A V admits left/right adjoin ts giv en b y tensoring with left/righ t duals of V in fg- b C A . Thus, tensoring ov er A is bi-exact. ( b ) ⇒ ( c ): This is immediate. ( c ) ⇒ ( b ): Denote D = fg - b C A . Since A is Artinian, Theorem 3.4 implies that D is an ab elian k -linear category in which ev ery ob ject has ﬁnite length. In D , every ob ject is a quotien t of a free module X ⊗ A for some X ∈ C , whic h is rigid. Th us, by a non-braided version of Etingof and Penneys’ lemma (see Theorem A.2 ), rigidit y of all simple ob jects implies rigidity of the whole category . □ Theorem 3.7. L et A ∈ b C b e a haploid, A rtinian algebr a that is c entr al. If either one of the c onditions in The or em 3.6 is satisﬁe d, then fg - b C A is a tensor c ate gory. If mor e over C is F r ob enius, then fg - b C A is a F r ob enius tensor c ate gory. Pr o of. By Theorem 3.4 , fg - b C A is an ab elian k -linear category where all ob jects ha v e ﬁnite length. Moreo v er, Theorem 3.6 implies that fg - b C A is rigid. As A is haploid, Hom fg- b C A ( A, A ) ∼ = Hom b C ( 1 , A ) ∼ = k . Consequen tly , Lemma 2.1 implies that fg- b C A is lo cally ﬁnite. The same argumen t as in Theorem 3.4 applies to the monoidal sub category fg - b C A and the claim follo ws. □ R emark 3.8 . W e can relax the assumption that Hom( 1 , A ) ∼ = k to the weak er assumptions that dim(Hom( 1 , A )) < ∞ and that A is connected 1 . Indeed, Hom b C ( 1 , A ) is a ﬁnite-dimensional con- nected algebra, and the unit ob ject of fg- b C A (namely A ) satisﬁes Hom fg- b C A ( A, A ) ∼ = Hom b C ( 1 , A ), whic h is a ﬁnite dimensional connected algebra. Hence, b y a similar argument as in [EGNO16, Theorem 4.3.1], Hom fg- b C A ( A, A ) is isomorphic to a direct sum of copies of k . Since it is also connected, w e get Hom fg- b C A ( A, A ) ∼ = k . 3.3. F rob enius algebras. In this subsection, we introduce F rob enius algebras in b C and discuss their prop erties. W e sho w that if C is ribb on and A is a F rob enius algebra, then the category of lo cal mo dules ov er A admits a ribb on structure. Recall that b C is closed monoidal and w e denote the righ t internal Hom as [ − , − ]. It satisﬁes Hom b C ( L ⊗ M , N ) ∼ = Hom b C ( L, [ M , N ]). 1 A is called connected if the ring A inv := Hom( 1 , A ) is a ﬁnite dimensional connected ring, that is, there are no non trivial idemp otents. Equiv alently , A is not isomorphic to A 1 ⊕ A 2 for some algebras A 1 and A 2 . 12 KENICHI SHIMIZU AND HARSHIT Y AD A V Deﬁnition 3.9. Let A b e an algebra in b C with multiplication µ , and let λ : A → 1 b e a morphism in b C . W e deﬁne φ : A → [ A, 1 ] to b e the morphism in b C corresp onding to λµ : A ⊗ A → 1 under the adjunction isomorphism Hom b C ( A ⊗ A, 1 ) ∼ = Hom b C ( A, [ A, 1 ]). W e say that A is a F r ob enius algebr a with F r ob enius form λ if the morphism φ is in vertible. As we will see in Section 4 , a ‘simple curren t algebra’ is such a F rob enius algebra (Theorem 4.15 ). W e exp ect co-F rob enius Hopf algebras to provide more examples, and w e will pursue this line of inquiry in future w ork. No w w e giv e prop erties of F rob enius algebras. Lemma 3.10 b elo w is an analogue of the fact that the free mo dule functor for an ordinary F rob enius algebra is right adjoint to the forgetful functor from the category of mo dules. Lemma 3.10. L et A b e a F r ob enius algebr a in b C . Then ther e is a natur al isomorphism Hom b C A ( M , X ⊗ A ) ∼ = Hom b C ( M , X ) (3.1) for M ∈ b C A and X ∈ C . Pr o of. There is a natural transformation f X,M : X ⊗ [ A, M ] → [ A, X ⊗ M ] for X , M ∈ b C making the functor [ A, − ] : b C → b C a ‘lax’ left b C -mo dule functor since [ A, − ] is a right adjoint of the left b C -mo dule functor − ⊗ A : b C → b C . W e note that f X,M is inv ertible if X is rigid. It is easy to see that the morphism φ : A → [ A, 1 ] in Deﬁnition 3.9 is an isomorphism of right A -mo dules. Hence, for X ∈ C , we hav e X ⊗ A ∼ = X ⊗ [ A, 1 ] ∼ = [ A, X ] as righ t A -mo dules. Now we hav e Hom b C A ( M , X ⊗ A ) ∼ = Hom b C A ( M , [ A, X ]) ∼ = Hom b C ( M ⊗ A A, X ) ∼ = Hom b C ( M , X ) for M ∈ b C A and X ∈ C . The pro of is done. □ W e deﬁne Hom A ( M , − ) to b e the right adjoint of the functor − ⊗ A M : b C A → b C . This can b e constructed in the same wa y as the internal Hom functor for A b C A (see [SY24]). Lemma 3.11. With the ab ove notation, we have a natur al isomorphism Hom A ( M , X ⊗ A ) ∼ = [ M , X ] of right A -mo dules for M ∈ A b C A and X ∈ C . Pr o of of L emma 3.11 . F or a righ t A -mo dule L , we hav e natural isomorphisms Hom A ( L, Hom A ( M , X ⊗ A )) ∼ = Hom A ( L ⊗ A M , X ⊗ A ) ∼ = Hom b C ( L ⊗ A M , X ) ∼ = Hom A ( L, [ M , X ]) , where w e hav e used Lemma 3.10 . Th us the claim follows from the Y oneda lemma. □ 3.4. Braided case. W e also record the analogous ﬁniteness consequences for the lo cal mo dule category . In this section, C is assumed to b e a braided tensor category . Prop osition 3.12. L et A ∈ b C b e a c ommutative A rtinian algebr a. Then we have ﬂ - b C loc A = fg - b C loc A = fp - b C loc A , and these c ate gories ar e ﬁnite length, ab elian, k -line ar, br aide d monoidal c ate gories. If, mor e over, fg - b C A has enough inje ctive obje cts, then so do these c ate gories. RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 13 Pr o of. The ﬁrst statemen t follo ws from Theorem 3.4 . As fg- b C loc A = fg - b C A ∩ b C loc A and both fg - b C A and b C loc A are ab elian k -linear subcategories of b C A , fg- b C loc A is an abelian k -linear sub category as w ell. Lastly , since fg- b C A is ﬁnite length and fg - b C loc A is a full sub category of fg- b C A , fg - b C loc A is ﬁnite length as w ell. Recall that the inclusion functor i : b C loc A → b C A admits a right adjoin t i R . If M ∈ fg- b C A = ﬂ- b C A , then i R ( M ) (which is a sub ob ject of M , see ( 2.2 )) is also of ﬁnite length, hence ﬁnitely generated. Th us, the right adjoint i R : b C A → b C loc A restricts to a functor fg- b C A → fg- b C loc A . Supp ose that fg - b C A has enough injective ob jects. Let M ∈ fg - b C loc A and choose an injection i ( M )  → I in fg- b C A with I injectiv e. Then, i R ( I ) is injective in b C loc A b ecause Hom b C loc A ( − , i R ( I )) ∼ = Hom b C A ( i ( − ) , I ) is exact. As M  → i R ( I ), we see that fg - b C loc A has enough injective ob jects as w ell. Lastly , when A is a comm utativ e algebra, the braided monoidal structure on b C loc A restricts to fg- b C loc A b ecause the tensor pro duct of t w o ﬁnitely generated lo cal mo dules is again ﬁnitely generated. □ 3.4.1. R igidity. Theorem 3.13. L et A ∈ b C b e an A rtinian, haploid c ommutative algebr a. If either one of the c onditions in The or em 3.6 is satisﬁe d, then fg - b C A is a tensor c ate gory and fg - b C loc A is a br aide d tensor c ate gory. If mor e over C is F r ob enius, then b oth ar e F r ob enius tensor c ate gories. Pr o of. The pro of is similar to that of Theorem 3.7 . The k ey point is that b C loc A is a closed monoidal sub category of b C A . Hence, fg- b C loc A is a closed braided monoidal subcategory of fg - b C A . Thus, if fg- b C A is rigid, so is fg- b C loc A . Since A is haploid, End fg- b C A ( A ) ∼ = Hom b C ( 1 , A ) is connected; as ab ov e, this iden tiﬁes the unit endomorphism ring with k , so these are tensor categories. W e recall from [SS23] that a tensor category is F rob enius if and only if it has enough injective ob jects, if and only if it has enough pro jectiv e ob jects. No w w e assume that C is F rob enius. Then, b y Theorem 3.4 , the tensor category fg - b C A is F rob enius. Hence, b y Prop osition 3.12 , fg- b C loc A is also F rob enius. □ 3.4.2. R ibb on structur e on lo c al mo dules. Let C b e a ribb on tensor category with braiding c and t wist θ . Then b C is a closed braided monoidal category . Moreov er, the t wist induces an automor- phism of the iden tity functor of b C that satisﬁes θ M ⊗ N = c N ,M c M ,N ( θ M ⊗ θ N ) for all M , N ∈ b C . W e denote by b C ′ the M ¨ uger cen ter of b C . Next we will sho w that, in the ind-completion of a ribb on category , the internal Hom is compatible with the ribb on structure. In particular, θ [ M , 1 ] = [ θ M , 1 ] for M ∈ b C . Lemma 3.14. F or al l obje cts M ∈ b C and X ∈ b C ′ , we have θ [ M ,X ] = [ θ M , θ X ] . Pr o of. W e write M as M = lim − → i ∈ I M i for M i ∈ C . Then w e hav e [ M , X ] = lim ← − i ∈ I [ M i , X ]. If W ∈ C , then we may , and do, identify [ W, X ] with X ⊗ W ∗ . Hence we hav e θ [ W,X ] = θ X ⊗ W ∗ = c W ∗ ,X c X,W ∗ ( θ X ⊗ θ W ∗ ) = θ X ⊗ ( θ W ) ∗ = [ θ W , θ X ] , where w e hav e used the assumption X ∈ b C ′ at the third equalit y . Since M i ∈ C for all i , θ [ M ,X ] = lim ← − i ∈ I θ [ M i ,X ] = lim ← − i ∈ I [ θ M i , θ X ] = [lim − → i ∈ I θ M i , θ X ] = [ θ M , θ X ] . □ Let A be a commutativ e F rob enius algebra in b C . Then the category D := fg- b C loc A of ﬁnitely- generated lo cal A -mo dules is a braided monoidal category . 14 KENICHI SHIMIZU AND HARSHIT Y AD A V Theorem 3.15. If θ A = id A and D is rigid, then D is a ribb on c ate gory. W e explain that this theorem can b e applied to simple current algebras in b C in c haracteristic zero in Section 4 . The condition θ A = id A is analogous to trivialit y of the Nak a yama automorphism in the ﬁnite-dimensional F rob enius case [FFRS06, Prop osition 2.25]. Pr o of. By the assumption that θ A = id A , w e can deﬁne a natural isomorphism θ A M : M → M for M ∈ D b y θ A M = θ M in the same wa y as [KJO02, Theorem 1.17 (2)]. With the use of the in ternal Hom functor, the dualit y functor of D is given by M 7→ M † := Hom A ( M , A ). It remains to show that ( θ A M ) † = θ A M † holds for all M ∈ D . By Lemma 3.11 , there is a natural isomorphism φ M : M † → [ M , 1 ] for M ∈ b C A . Now we hav e φ M ◦ θ A M † = θ A [ M , 1 ] ◦ φ M = [ θ A M , 1 ] ◦ φ M = φ M ◦ ( θ A M ) † for M ∈ D , where the ﬁrst equality follows from the naturality of θ , the second from the previous lemma, and the last from the naturality of φ . The pro of is done. □ 4. Represent a tion theor y of simple current algebras Throughout this section, C denotes a lo cally ﬁnite k -linear ab elian closed monoidal category with simple unit; additional hypotheses (rigidit y , braiding, etc.) will b e imp osed when needed. W e also ﬁx a subgroup Γ of the group Inv ( C ) of in vertible ob jects of C and, for eac h g ∈ Γ, c ho ose an ob ject E g ∈ C representing g . In view of Lemma 2.9 (a), we assume that there is a 2-co c hain η ∈ C 2 (Γ) such that ∂ 3 ( η ) = ω | Γ × Γ × Γ , where ω is the 3-co cycle on Inv ( C ) arising from the asso ciator, and construct a simple curren t algebra A = L g ∈ Γ E g b y the 2-co chain η . The aim of this section is to inv estigate the represen tation theory of A . In this section, we v erify that A satisﬁes the h yp otheses of Theorem 3.1 under some assumptions on Γ and C . As a consequence, w e obtain criteria ensuring the category of ﬁnitely-generated lo cal modules o v er A is a ribb on (F rob enius) tensor category (Theorem 4.24 ). 4.1. Classiﬁcation of simple mo dules. In this subsection, w e giv e a classiﬁcation of simple righ t mo dules of the algebra A = L g ∈ Γ E g in b C . T o do this, w e study the restriction of A -modules to subalgebras of A , whic h is similar in spirit to Cliﬀord theory . The inv ertible summands E g act on simples of b C b y tensor pro duct, so any simple A -mo dule M is supp orted on a single Γ-orbit of a simple constituen t X ⊂ M . One then passes from the full algebra A to the stabilizer subgroup S = Stab Γ ( X ) and the corresp onding subalgebra A X = L s ∈ S E s , whic h captures the part of the action that preserves X . The asso ciativity constraints for an A X -action on X pro duce a canonical 2-co cycle ξ X on S ; this cocycle measures the obstruction to choosing the identiﬁcations X ⊗ E s ≃ X compatibly , and it forces the m ultiplicit y space to carry a ξ X -pro jectiv e representation of S . Thus simple A X -mo dules whose underlying ob ject is isomorphic to X ⊕ · · · ⊕ X are parametrized by irreducible ξ X -pro jectiv e represen tations W of S , and the corresp onding simple A -mo dules are obtained by induction M ( X , W ) = ( W ⊗ X ) ⊗ A X A . Finally , isomorphism classes are obtained b y the ob vious orbit relation: t wisting X by E g replaces ( S, ξ X , W ) by the conjugate data, yielding the same induced A -module. 4.1.1. Pr oje ctive r epr esentations. W e recall basics on pro jective represen tations of a group. Let S b e a group and ξ ∈ C 2 ( S ) a normalized 2-co cycle. A pr oje ctive r epr esentation of S with multiplier ξ is a ﬁnite-dimensional vector space V together with a map ρ : S → GL( V ) satisfying ρ ( s ) ρ ( t ) = ξ ( s, t ) ρ ( st ) for all s, t ∈ S. RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 15 Morphisms are linear maps that intert wine the S -actions. W e denote by Rep( S, ξ ) the category of such representations. It is equiv alen t to the category of ﬁnite-dimensional mo dules ov er the t wisted group algebra k ξ [ S ], which has k -basis { e s } s ∈ S and multiplication e s · e t = ξ ( s, t ) e st . If ξ and ξ ′ are cohomologous, then Rep( S, ξ ) ≃ Rep( S, ξ ′ ), so the category dep ends only on the class [ ξ ] ∈ H 2 ( S ). 4.1.2. Mo dules over the stabilizer. F or a simple ob ject X ∈ C , we deﬁne Stab Γ ( X ) = { g ∈ Γ | X ⊗ E g ∼ = X } and call it the stabilizer of X . It is ob vious that Stab Γ ( X ) is a subgroup of Γ. W e ﬁx a simple ob ject X ∈ C and write S = Stab Γ ( X ). Later, we will construct a simple righ t A -mo dule by induction from the subalgebra A X := M s ∈ S E s of A . W e ﬁrst remark that A X is in fact an algebra in C . Namely , Lemma 4.1. The gr oup S is ﬁnite. Pr o of. W e denote b y [ − , − ] l the left in ternal Hom functor of C (th us [ X , − ] l is righ t adjoin t to X ⊗ − ). If g ∈ S , then w e hav e Hom C ( E g , [ X , X ] l ) ∼ = Hom C ( X ⊗ E g , X ) ∼ = Hom C ( X , X )  = 0. This means that E g is a simple sub ob ject of [ X , X ] l . Hence the cardinality of S is b ounded b y the length of the so cle of [ X , X ] l ∈ C , which is at most ﬁnite. □ W e c ho ose a family ˜ φ s : X ⊗ E s → X ( s ∈ S ) of isomorphisms in C such that ˜ φ 1 is the unit isomorphism, and deﬁne the map η X : S × S → k × so that the following diagram is commutativ e: ( X ⊗ E s ) ⊗ E t X ⊗ E t X X ⊗ ( E s ⊗ E t ) X ⊗ E st X ˜ ϕ s ⊗ id α X,E s ,E t ˜ ϕ t η X ( s,t ) id id ⊗ ϕ s,t ˜ ϕ st (4.1) for all s, t ∈ S . Lemma 4.2. ∂ 3 ( η − 1 X ) = ω | S × S × S . Pr o of. There is a commutativ e diagram given b y Figure 1 (in the ﬁgure, the tensor product of ob jects of C is expressed by juxtap osition to sa ve space). The comp osition along the ﬁrst column is equal to the iden tity morphism by the p entagon axiom. Th us we hav e η X ( s, t ) − 1 η X ( st, u ) − 1 ω ( s, t, u ) − 1 η X ( s, tu ) η X ( t, u ) = 1 ( s, t, u ∈ S ) , (4.2) whic h implies the claim (see the deﬁnition ( 2.4 ) of ∂ 3 ). □ By this lemma, w e hav e a 2-co cycle ξ X on S giv en b y ξ X ( s, t ) = η ( s, t ) · η X ( s, t ) ( s, t ∈ S ) . (4.3) W e call ξ X the 2-c o cycle asso ciate d to { ˜ φ s } s ∈ S . The 2-co cycle ξ X dep ends on the c hoice of the family { ˜ φ s } s ∈ S of isomorphisms (and, in fact, also dep ends on c hoices of φ s,t and η ); how ever, its cohomology class dep ends only on X . No w we consider the tensor category v ec S of ﬁnite-dimensional S -graded vector spaces o v er k . Let k s ∈ v ec S denote the one-dimensional v ector space k graded b y s ∈ S . W e denote b y M X the 16 KENICHI SHIMIZU AND HARSHIT Y AD A V (( X E s ) E t ) E u ( X E t ) E u X E u X X ( E t E u ) ( X E s )( E t E u ) X E tu X ( X E s ) E tu X ( E s ( E t E u )) X ( E s E tu ) X E stu X X (( E s E t ) E u ) X ( E st E u ) X E stu X ( X ( E s E t )) E u ( X E st ) E u X E u X (( X E s ) E t ) E u ( X E t ) E u X E u X ( ˜ ϕ s ⊗ id) ⊗ id α X E s ,E t ,E u ˜ ϕ t ⊗ id α X,E t ,E u Diagram ( 4.1 ) ˜ ϕ u η X ( t,u ) id id ⊗ ϕ t,u ˜ ϕ s ⊗ (id ⊗ id) (id ⊗ id) ⊗ ϕ t,u α X,E s ,E t E u ˜ ϕ tu Diagram ( 4.1 ) η X ( s,tu ) id ˜ ϕ s ⊗ id α X,E s ,E tu id ⊗ (id ⊗ ϕ t,u ) id ⊗ α − 1 E s ,E t ,E u Diagram ( 2.5 ) id ⊗ ϕ s,tu ˜ ϕ stu ω ( s,t,u ) − 1 id ω ( s,t,u ) − 1 id id ⊗ ( ϕ st ⊗ id) α − 1 X,E s E t ,E u id ⊗ ϕ st,u α − 1 X,E st ,E u Diagram ( 4.1 ) ˜ ϕ stu η X ( st,u ) − 1 id α − 1 X,E s ,E t ⊗ id (id ⊗ ϕ s,t ) ⊗ id Diagram ( 4.1 ) ˜ ϕ st ⊗ id η X ( s,t ) − 1 id ˜ ϕ u η X ( s,t ) − 1 id ( ˜ ϕ s ⊗ id) ⊗ id ˜ ϕ t ⊗ id ˜ ϕ u Figure 1. Pro of of Equation ( 4.2 ) category v ec made in to a righ t mo dule category ov er vec S b y the action giv en by V  W = V ⊗ W for V ∈ v ec and W ∈ v ec S and the mo dule asso ciator ˜ α determined by ˜ α V , k s , k t = ξ X ( s, t ) id V : ( V  k s )  k t → V  ( k s ⊗ k t ) (4.4) for s, t ∈ S . There is a unique (up to isomorphism) linear functor E : vec S → C suc h that E ( k s ) = E s for all elements s ∈ S . The functor E is a monoidal functor with the monoidal structure induced b y the morphism µ s,t giv en by ( 2.7 ). Hence C is a right mo dule category ov er vec S through the monoidal functor E . W e note that the mo dule asso ciator ˜ α of C is given by ˜ α V , k s , k t = (id V ⊗ µ s,t ) ◦ α V ,E s ,E t : ( V  k s )  k t → V  ( k s ⊗ k t ) (4.5) for V ∈ C and s, t ∈ S . F rom now on, w e view vec as a full sub category of C by identifying the unit ob ject of C with the v ector space k so that V ⊗ Y for V ∈ v ec and Y ∈ C makes sense. Lemma 4.3. The functor F X : M X → C , F X ( W ) = W ⊗ X is a right vec S -mo dule functor by the mo dule structur e f determine d by f W, k s = (id E ( W ) ⊗ ˜ φ s ) ◦ α W,X,E s : F X ( W )  k s → F X ( W  k s ) (4.6) for s ∈ S . Pr o of. It is easy to see f W, k 1 = id W ⊗ id X for W ∈ M X . T o complete the pro of, it suﬃces to sho w that the equation F X ( ˜ α W, k s , k t ) ◦ f W ◁ k s , k t ◦ ( f W, k s  k t ) = f W, k s ⊗ k t ◦ ˜ α F X ( W ) ,E s ,E t (4.7) RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 17 (( W X ) E s ) E t ( W X )( E s E t ) ( W X ) E st ( W ( X E s )) E t ( W X ) E t W (( X E s ) E t ) W ( X ( E s E t )) W ( X E st ) W ( X E t ) W ( X E st ) W ( X E st ) W X W X W X α W ⊗ X,E s ,E t α W,X,E s ⊗ id (id ⊗ id) ⊗ µ s,t α W,X,E s ⊗ E t (pentagon axiom) α W,X,E st α W,X ⊗ E s ,E t (id ⊗ ˜ ϕ s ) ⊗ id α W,X,E t id ⊗ ( ˜ ϕ s ⊗ id) id ⊗ α X,E s ,E t id ⊗ (id ⊗ µ s,t ) id ⊗ (id ⊗ ϕ s,t ) ( 2.7 ) id ⊗ ˜ ϕ t Diagram ( 4.1 ) id ⊗ ˜ ϕ st η ( s,t ) id id ⊗ ˜ ϕ st η X ( s,t ) id η ( s,t ) id Figure 2. Pro of of Lemma 4.3 holds for all W ∈ M X and s, t ∈ S . W e consider the commutativ e diagram giv en by Figure 2 . By ( 4.4 ) and ( 4.6 ), the left hand side of ( 4.7 ) is equal to the comp osition of morphisms along the counter-clockwise path from the upp er left corner to the b ottom right one in the diagram of Figure 2 . By ( 4.5 ) and ( 4.6 ), the clo ckwise path yields the right hand side of ( 4.7 ). Thus w e obtain ( 4.7 ). □ Giv en a righ t mo dule category M o v er a monoidal category D and an algebra R in D with m ultiplication m and unit u , the category of right R -mo dules in M is deﬁned. W e recall that an ob ject of this category is an ob ject M ∈ M together with a morphism a : M  R → M such that the follo wing diagrams are commutativ e: ( M  R )  R M  R M M  ( R ⊗ R ) M  R M ∼ = a◁ id a id id ◁m a M  1 M  R M M . id ◁u ∼ = a id W e consider the group algebra R = k S , whic h is an algebra in v ec S b y the grading giv en b y R s = k s ( s ∈ S ). When we view C as a right mo dule category ov er v ec S via the monoidal functor E : vec S → C , the category of right R -mo dules in C is iden tical to C A X since E ( k S ) = A X as algebras. T aking the form ula ( 4.5 ) of the mo dule associator into account, w e see that a righ t R -mo dule in M X is the same thing as a ﬁnite-dimensional vector space W together with a family { ρ ( s ) } s ∈ S of linear endomorphisms on W such that the equations ρ (1) = id W and ξ X ( s, t ) ρ ( st ) = ρ ( t ) ρ ( s ) hold for all s, t ∈ S . Therefore the category of righ t R -mo dules in M X is iden tiﬁed with the category Rep( S op , ξ op X ) of pro jectiv e representations of S op with multiplier ξ op X , where ξ op X ( t, s ) = ξ X ( s, t ) ( s, t ∈ S ). W e use this observ ation to prov e: Lemma 4.4. Given ρ ∈ Rep( S op , ξ op X ) with r epr esentation sp ac e W , we deﬁne L ( X , { ˜ φ s } s ∈ S , ρ ) = W ⊗ X and make it a right A X -mo dule in C by the action induc e d by ( W ⊗ X ) ⊗ E s α W,X,E s − − − − − − → W ⊗ ( X ⊗ E s ) ρ ( s ) ⊗ ˜ ϕ s − − − − − − → W ⊗ X . 18 KENICHI SHIMIZU AND HARSHIT Y AD A V This c onstruction gives rise to a functor L ( X , { ˜ φ s } s ∈ S , − ) : Rep( S op , ξ op X ) → C A X , which induc es an e quivalenc e fr om Rep( S op , ξ op X ) to the c ate gory of right A X -mo dules in C whose underlying obje ct is isomorphic to the dir e ct sum of ﬁnitely many c opies of X . Pr o of. The right v ec S -mo dule functor F X : M X → C induces a functor from the category of righ t R -mo dules in M X to the category of those in C , and the induced functor is nothing but the functor L ( X , { ˜ φ s } s ∈ S , − ) in question. The last claim of this lemma follows be cause F X induces an equiv alence from M X to the full sub category of C consisting of ﬁnite direct sums of copies of X . □ Let ( ρ, W ) b e as in Lemma 4.4 , and let ( ρ ′ , W ′ ) b e a pro jectiv e representation of S op that is pro jectiv ely equiv alent to ρ . By deﬁnition, there are an isomorphism T : W → W ′ and a map θ : S op → k × suc h that ρ ′ ( s ) = θ ( s ) T ρ ( s ) T − 1 for all s ∈ S op . Since the multiplier of ρ ′ is ξ ′ ( t, s ) = θ ( s ) θ ( t ) θ ( st ) − 1 ξ X ( s, t ) ( s, t ∈ S op ) and may not be equal to ξ op X , the righ t A X -mo dule L ( X , { ˜ φ s } s ∈ S , ρ ′ ) may not be deﬁned. How ev er, w e can use the map θ to introduce a new family ˜ φ ′ s = θ ( s ) − 1 ˜ φ s : X ⊗ E s → X ( s ∈ S ) of isomorphisms. Then the right A X -mo dule L ( X , { ˜ φ ′ s } s ∈ S , ρ ′ ) is deﬁned and L ( X , { ˜ φ ′ s } s ∈ S , ρ ′ ) ∼ = L ( X , { ˜ φ s } s ∈ S , ρ ) as righ t A X -mo dules. Thus we introduce the following notation: Deﬁnition 4.5. Given a pro jective representation ρ ′ of S op whose multiplier is cohomologous to the 2-co cycle ξ op X , we deﬁne L ( X , ρ ′ ) := L ( X , { ˜ φ ′ s } s ∈ S , ρ ′ ), where { ˜ φ ′ s : X ⊗ E s → X } s ∈ S is a family of isomorphisms in C suc h that the asso ciated 2-co cycle is equal to the multiplier of ρ ′ . 4.1.3. Construction of simple mo dules. Let X b e a simple ob ject of C , and let S be the stabilizer of X . W e ﬁx a family ˜ φ s : X ⊗ E s → X ( s ∈ S ) of isomorphisms and deﬁne ξ X b y ( 4.3 ). Deﬁnition 4.6. W e deﬁne the follo wing functor M ( X , − ) : Rep( S op , ξ op X ) → b C A , ρ 7→ L ( X , ρ ) ⊗ A X A. Lemma 4.7. The functor M ( X , − ) intr o duc e d in the ab ove is exact and faithful. Pr o of. The functor M ( X , − ) is decomp osed as follows: Rep( S op , ξ op X ) L ( X, − ) − − − − − − − − → X i − − − − → b C A X F − − − − → b C A , where X is the category of righ t A X -mo dules in C whose underlying ob ject is isomorphic to a direct sum of ﬁnitely many copies of X , i is the inclusion functor, and F = − ⊗ A X A is the induction functor. By Lemma 4.4 , the ﬁrst arro w is exact and faithful. It is ob vious that the second arrow is also exact and faithful. Since A is free ov er A X , the functor F is also exact and faithful. Thus M ( X , − ) is exact and faithful as the comp osition of such functors. □ W e ﬁx coset representativ es { g i } i ∈ I of S \ Γ. Then we hav e an isomorphism A = M i ∈ I M s ∈ S E sg i L i ∈ I L s ∈ S µ − 1 s,g i − − − − − − − − − − − − − − → ∼ = M i ∈ I M s ∈ S E s ⊗ E g i = M i ∈ I A X ⊗ E g i RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 19 of left A X -mo dules. Thus we hav e M ( X , ρ ) ∼ = M i ∈ I ( W ⊗ X ) ⊗ E g i (4.8) as an ob ject of b C , where W is the representation space of ρ . The action of A on the right hand side induced b y ( 4.8 ) is giv en b y (( W X ) E g i ) E γ ∼ = ( W X )( E g i E γ ) (id ⊗ id) ⊗ µ − 1 s,g j µ g i ,γ − − − − − − − − − − − − − − − − → ( W X )( E s E g j ) ∼ = ( W ( X E s )) E g j ( ρ ( s ) ⊗ ˜ ϕ s ) ⊗ id − − − − − − − − − − − − − − − − → ( W X ) E g j (4.9) for γ ∈ Γ, where the tensor pro duct of ob jects is expressed by juxtap osition, s ∈ S and j ∈ I are elemen ts determined by sg j = g i γ , and ∼ = ’s are canonical isomorphisms obtained b y the asso ciator. W e now give some results on the structure of M ( X , ρ ). Lemma 4.8. The right A -mo dule M ( X, ρ ) is isomorphic to A if and only if X ∼ = E g for some g ∈ Γ . Pr o of. Consider X = E g for some g ∈ Γ. Then, the stabilizer in Γ is trivial, hence A X = 1 and M ( E g , ρ ) ∼ = E g ⊗ A as right A -modules (necessarily with ρ the trivial one-dimensional represen tation). Finally , multiplication in A induces an isomorphism E g ⊗ A ∼ = A of right A - mo dules, since it iden tiﬁes each summand E g ⊗ E ′ g ∼ = E g g ′ with the corresp onding summand of A . F or the con v erse, we assume that M ( X, ρ ) ∼ = A in b C A and let Y be a simple sub ob ject of M ( X , ρ ) as an ob ject of b C . Then Y ∼ = E h for some h ∈ Γ by the deﬁnition of A , while Y ∼ = X ⊗ E g i for some i ∈ I by the decomp osition ( 4.8 ). Th us X ∼ = E h ⊗ E ∗ g i ∼ = E hg − 1 i , as desired. □ Lemma 4.9. The right A -mo dule M ( X, ρ ) in b C is simple if and only if the pr oje ctive r epr esen- tation ρ is irr e ducible. Pr o of. W e assume that ρ is irreducible. Let M ′ b e a non-zero submo dule of M ( X , ρ ), and let Y b e a simple sub ob ject of M ′ as an ob ject of b C . Then Y ∼ = X ⊗ E g i for some i ∈ I and b y the same argumen t as in Lemma 4.8 , we ﬁnd that X is also a sub ob ject of M ′ . Since ρ is irreducible, M ′ con tains the sub ob ject corresp onding to ( W ⊗ X ) ⊗ E 1 in the decomp osition ( 4.8 ). By ( 4.9 ), it is easy to see that M ′ con tains ( W ⊗ X ) ⊗ E g j for all j ∈ I . Th us M ( X, ρ ) is simple. W e no w consider the case where ρ is reducible. Then there is an exact sequence 0 → ρ ′ → ρ → ρ ′′ → 0 in the category Rep( S op , ξ op X ) with ρ ′ and ρ ′′ non-zero. By Lemma 4.7 , w e ha v e an exact sequence 0 → M ( X , ρ ′ ) → M ( X , ρ ) → M ( X , ρ ′′ ) → 0 in b C A with M ( X , ρ ′ ) and M ( X, ρ ′′ ) non-zero. This means that M ( X , ρ ) is not a simple right A -mo dule. The pro of is done. □ Lemma 4.10. The functor M ( X , − ) pr eserves the length. Pr o of. Lemma 4.9 implies that M ( X , − ) sends a simple ob ject to a simple ob ject. By using Lemma 4.7 , one can prov e that M ( X , ρ ) and ρ hav e the same length by induction on the length of a pro jectiv e represen tation ρ . □ 4.1.4. Classiﬁc ation of simple mo dules. Theorem 4.11. L et M b e a simple right A -mo dule in b C , and let X b e a simple sub obje ct of M as an obje ct of b C . Then, M is isomorphic to M ( X , ρ ) for some irr e ducible pr oje ctive r epr esentation ρ of Stab Γ ( X ) op with multiplier ξ op X . 20 KENICHI SHIMIZU AND HARSHIT Y AD A V Pr o of. W e c ho ose a simple sub ob ject X of M in b C . Let L b e the image of X ⊗ A X under the action M ⊗ A → M . Then L is a non-zero A X -submo dule of M . W e choose a simple A X - submo dule L ′ of L . Since X ⊗ A X is the ﬁnite direct sum of copies of X , so is L , and therefore so is L ′ . Th us, by Lemma 4.4 , the right A X -mo dule L ′ is isomorphic to L ( X , ρ ) for some irreducible ρ ∈ Rep( S op , ξ op X ). By the universal prop erty of the induction, there is a non-zero morphism of righ t A -mo dules, sa y f , from M ( X , ρ ) to M . Since both M ( X , ρ ) and M are simple, the morphism f is in fact an isomorphism by Sch ur’s lemma. The pro of is done. □ Giv en an ob ject M ∈ b C and a simple ob ject X ∈ C , w e denote by M | X the sum of all subob jects of M isomorphic to X . Let M b e a simple righ t A -mo dule in b C , and let X b e a simple sub ob ject of M in b C . As sho wn in the ab ov e theorem, M is isomorphic to M ( X , ρ ) for some ρ ∈ Rep( S op , ξ op X ). In view of the decomp osition ( 4.8 ) and the pro of of the ab ov e theorem, the sub ob ject M | X is an A X -submo dule of M such that M | X ∼ = L ( X , ρ ), and th us w e can reco ver the pro jective equiv alence class of ρ from M | X b y Lemma 4.4 . W e use this observ ation to establish: Theorem 4.12. M ( X , ρ ) and M ( X ′ , ρ ′ ) ar e isomorphic as right A -mo dules if and only if ther e exists an element g ∈ Γ such that X ′ is isomorphic to X ⊗ E g and ρ ′ is pr oje ctively e quivalent to ρ g . Here, ρ g is deﬁned b y ρ g ( s ′ ) = ρ ( g s ′ g − 1 ) for s ′ ∈ S ′ . W e assume that X ′ is isomorphic to X ⊗ E g and write S = Stab Γ ( X ) and S ′ = Stab Γ ( X ′ ). It is easy to see that S ′ = g − 1 S g . Thus ρ g is a w ell-deﬁned pro jectiv e representation of S ′ op . Pr o of. The ‘if ’ part is easy . W e prov e the ‘only if ’ part. W e assume that M ( X , ρ ) and M ( X ′ , ρ ′ ) are isomorphic as righ t A -mo dules. Then, X ′ is a sub ob ject of M ( X , ρ ). Thus, by ( 4.8 ), X ′ is isomorphic to X ⊗ E g for some g ∈ Γ. By the discussion preceding this theorem, w e ha v e isomorphisms L ( X ′ , ρ ′ ) ∼ = M ( X ′ , ρ ′ ) | X ′ ∼ = M ( X , ρ ) | X ′ (4.10) of righ t A X ′ -mo dules in C . No w w e ﬁx an isomorphism ψ : X ⊗ E g → X ′ and consider the diagram giv en by Figure 3 , where s ′ ∈ S ′ and s = g s ′ g − 1 ∈ S so that sg = g s ′ . The diagram is comm utativ e except the cell lab eled ( ♡ ). W e can, and do, choose isomorphisms ˜ φ ′ s ′ : X ′ ⊗ E s ′ → X ′ ( s ′ ∈ S ′ ) and put it on the dashed arro w of the diagram so that the cell ( ♡ ) is comm utativ e. In view of ( 4.9 ), the left column of the diagram is the action of A X ′ on M ( X , ρ ) | X ′ . Thus, b y the diagram, w e obtain an isomorphism (id W ⊗ ψ ) α W,X,E g : M ( X , ρ ) | X ′ → L ( X ′ , { ˜ φ ′ s ′ } s ′ ∈ S ′ , ρ g ) of righ t A X ′ -mo dules. By ( 4.10 ) and Lemma 4.4 , we conclude that ρ g is pro jectiv ely equiv alent to ρ ′ . The pro of is done. □ 4.1.5. Finiteness of the numb er of simple mo dules. W e give a formula of the n um b er of isomor- phism classes of simple ob jects of b C A and, in particular, determine when it is ﬁnite. Given an ab elian category A , we denote b y Irr( A ) the set of isomorphism classes of simple ob jects of A . The group Γ acts on Irr( C ) b y [ X ] · g = [ X ⊗ E g ] for g ∈ Γ and [ X ] ∈ Irr( C ) (this action is w ell-deﬁned since ( − ) ⊗ E g is an auto equiv alence and preserves simplicit y). RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 21 (( W X ) E g ) E s ′ ( W ( X E g )) E s ′ ( W X ′ ) E s ′ ( W X )( E g E s ′ ) W (( X E g ) E s ′ ) W ( X ′ E s ′ ) ( W X ) E g s ′ W ( X ( E g E s ′ )) ( W X )( E s E g ) W ( X E g s ′ ) (( W X ) E s ) E g W ( X ( E s E g )) ( W ( X E s )) E g W (( X E s ) E g ) W ( X E g ) W X ′ ( W X ) E g W ( X E g ) W X ′ α W,X,E g ⊗ id α W ⊗ X,E g ,E s ′ (pentagon axiom) (id ⊗ ψ ) ⊗ id α W,X ⊗ E g ,E s ′ α W,X ′ ,E s ′ α W,X,E g ⊗ E s ′ (id ⊗ id) ⊗ µ g,s ′ id ⊗ ( ψ ⊗ id) id ⊗ α X,E g ,E s ′ ( ♡ ) (id ⊗ id) ⊗ µ − 1 s,g α W,X,E gs ′ id ⊗ (id ⊗ µ g,s ′ ) α W,X,E s ⊗ E g α − 1 W ⊗ X,E g ,E s id ⊗ µ − 1 s,g α W,X,E s ⊗ id (pentagon axiom) id ⊗ α − 1 X,E s ,E g α W,X ⊗ E s ,E g ρ ( s ) ⊗ ˜ ϕ s id ⊗ ( ˜ ϕ s ⊗ id) ρ ( s ) ⊗ id id ⊗ ψ ρ ( s ) ⊗ id α W,X,E g id ⊗ ψ Figure 3. Pro of of Theorem 4.12 Corollary 4.13. F or br evity, we set I X = Irr  Rep(Stab Γ ( X ) op , ξ op X )  for a simple obje ct X ∈ C , wher e ξ X is the 2 -c o cycle determine d by ( 4.3 ) . Then ther e is a bije ction G X ∈O I X → Irr( b C A ) , [ ρ ] 7→ [ M ( X , ρ )] ([ ρ ] ∈ I X ) , wher e O is the set of orbit r epr esentatives of Irr( C ) / Γ . The bijection implies # Irr( b C A ) = P [ X ] ∈O # I X . Since the set I X is non-empty and ﬁnite for all simple X ∈ C (since the stabilizer Stab Γ ( X ) is ﬁnite), we ha v e that Irr( b C A ) is ﬁnite if and only if Irr( C ) / Γ is ﬁnite. Pr o of. W e ﬁrst prov e that the map is surjectiv e. By Theorem 4.11 , ev ery simple right A -mo dule is isomorphic to M ( X ′ , ρ ′ ) for some simple X ′ ∈ C and some [ ρ ′ ] ∈ I X ′ . By the deﬁnition of the action of Γ on Irr( C ), there exists X ∈ O and g ∈ Γ suc h that X ′ ∼ = X ⊗ E g . Letting ρ = ( ρ ′ ) g − 1 , w e ha ve M ( X ′ , ρ ′ ) ∼ = M ( X , ρ ) by Theorem 4.12 . This means that the map is surjective. T o sho w the injectivity , w e let X , Y ∈ O , [ ρ ] ∈ I X and [ σ ] ∈ I Y and assume that [ M ( X, ρ )] = [ M ( Y , σ )]. By Theorem 4.12 , there is an elemen t g ∈ Γ such that Y ∼ = X ⊗ E g and σ ∼ ρ g . Since X and Y are representativ es of some orbits, w e hav e X = Y and th us g ∈ Stab Γ ( X ). Hence σ ∼ ρ g ∼ ρ . This sho ws that the map is injective. The pro of is done. □ 4.2. T ensor category fg - b C A . F rom here on, C is a tensor category . As ab o v e, A = L g ∈ Γ E g is a simple current algebra ov er a subgroup Γ < Inv ( C ). With some additional assumptions, w e will 22 KENICHI SHIMIZU AND HARSHIT Y AD A V v erify that A satisﬁes the assumptions of Theorem 3.7 , and th us the conclusion of this theorem holds. 4.2.1. A rtinian pr op erty. Prop osition 4.14. The algebr a A is Artinian. Pr o of. Since a ﬁnitely generated ob ject of fg- b C A is a quotient of X ⊗ A for some X ∈ C , and since a quotient of a ﬁnite length ob ject is also of ﬁnite length, it suﬃces to sho w that the free mo dule X ⊗ A for X ∈ C is of ﬁnite length. W e prov e this by induction on the length of X as an ob ject of C . The claim is trivial if  ( X ) = 0. W e assume that  ( X ) = 1 or, equiv alently , X is a simple ob ject. Since X ⊗ A X is a righ t A X -mo dule whose underlying ob ject is isomorphic to a direct sum of ﬁnitely man y copies of X , by Lemma 4.4 , there exists ρ ∈ Rep( S op , ξ op X ) suc h that X ⊗ A X ∼ = L ( X , ρ ) as righ t A X -mo dules. There are isomorphisms X ⊗ A ∼ = ( X ⊗ A X ) ⊗ A X A ∼ = L ( X , ρ ) ⊗ A X A = M ( X, ρ ) of righ t A -mo dules. Thus, by Lemma 4.10 , X ⊗ A is of ﬁnite length. No w we assume that  ( X ) > 1 and Y ⊗ A is of ﬁnite length for all Y ∈ C with  ( Y ) <  ( X ). W e c ho ose a simple sub ob ject X ′ ⊂ X and let X ′′ = X/X ′ so that there is a short exact sequence 0 → X ′ → X → X ′′ → 0 in C . By applying the free mo dule functor to this sequence, we obtain a short exact sequence 0 → X ′ ⊗ A → X ⊗ A → X ′′ ⊗ A → 0 in b C A . Since X ′ ⊗ A and X ′′ ⊗ A are of ﬁnite length, so is X ⊗ A . This prov es that A is Artinian. □ 4.2.2. F r ob enius algebr a structur e. In this subsection, we pro v e that A is F rob enius. W e recall Io v anov’s result [Io v06], which plays a k ey role. Let A b e a complete and co complete ab elian category , and let { M i } i ∈ I b e an inﬁnite family of ob jects of A . Then w e ha ve tw o ob jects: The pro duct Π := Q i ∈ I M i and the copro duct Σ := L i ∈ I M i . There is a canonical morphism Σ → Π. When A is the category of mo dules o v er a ring and M i  = 0 for inﬁnitely man y i , the canonical morphism is nev er an isomorphism. Ho w ev er, according to Io v anov [Io v06, Example 2.9], the canonical morphism Σ → Π is an isomorphism if A is an ind-completion of a lo cally ﬁnite ab elian category and Σ is quasi-ﬁnite. W e use this to pro ve: Theorem 4.15. The algebr a A is F r ob enius with the F r ob enius form λ : A → 1 given by the pr oje ction to E 1 = 1 . Pr o of. F or a family { M i } i ∈ I of ob jects of b C , w e ha ve [ L i ∈ I M i , N ] ∼ = Q i ∈ I [ M i , N ]. Thus, [ A, 1 ] ∼ = Y g ∈ Γ [ g , 1 ] ∼ = Y g ∈ Γ g − 1 ∼ = Y g ∈ Γ g . Un winding the deﬁnition of the morphism φ : A → [ A, 1 ] in Deﬁnition 3.9 , one c hec ks that under the ab o v e identiﬁcations it coincides with the canonical morphism L g ∈ Γ g → Q g ∈ Γ g : indeed, writing A = L g ∈ Γ g , the comp osite λµ v anishes on g ⊗ h unless h ≃ g − 1 , in which case it is iden tiﬁed (up to the chosen identiﬁcations g ⊗ g − 1 ∼ = 1 ) with the ev aluation pairing. By adjunction, this yields a morphism g → [ A, 1 ] ∼ = Q h ∈ Γ h whose only non-zero comp onen t is the canonical map g → g into the h = g factor. Summing o ver g gives the canonical morphism L g ∈ Γ g → Q g ∈ Γ g . This is an isomorphism b y Io v anov’s result cited ab o v e. □ RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 23 4.2.3. R igidity of simple mo dules. W e retain the notation in the previous subsection. W e ﬁx a simple ob ject X of C with stabilizer S and a pro jective representation ρ of S op with m ultiplier ξ op X . Theorem 4.16. Assume that | S |  = 0 in the b ase ﬁeld k . Then M ( X, ρ ) is a dir e ct summand of the fr e e mo dule X ⊕ m ⊗ A for some p ositive inte ger m . Pr o of. Since A X is separable under the assumption of this theorem [FRS04, Lemma 3.1], the quotien t morphism M ⊗ N → M ⊗ A X N for a righ t A X -mo dule M and a left A X -mo dule N in b C has a section that is natural in M and N . Thus, when N is an A X - B -bimodule for some algebra B in b C , the righ t B -module M ⊗ A X N is a direct summand of the right B -module M ⊗ N . By applying this argumen t to M = L ( X, ρ ), N = A and B = A , the claim follo ws. □ Corollary 4.17. Supp ose that X ∈ C is rigid and A is a c entr al c ommutative algebr a in b C . If | S |  = 0 in k , M ( X , ρ ) is a rigid obje ct in the c ate gory of right A -mo dules in b C . Pr o of. The previous theorem sa ys that M ( X , ρ ) is a direct summand of a free mo dule X ⊕ m ⊗ A for some m . Since X ⊗ A is rigid (with left dual X ∗ ⊗ A and righ t dual ∗ X ⊗ A ), the claim follows from the fact that a direct summand of a rigid ob ject is rigid. □ In view of the ab ov e corollary , we consider the following condition: | Stab Γ ( X ) |  = 0 in k for every simple ob ject X ∈ C . (4.11) The condition ( 4.11 ) holds if, for example, k is of characteristic zero. Even in the case where k has p ositiv e characteristic p > 0, the condition ( 4.11 ) holds if g p  = 1 for every non-identit y elemen t g ∈ Γ. In particular, ( 4.11 ) holds if Γ is torsion free. Corollary 4.18. If C is a tensor c ate gory and the simple curr ent algebr a A is c entr al c ommutative such that the c ondition ( 4.11 ) is satisﬁe d, then fg - b C A is a tensor c ate gory. Pr o of. By the previous corollary , every simple ob ject of fg- b C A is rigid. Moreov er, A is a haploid comm utativ e algebra. Thus, by Theorem 3.7 , fg - b C A is a tensor category . □ 4.2.4. Existenc e of enough pr oje ctives and ﬁniteness. Assume that C has enough pro jective ob- jects. Then, by Theorem 3.4 , Artinian prop erty of A implies that fg- b C A has enough pro jectiv e ob jects. W e give a more explicit construction of pro jective ob jects. Let X ∈ C b e a simple ob ject with pro jectiv e cov er π : P → X , and let ρ b e an irreducible pro jectiv e represen tation of Stab Γ ( X ) op with m ultiplier ξ op X . W e set P ′ ( X , ρ ) := ( P ⊗ W ) ⊗ A X (the free righ t A X -mo dule), where W is the represen tation space of ρ . Then P ′ ( X , ρ ) is a pro jective righ t A X -mo dule and there is an epimorphism P ′ ( X , ρ ) ( π ⊗ id) ⊗ id − − − − − − − − − − → L ( X , ρ ) ⊗ A X action − − − − − − − → L ( X , ρ ) of righ t A X -mo dules. Now we set P ( X , ρ ) := P ′ ( X , ρ ) ⊗ A X A ∼ = ( P ⊗ W ) ⊗ A. Then P ( X, ρ ) is a pro jectiv e ob ject in the category of righ t A -mo dules in b C and the simple module M ( X , ρ ) = L ( X , ρ ) ⊗ A X A is a quotien t of P ( X , ρ ). Th us fg - b C A is a lo cally ﬁnite ab elian category where ev ery simple ob ject is a quotien t of a pro jectiv e ob ject. This implies that fg- b C A has enough pro jectiv e ob jects. 24 KENICHI SHIMIZU AND HARSHIT Y AD A V Theorem 4.19. L et C b e a F r ob enius tensor c ate gory. Supp ose that A is a c entr al c ommutative algebr a in b C such that the c ondition ( 4.11 ) is satisﬁe d. Then, fg - b C A is a F r ob enius tensor c ate gory. Mor e over, fg - b C A is a ﬁnite tensor c ate gory if and only if Irr( C ) / Γ is ﬁnite. Pr o of. By Corollary 4.18 , fg - b C A is a tensor category . As C is F rob enius and A is Artinian, b y Theorem 3.4 , fg- b C A has enough pro jectiv e ob jects. Thus, it is a F rob enius tensor category . The claim ab out ﬁniteness follows by Corollary 4.13 . □ Question 4.20. Assuming that fg - b C A is a ﬁnite tensor c ate gory, what ar e the F r ob enius-Perr on dimensions of simple obje cts and their pr oje ctive c overs? What is the FPdim of the c ate gory? 4.3. The sub category of lo cal modules. F rom now on, C is a braided tensor category and A is a comm utativ e simple current algebra o ver a subgroup Γ < Inv ( C ). By Lemma 2.9 and equation ( 2.8 ), this implies that the in v ertible summands of A cen tralize each other, that is, the following equation holds: c E h ,E g c E g ,E h = id E g ⊗ E h ( g , h ∈ Γ) . (4.12) 4.3.1. Classiﬁc ation of simple lo c al mo dules. Fix a simple ob ject X ∈ C and an irreducible pro- jectiv e represen tation ρ of the stabilizer S = Stab Γ ( X ) op . Let C Γ := { Y ∈ C | c E g ,Y c Y ,E g = id Y ⊗ E g for all g ∈ Γ } (4.13) b e the M ¨ uger cen tralizer of the p ointed tensor sub category ⟨ E g | g ∈ Γ ⟩ ⊂ C . Prop osition 4.21. Then M ( X , ρ ) is a lo c al A -mo dule if and only if X ∈ C Γ . Pr o of. Put M := M ( X , ρ ) and let a M : M ⊗ A → M b e its action. Assume c E g ,X c X,E g = id X ⊗ E g for all g ∈ Γ. Then the free mo dule X ⊗ A is lo cal. As M is a quotien t of X ⊗ A and b C loc A is closed under cok ernels in b C A , M is lo cal. Con v ersely , assume ﬁrst that M is lo cal. By ( 4.8 ), c ho ose coset representativ es { g i } i ∈ I of S \ Γ with g 1 = 1, so one summand of M is W ⊗ X . F or each g ∈ Γ, the (1 , g )-block of the action is an isomorphism a 1 ,g : (( W ⊗ X ) ⊗ E 1 ) ⊗ E g ∼ − − → ( W ⊗ X ) ⊗ E g j , where S g j = S g (for S = Stab Γ ( X )). Restricting lo calit y a M ◦ c A,M = a M ◦ c − 1 M ,A to this blo c k and p ostcomp osing with a − 1 1 ,g giv es c E g ,W ⊗ X = c − 1 W ⊗ X ,E g . Since W is a ﬁnite-dimensional v ector space ob ject (hence a direct sum of copies of 1 ), this is equiv alent to c E g ,X = c − 1 X,E g , i.e. c E g ,X c X,E g = id X ⊗ E g for all g ∈ Γ. □ Corollary 4.22. F or br evity, we set I X = Irr  Rep(Stab Γ ( X ) op , ξ op X )  for a simple obje ct X ∈ C , wher e ξ X is the 2 -c o cycle determine d by ( 4.3 ) . Then ther e is a bije ction G X ∈O Γ I X → Irr( b C loc A ) , [ ρ ] 7→ [ M ( X , ρ )] ([ ρ ] ∈ I X ) , wher e O Γ is orbit r epr esentatives of Irr( C Γ ) / Γ . The bijection implies # Irr( b C loc A ) = P [ X ] ∈O Γ # I X . Since the set I X is non-empt y and ﬁnite for all simple X ∈ C , we hav e that Irr( b C loc A ) is ﬁnite if and only if Irr( C Γ ) / Γ is ﬁnite. RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 25 Pr o of. By Prop osition 4.21 , the simple ob jects of b C loc A are exactly those M ( X , ρ ) with X ∈ Irr( C Γ ) and ρ irreducible. The remaining argument is the same as Corollary 4.13 . □ 4.3.2. Br aiding b etwe en simple lo c al mo dules. Lemma 4.23. L et M ( X , ρ ) and M ( Y , σ ) b e simple lo c al A -mo dules. Then the double br aiding in fg - b C loc A b etwe en M ( X , ρ ) and M ( Y , σ ) is trivial if and only if the double br aiding in C b etwe en X and Y is trivial. Pr o of. ( ⇒ ) W e ﬁrst introduce some notations: W e set S = Stab Γ ( X ) and T = Stab Γ ( Y ), and let { x i } i ∈ I , { y j } j ∈ J and { z k } k ∈ K b e represen tativ es of Γ /S , Γ /T and T \ Γ /S , resp ectively . W e assume that the index sets I , J and K hav e a sp ecial element 0 such that x 0 = y 0 = z 0 = 1. By ( 4.8 ) and the braiding, we obtain an isomorphism M ( X , ρ ) ⊗ M ( Y , σ ) ∼ = M i ∈ I M j ∈ J ( L ( X , ρ ) ⊗ L ( Y , σ )) ⊗ ( E x i ⊗ E y j ) . (4.14) Since A ∼ = L k ∈ K A Y ⊗ ( E z k ⊗ A X ) as A Y - A X -bimo dules, we hav e M ( Y , σ ) = L ( Y , σ ) ⊗ A Y A ∼ = M k ∈ K L ( Y , σ ) ⊗ ( E z k ⊗ A X ) ∼ = M k ∈ K A X ⊗ ( L ( Y , σ ) ⊗ E z k ) as A X -mo dules. Hence, M ( X , ρ ) ⊗ A M ( Y , σ ) = ( L ( X, ρ ) ⊗ A X A ) ⊗ A M ( Y , σ ) ∼ = L ( X , ρ ) ⊗ A X M ( Y , σ ) ∼ = M k ∈ K ( L ( X , ρ ) ⊗ L ( Y , σ )) ⊗ E z k (4.15) as ob jects of b C . Let π : M ( X , ρ ) ⊗ M ( Y , σ ) → M ( X , ρ ) ⊗ A M ( Y , σ ) be the canonical epimorphism, and deﬁne the epimorphism ˜ π so that the following diagram is comm utativ e: M ( X , ρ ) ⊗ M ( Y , σ ) L i ∈ I L j ∈ J ( L ( X , ρ ) ⊗ L ( Y , σ )) ⊗ ( E x i ⊗ E y j ) M ( X , ρ ) ⊗ A M ( Y , σ ) L k ∈ K ( L ( X , ρ ) ⊗ L ( Y , σ )) ⊗ E z k π ( 4.14 ) ∼ = ˜ π ( 4.15 ) ∼ = The epimorphism e π maps the ( i, j )-th comp onen t ( L ( X , ρ ) ⊗ L ( Y , σ )) ⊗ ( E x i ⊗ E y j ) to the k -th comp onen t, where k is the unique index such that x i y j ∈ T z k S . W e let ˜ π i,j : ( L ( X , ρ ) ⊗ L ( Y , σ )) ⊗ ( E x i ⊗ E y j ) → ( L ( X, ρ ) ⊗ L ( Y , σ )) ⊗ E z k ( x i y j ∈ T z k S ) b e the morphism induced by ˜ π . Although the morphism ˜ π ij ma y b e tedious to b e written down explicitly in general (as it inv olv es the actions of t ∈ T and s ∈ S such that x i y j = tz k s ), one can easily v erify that ˜ π 00 is the iden tity morphism. W e consider the double braiding M for M ( X , ρ ) and M ( Y , σ ). Since X , Y ∈ C Γ , and since the in v ertible summands of A centralize each other, we hav e a commutativ e diagram M ( X , ρ ) ⊗ M ( Y , σ ) L i ∈ I L j ∈ J ( L ( X , ρ ) ⊗ L ( Y , σ )) ⊗ ( E x i ⊗ E y j ) M ( X , ρ ) ⊗ M ( Y , σ ) L i ∈ I L j ∈ J ( L ( X , ρ ) ⊗ L ( Y , σ )) ⊗ ( E x i ⊗ E y j ) M M ( X,ρ ) , M ( Y ,σ ) ( 4.14 ) ∼ = L i ∈ I L j ∈ J M L ( X,ρ ) , L ( Y ,σ ) ⊗ id ( 4.14 ) ∼ = 26 KENICHI SHIMIZU AND HARSHIT Y AD A V Let M A denote the double braiding of the category b C loc A of lo cal mo dules. By deﬁnition, M A is c haracterized by the equation π ◦ M M ( X,ρ ) , M ( Y ,σ ) = M A M ( X,ρ ) , M ( Y ,σ ) ◦ π . By the ab ov e comm utativ e diagram, w e see that there are isomorphisms m k ( k ∈ K ) suc h that the follo wing diagram is comm utative: M ( X , ρ ) ⊗ A M ( Y , σ ) L k ∈ K ( L ( X , ρ ) ⊗ L ( Y , σ )) ⊗ E z k M ( X , ρ ) ⊗ A M ( Y , σ ) L k ∈ K ( L ( X , ρ ) ⊗ L ( Y , σ )) ⊗ E z k M A M ( X,ρ ) , M ( Y ,σ ) ( 4.15 ) ∼ = L k ∈ K m k ( 4.15 ) ∼ = By the ab ov e commutativ e diagrams and the deﬁnition of M A , w e hav e m 0 = M L ( X,ρ ) , L ( Y ,σ ) ⊗ id 1 . No w we assume that M A M ( X,ρ ) , M ( Y ,σ ) is the identit y morphism. Then m k is the identit y mor- phism for all k ∈ K . Thus, in particular, m 0 = id. Since L ( X , ρ ) and L ( Y , σ ) are direct sums of ﬁnitely many copies of X and Y , resp ectiv ely , m 0 = id implies that M X,Y = id. W e hav e prov ed the ‘only if ’ part of this theorem. ( ⇐ ) F or the con v erse, assume that M X,Y = id X ⊗ Y . Since L ( X, ρ ) and L ( Y , σ ) are direct sums of ﬁnitely man y copies of X and Y , resp ectively , we ha v e M L ( X,ρ ) , L ( Y ,σ ) = id. Also, b y ( 4.12 ), M E x i ,E y j = id for all i ∈ I and j ∈ J . Hence the righ t vertical morphism in the ab ov e comm utativ e diagram is the iden tity , and therefore M M ( X,ρ ) , M ( Y ,σ ) = id. Now the relation π ◦ M M ( X,ρ ) , M ( Y ,σ ) = M A M ( X,ρ ) , M ( Y ,σ ) ◦ π implies, since π is an epimorphism, that M A M ( X,ρ ) , M ( Y ,σ ) = id . This pro v es the ‘if ’ part. □ 4.3.3. R ibb on structur e and non-de gener acy. Let G := ⟨ E g | g ∈ Γ ⟩ ⊂ C . By ( 4.12 ), G is symmet- ric, hence G ⊂ C Γ = Z (2) ( G ⊂ C ). Now we can state the main result of this section. Theorem 4.24. L et C b e a (F r ob enius) br aide d tensor c ate gory such that c ondition ( 4.11 ) is satisﬁe d. (a) Then, fg - b C loc A is a (F r ob enius) br aide d tensor c ate gory. (b) If C is ribb on and θ E g = id E g for al l g ∈ Γ , then fg - b C loc A is a ribb on tensor c ate gory. (c) When C has enough pr oje ctives, fg - b C loc A is a br aide d ﬁnite tensor c ate gory if and only if Irr( C Γ ) / Γ is ﬁnite. Pr o of. (a) Recall that A is Artinian and haploid. Moreov er, b y Corollary 4.18 , all simple A - mo dules are rigid. Thus, by Theorem 3.13 , fg - b C loc A is a braided (F rob enius) tensor category . (b) Note that A is F rob enius by Theorem 4.15 , and that θ A = id A b y the assumption on twists. Since fg- b C loc A is rigid b y part (a), Theorem 3.15 implies that it is ribb on. (c) By part (a), fg- b C loc A is a locally ﬁnite ab elian category with enough pro jectiv es. Hence, it is braided ﬁnite if and only if it has ﬁnitely many simple ob jects. The latter is equiv alen t to Irr( C Γ ) / Γ b eing ﬁnite by Corollary 4.22 . □ RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 27 Theorem 4.25. L et C b e a F r ob enius br aide d tensor c ate gory and char( k ) = 0 . If the simple obje cts in C Γ that trivial ly double br aid with al l other simples in C Γ ar e pr e cisely E g for g ∈ Γ , then fg - b C loc A is non-de gener ate. Pr o of. As c har( k ) = 0, condition ( 4.11 ) is satisﬁed. Thus, Theorem 4.24 (a) implies that fg- b C loc A is a F robenius tensor category . Hence, by Lemma 2.2 , fg - b C loc A is non-degenerate if and only if the only simple ob ject in its M¨ uger cen ter is A . But, simple ob jects in fg - b C loc A are M ( X, ρ ) for X ∈ C Γ and for such an ob ject to lie in the M ¨ uger center, it has to double braid trivially with all other simple ob jects. By Lemma 4.23 , this implies that X trivially double braids with all simple ob jects of C Γ . By assumption of the theorem, X ∼ = E g for some g ∈ Γ. Now, Lemma 4.8 implies that M ( X, ρ ) ∼ = A . Thus, the M ¨ uger center of fg - b C loc A is trivial, i.e. fg- b C loc A is non-degenerate. □ 5. Examples from quantum (super)gr oup ca tegories In this section we study tw o families of examples coming from unrolled quantum groups U H ( g ) and the unrolled quantum sup ergroups gl (1 | 1). W e analyze the corresp onding categories of ﬁnite- dimensional w eigh t mo dules and the simple-curren t local-mo dule categories they pro duce. W e follo w the same program: (a) identify the inv ertible simple ob jects and their tensor pro duct la w; (b) compute the relev an t braiding and t wist data on these inv ertibles; (c) choose commutativ e simple current algebras and extract consequences for the corresp ond- ing category of lo cal mo dules. The categorical consequences in step (c) are obtained b y applying Theorem 4.24 to the explicit simple-curren t data. Let Z / 2 Z = { ¯ 0 , ¯ 1 } b e the additive group of order 2. W e will sometimes write M S,T to denote the double braiding b etw een tw o ob jects S and T . 5.1. Unrolled quantum group categories. The categorical input follo ws Creutzig–Rup ert [CR22] (and references therein); w e recall only the data needed for applying the simple-curren t mac hinery of Section 4 . The setting for general sublattices L ′ ⊆ Λ inv (Prop osition 5.1 and its corollaries) is new. Fix a simple complex Lie algebra g with Cartan matrix A = ( a ij ), ro ot lattice Q ⊂ h ∗ , and w eigh t lattice P ⊂ h ∗ . Assume q is a primitiv e  -th ro ot of unity suc h that ord( q 2 ) > d i for all i . W e ﬁx log q and set q z = exp( z log q ) for z ∈ C . W e also set r =  if  is o dd , r = / 2 if  is even . (5.1) Let C := Rep wt ( U H q ( g )) b e the category of ﬁnite-dimensional weigh t mo dules for Rup ert’s unrolled restricted quantum group, obtained from the unrolled quantum group by quotien ting by the Hopf ideal generated by the prescrib ed p ow ers of the ro ot vectors. Concretely , U H q ( g ) is generated b y E i , F i , H i together with group-lik e elemen ts K γ indexed b y a ﬁxed lattice betw een Q and P , and a w eigh t mo dule is a ﬁnite-dimensional mo dule that decomp oses in to common H i -eigenspaces such that, for γ = P i k i α i , the op erator K γ acts on the λ -weigh t space b y the scalar q P i k i d i λ ( H i ) . Let λ ( H i ) denote ev aluation of λ ∈ h ∗ on H i . By Rupert [Rup22] (see also [CR22]), C is a ribb on F rob enius tensor category with enough pro jectives (equiv alen tly , pro jectiv e co v ers), and its M ¨ uger cen ter is trivial. 28 KENICHI SHIMIZU AND HARSHIT Y AD A V Let M λ denote the simple quotien t of the V erma mo dule with highest weigh t λ ∈ h ∗ . Then Irr( C ) = { M λ | λ ∈ h ∗ } . The mo dule M λ is one-dimensional, hence inv ertible, if and only if E i and F i act b y zero for all i , whic h is equiv alent to λ ( H i ) ∈ ℓ 2 d i Z for all i . In that case we write C λ := M λ . Hence Λ inv :=  λ ∈ h ∗     λ ( H i ) ∈  2 d i Z ∀ i  is isomorphic to the additive group of inv ertible simple ob jects in C , via λ 7→ C λ , and for λ, µ ∈ Λ inv one has C λ ⊗ C µ ∼ = C λ + µ . On inv ertibles one has c C λ ,C µ = q ⟨ λ,µ ⟩ τ , M C λ ,C µ = q 2 ⟨ λ,µ ⟩ id . where τ is the braiding of V ec and ⟨ λ, µ ⟩ := X i,j d i ( A − 1 ) ij λ ( H i ) µ ( H j ) for the in v ariant bilinear form on h ∗ . Here d i = ⟨ α i , α i ⟩ / 2 are the symmetrizers. The t wist is [CR22, Eq. (4.12)]: θ C λ = q ⟨ λ,λ +2(1 − r ) ρ ⟩ id C λ , (5.2) where ρ is the W eyl vector and r is as in ( 5.1 ). Th us, for an additiv e subgroup L ′ ⊆ Λ inv , the corresp onding simple current algebra is A L ′ = M λ ∈ L ′ C λ ∈ b C . In the notation of Section 4 , write β ( λ, µ ) = q ⟨ λ,µ ⟩ , b ( λ, µ ) = q 2 ⟨ λ,µ ⟩ , q ( λ ) = q ⟨ λ,λ ⟩ = β ( λ, λ ) . F or an additive subgroup L ′ ⊆ h ∗ , w e write ( L ′ ) ∗ := { γ ∈ h ∗ | ⟨ γ , λ ⟩ ∈ Z for all λ ∈ L ′ } for the dual lattice with resp ect to ⟨− , −⟩ . Prop osition 5.1. (a) The obje ct A L ′ is a haploid algebr a in b C . It is c ommutative if and only if q | L ′ ≡ 1 . Equivalently, ⟨ λ, λ ⟩ ∈  Z for λ ∈ L ′ . (b) L et C L ′ :=  X ∈ C   c C λ ,X ◦ c X,C λ = id X ⊗ C λ ∀ λ ∈ L ′  . Then C L ′ = C int L ′ , wher e C int L ′ is the ful l sub c ate gory of obje cts al l of whose weights lie in ℓ 2 ( L ′ ) ∗ . (c) Assume that A L ′ is c ommutative. Then the orbit set Irr( C L ′ ) /L ′ , wher e L ′ acts by tensoring with the invertibles C λ , is identiﬁe d with Λ( L ′ ) := ℓ 2 ( L ′ ) ∗ L ′ . Note that when A L ′ is commutativ e, equiv alen tly ⟨ λ, λ ⟩ ∈  Z for all λ ∈ L ′ , we get L ′ ⊆ ℓ 2 ( L ′ ) ∗ , so the quotien t Λ( L ′ ) is w ell deﬁned. Pr o of. (a) On inv ertibles, the braiding scalar is β ( λ, µ ) = q ⟨ λ,µ ⟩ , and this is a bicharacter on Λ inv . Hence the asso ciated ab elian 3-cocycle on the p oin ted sub category of inv ertibles is represented RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 29 b y (1 , β ), so the ordinary asso ciator cocycle ω restricts trivially to L ′ . By Lemma 2.9 (a), this giv es an algebra structure on A L ′ for any additiv e subgroup L ′ ⊆ Λ inv . Then Lemma 2.9 (b) implies that A L ′ is commutativ e if and only if q | L ′ ≡ 1. Since q ( λ ) = q ⟨ λ,λ ⟩ , this is equiv alent to ⟨ λ, λ ⟩ ∈  Z for all λ ∈ L ′ . That A is haploid follows from Hom C ( 1 , C λ ) = 0 for λ  = 0, hence Hom b C ( 1 , A L ′ ) ∼ = Hom C ( 1 , C 0 ) ∼ = k . (b) Let X ∈ C be a weigh t mo dule. F or a weigh t v ector w γ ∈ X one has  c C λ ,X ◦ c X,C λ  ( v λ ⊗ w γ ) = q 2 ⟨ λ,γ ⟩ ( v λ ⊗ w γ ) . Hence X ∈ C L ′ iﬀ 2 ⟨ λ, γ ⟩ ∈  Z for every λ ∈ L ′ and every weigh t γ of X , i.e. iﬀ all weigh ts lie in ℓ 2 ( L ′ ) ∗ . (c) Since A L ′ is commutativ e by (a), Theorems 4.11 and 4.12 together with Prop osition 4.21 sho w that simple lo cal A L ′ -mo dules are classiﬁed b y pairs ( M γ , ρ ), where M γ ∈ C L ′ and ρ is an irreducible pro jectiv e represen tation of Stab L ′ ( M γ ) := { η ∈ L ′ | M γ ⊗ C η ∼ = M γ } op . Since M γ ⊗ C η ∼ = M γ + η and M λ ∼ = M µ only for λ = µ , this stabilizer is trivial. Hence ρ is necessarily trivial, so simple lo cal A L ′ -mo dules are parametrized exactly by L ′ -orbits in Irr( C L ′ ). Moreo v er, Q ⊆ ℓ 2 ( L ′ ) ∗ since ⟨ η , α i ⟩ = d i η ( H i ) ∈ ℓ 2 Z for η ∈ L ′ ⊆ Λ inv and each simple ro ot α i ; hence if γ ∈ ℓ 2 ( L ′ ) ∗ , then ev ery weigh t of M γ also lies in ℓ 2 ( L ′ ) ∗ . As Irr( C ) = { M γ | γ ∈ h ∗ } , part (b) giv es that the simple ob jects of C L ′ are exactly { M γ | γ ∈ ℓ 2 ( L ′ ) ∗ } . Moreo v er, M γ ⊗ C η ∼ = M γ + η for η ∈ L ′ , so tw o such simples lie in the same L ′ -orbit if and only if their parameters diﬀer b y an element of L ′ . This identiﬁes Irr( C L ′ ) /L ′ with ℓ 2 ( L ′ ) ∗ L ′ = Λ( L ′ ). □ Lemma 5.2. L et M γ ∈ C L ′ b e simple. If M γ is non-invertible, then ther e exists a simple obje ct M α ∈ C L ′ such that c M α ,M γ ◦ c M γ ,M α  = id . In p articular, every simple obje ct of Z (2) ( C L ′ ) is invertible. Pr o of. As observed in the pro of of Prop osition 5.1 (c), one has Q ⊆ ℓ 2 ( L ′ ) ∗ . No w let v γ and v α i b e highest weigh t vectors of M γ and M α i . W rite the braiding as τ ◦ ˇ R with ˇ R = H e R , where H acts on w eigh t v ectors by q ⟨− , −⟩ and e R = 1 + X ν ∈ Q > 0 X s c ν,s E ν,s ⊗ F ν,s is a PBW expansion with each E ν,s of p ositive ro ot weigh t ν > 0. Since v γ and v α i are highest w eigh t vectors, every nontrivial PBW term kills the ﬁrst tensor factor, so e R ( v γ ⊗ v α i ) = v γ ⊗ v α i , e R ( v α i ⊗ v γ ) = v α i ⊗ v γ . Therefore  c M α i ,M γ ◦ c M γ ,M α i  ( v γ ⊗ v α i ) = q 2 ⟨ γ ,α i ⟩ ( v γ ⊗ v α i ) = q 2 γ ( H i ) i ( v γ ⊗ v α i ) . If M γ is non-in v ertible, then γ ( H i ) / ∈ ℓ 2 d i Z for some i , hence q 2 γ ( H i ) i  = 1. F or this i , w e hav e α i ∈ Q ⊆ ℓ 2 ( L ′ ) ∗ , so M α i ∈ C L ′ and the ab ov e mono dromy is not the identit y . The ﬁnal statement follo ws immediately . □ 30 KENICHI SHIMIZU AND HARSHIT Y AD A V Corollary 5.3. Assume that A L ′ is c ommutative. (a) fg - b C loc A L ′ is ribb on if and only if 2(1 − r ) ⟨ λ, ρ ⟩ ∈  Z for al l λ ∈ L ′ . (b) fg - b C loc A L ′ is br aide d ﬁnite if and only if Λ( L ′ ) is ﬁnite, e quivalently rank( L ′ ) = rank( P ) . (c) fg - b C loc A L ′ is non-de gener ate if the induc e d bichar acter on Λ( L ′ ) given by b ([ λ ] , [ µ ]) = q 2 ⟨ λ,µ ⟩ has trivial r adic al. Pr o of. (a) By Theorem 4.24 (b), b eing ribb on is equiv alent to θ C λ = id for all λ ∈ L ′ . F rom ( 5.2 ), θ C λ = q ⟨ λ,λ ⟩ q 2(1 − r ) ⟨ λ,ρ ⟩ id C λ . By commutativit y of A L ′ , ⟨ λ, λ ⟩ ∈  Z . So, the ﬁrst factor is 1, and the criterion is exactly 2(1 − r ) ⟨ λ, ρ ⟩ ∈  Z for all λ ∈ L ′ . (b) The general criterion is Theorem 4.24 (c): braided ﬁniteness is equiv alen t to ﬁniteness of Irr( C L ′ ) /L ′ . By Prop osition 5.1 (c), this orbit set is Λ( L ′ ). The equiv alence Λ( L ′ ) ﬁnite ⇐ ⇒ rank( L ′ ) = rank( P ) is the lattice computation in [CR22, Theorem 4.2]. (c) By Theorem 4.25 , non-degeneracy holds if every s imple ob ject of Z (2) ( C L ′ ) lies in ⟨ C λ | λ ∈ L ′ ⟩ . By Lemma 5.2 , every simple ob ject of Z (2) ( C L ′ ) is inv ertible. Th us Z (2) ( C L ′ ) consists only of in v ertibles C γ with γ ∈ Λ inv ∩ ℓ 2 ( L ′ ) ∗ . F or suc h C γ , the double braiding with M δ ∈ C L ′ is q 2 ⟨ γ ,δ ⟩ id, so C γ ∈ Z (2) ( C L ′ ) if and only if q 2 ⟨ γ ,δ ⟩ = 1 for all δ ∈ ℓ 2 ( L ′ ) ∗ , i.e. [ γ ] lies in the radical of b on Λ( L ′ ). Hence, if the radical of b is trivial, then every simple ob ject of Z (2) ( C L ′ ) lies in ⟨ C λ | λ ∈ L ′ ⟩ , and the claim follo ws from Theorem 4.25 . □ 5.2. General linear sup ergroup gl (1 | 1) . Let U E q ( gl (1 | 1)) denote the unrolled quantum sup er- group of gl (1 | 1) at a ro ot of unity q = e 2 π i/r ( r ≥ 3) [GY25, § 2.2]; it is a Hopf sup eralgebra with ev en generators E , G, K ± 1 and o dd generators X , Y . Let C := D q , int denote the category of ﬁnite-dimensional integral weigh t U E q ( gl (1 | 1))-sup ermo dules. By a weigh t mo dule we mean a ﬁnite-dimensional sup ermo dule on which the commuting even generators E and G act semisimply; th us ev ery weigh t vector v has a weigh t λ = ( λ E , λ G ) ∈ C 2 c haracterized b y E v = λ E v , Gv = λ G v . The adjectiv e “in tegral” means that all G -w eigh ts satisfy λ G ∈ Z . Then C is a ribbon tensor category with enough pro jectives [GY25, Theorem 2.14]. Throughout, q z := exp( z log q ) with log q = 2 π i/r . 5.2.1. Simple obje cts and their fusion. The simple ob jects of C are [GY25, § 2.3.2]: • the one-dimensional mo dules ε  nr 2 , b  ¯ p , where n ∈ Z , b ∈ Z and ¯ p ∈ Z / 2 Z : it has a basis v ector v of parity ¯ p suc h that E v = nr 2 v , Gv = bv , X v = 0 , Y v = 0 . • the 2-dimensional quantum Kac modules V ( α, a ) ¯ p , where α ∈ C \ r 2 Z , a ∈ Z and ¯ p ∈ Z / 2 Z : it has t wo basis vectors v (of degree ¯ p ) and v ′ (of degree ¯ p + ¯ 1) suc h that E v = αv , Gv = av , X v = 0 , Y v = v ′ , E v ′ = αv ′ , Gv ′ = ( a − 1) v ′ , X v ′ = [ α ] q v , Y v ′ = 0 . (5.3) Note that the unit ob ject is ε (0 , 0) ¯ 0 . The tensor pro ducts b etw een simple ob jects are computed in [GY25, § 2.3.3]. Here w e only recall the formulas relev ant for the subsequen t analysis of simple curren t extensions. F or n, n ′ , b, b ′ , a ∈ Z , RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 31 ¯ p, ¯ p ′ , ¯ q ∈ Z / 2 Z and α ∈ C \ r 2 Z one has ε  nr 2 , b  ¯ p ⊗ ε  n ′ r 2 , b ′  ¯ p ′ ∼ = ε  ( n + n ′ ) r 2 , b + b ′  ¯ p + ¯ p ′ , V ( α, a ) ¯ p ⊗ ε  nr 2 , b  ¯ q ∼ = V  α + nr 2 , a + b  ¯ p + ¯ q , where all sums of ¯ p ’s are tak en in Z / 2 Z . 5.2.2. Br aiding b etwe en simples. The braiding in C is given by c V ,W = τ V ,W ◦ e R V ,W ◦ Υ V ,W , where, for w eight vectors v ∈ V of weigh t λ and w ∈ W of weigh t µ , we hav e Υ V ,W ( v ⊗ w ) = q − λ E µ G − λ G µ E ( v ⊗ w ) , τ V ,W ( v ⊗ w ) = ( − 1) ¯ v ¯ w ( w ⊗ v ) , and e R is left multiplication by 1 + ( q − q − 1 )( X ⊗ Y )( K ⊗ K − 1 ). F or the one-dimensional mo dules we ha v e X = Y = 0, hence e R = id. Thus for ε = ε ( nr / 2 , b ) ¯ p and ε ′ = ε ( n ′ r / 2 , b ′ ) ¯ p ′ with basis v ectors v , v ′ , resp ectively , we obtain c ε,ε ′ ( v ⊗ v ′ ) = ( − 1) ¯ p ¯ p ′ q ( − nb ′ − bn ′ ) r 2 ( v ′ ⊗ v ) . (5.4) In particular, the double braiding is scalar: c ε ′ ,ε ◦ c ε,ε ′ = q − ( nb ′ + n ′ b ) r id ε ⊗ ε ′ = id ε ⊗ ε ′ . (5.5) Let V = V ( α, a ) ¯ p b e a 2-dimensional quan tum Kac mo dule and let ε = ε ( nr/ 2 , b ) ¯ q b e 1- dimensional with basis vector w . Cho ose a homogeneous weigh t basis { v , v ′ } of V satisfying ( 5.3 ). Since X w = Y w = 0, the correction term in e R v anishes on b oth V ⊗ ε and ε ⊗ V , hence e R = id in b oth cases. Therefore c V ,ε ( v ⊗ w ) = ( − 1) ¯ p ¯ q q − αb − a nr 2 ( w ⊗ v ) , c ε,V ( w ⊗ v ) = ( − 1) ¯ p ¯ q q − nr 2 a − bα ( v ⊗ w ) , c V ,ε ( v ′ ⊗ w ) = ( − 1) ( ¯ p + ¯ 1) ¯ q q − αb − ( a − 1) nr 2 ( w ⊗ v ′ ) , c ε,V ( w ⊗ v ′ ) = ( − 1) ( ¯ p + ¯ 1) ¯ q q − nr 2 ( a − 1) − bα ( v ′ ⊗ w ) . In particular, the parit y signs cancel in the double braiding. Th us, the double braiding is diagonal on the w eight basis, and since q r = 1 w e get c ε,V ◦ c V ,ε = q − 2 αb id V ⊗ ε . (5.6) 5.2.3. M¨ uger c enter of C . Theorem 5.4. The M¨ uger c enter Z (2) ( C ) is a ful l tensor sub c ate gory whose simple obje cts ar e ε  nr 2 , 0  ¯ p ( n ∈ Z , ¯ p ∈ Z / 2 Z ) . Pr o of. Let S b e a transparent simple ob ject. If S = V ( α, a ) ¯ p , tak e T = ε (0 , 1) ¯ 0 . By ( 5.6 ), M S,T = q − 2 α id S ⊗ T . Since α / ∈ r 2 Z , we hav e q − 2 α  = 1, contradiction. Hence S ∼ = ε ( nr 2 , b ) ¯ p . Using ( 5.6 ) with T = V ( α, 0) ¯ 0 giv es M S,T = q − 2 αb id S ⊗ T . T ransparency for all such T forces b = 0. So every transparent simple is of the form ε ( nr 2 , 0) ¯ p . Con v ersely , let S = ε  nr 2 , 0  ¯ p , and let T ∈ C b e arbitrary . Cho ose a basis v ector v of S , and let w ∈ T b e a homogeneous weigh t vector of weigh t ( α , a ) and parity ¯ q . Since X and Y act trivially 32 KENICHI SHIMIZU AND HARSHIT Y AD A V on S , the correction term in e R v anishes on b oth S ⊗ T and T ⊗ S . Hence c S,T ( v ⊗ w ) = ( − 1) ¯ p ¯ q q − nr 2 a ( w ⊗ v ) , c T ,S ( w ⊗ v ) = ( − 1) ¯ p ¯ q q − a nr 2 ( v ⊗ w ) . Therefore M S,T ( v ⊗ w ) = q − nra ( v ⊗ w ) = ( q r ) − na ( v ⊗ w ) = v ⊗ w . Since w eigh t vectors span T , it follows that M S,T = id S ⊗ T . Thus S ∈ Z (2) ( C ). □ 5.2.4. The gr oup of invertibles. The in vertible simple ob jects of C are the one-dimensional mo dules ε  nr 2 , b  ¯ p with n, b ∈ Z and ¯ p ∈ { ¯ 0 , ¯ 1 } . The group of simple inv ertible ob jects of C is (non- canonically) isomorphic to Z × Z × Z / 2 Z under addition of parameters ( n, b, ¯ p ). F or ε = ε  nr 2 , b  ¯ p with n, b ∈ Z , c ε,ε = ( − 1) ¯ p q − nrb id ε ⊗ ε = ( − 1) ¯ p id ε ⊗ ε . (5.7) Let θ denote the ribb on twist of C . F or ε ( α , b ) ¯ p , one can compute θ directly from the deﬁnition using the explicit righ t ev aluation/co ev aluation maps: ev V ( v ⊗ f ) = ( − 1) ¯ f ¯ v f ( K v ) , co ev V (1) = X i ( − 1) ¯ v i v ∗ i ⊗ K − 1 v i . Using the same calculation as in [GY25, Theorem 2.14], we obtain θ ε ( α,b ) ¯ p = q − 2 αb + α id ε ( α,b ) ¯ p . (5.8) F or α = nr 2 (with n, b ∈ Z ), this b ecomes θ ε ( nr 2 ,b ) ¯ p = q − nrb + nr 2 id = q nr 2 id = ( − 1) n id , where w e used q r = 1, so q − nrb = 1, and q nr/ 2 = e π in = ( − 1) n . W e now determine exactly which simple current algebras are comm utativ e. Deﬁne the quadratic form q : Z × Z × Z / 2 Z → C × b y q ( n, b, ¯ p ) = ( − 1) ¯ p . (5.9) Hence q | Γ ≡ 1 ⇐ ⇒ Γ = L × { ¯ 0 } for some L ≤ Z 2 . (5.10) Th us the comm utativ e simple current algebras in b C are precisely those indexed by subgroups L ≤ Z 2 : A Γ = M ( n,b ) ∈ L ε  nr 2 , b  ¯ 0 (Γ = L × { ¯ 0 } ) . W e denote by π n and π b the pro jections on to the ﬁrst and second co ordinate of Z × Z × Z / 2 Z . Prop osition 5.5. L et Γ = L × { ¯ 0 } ≤ Inv ( C ) with L ≤ Z 2 , and set A Γ = L γ ∈ Γ γ ∈ b C . Write e ach γ ∈ Γ as γ = ε  n ( γ ) r 2 , b ( γ )  ¯ 0 . Then, fg - b C loc A Γ is br aide d tensor. It is ribb on iﬀ θ γ = id γ for al l γ ∈ Γ , e quivalently iﬀ n ( γ ) is even for every γ ∈ Γ . Pr o of. By Theorem 4.24 (a), fg - b C loc A Γ is braided tensor. By Theorem 4.24 (b), it is ribbon iﬀ θ γ = id γ for all γ ∈ Γ. F or γ = ε ( n ( γ ) r 2 , b ( γ )) ¯ 0 , the t wist formula ab o ve gives θ γ = ( − 1) n ( γ ) id γ , so this is equiv alent to n ( γ ) b eing even for every γ ∈ Γ. □ 5.2.5. Finiteness of fg - b C A and fg - b C loc A . The full mo dule category fg- b C A is never ﬁnite, but the lo cal sub category ma y b e. Let Γ = L × { ¯ 0 } ≤ Inv ( C ) with L ≤ Z 2 , and let A Γ = L γ ∈ Γ γ . The RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 33 Γ-orbits on Kac simples hav e a contin uous α -parameter modulo the discrete subgroup r 2 π n ( L ), hence Irr( C ) / Γ is inﬁnite and fg- b C A Γ has inﬁnitely man y simple ob jects b y Theorem 4.19 . F or γ = ε ( nr 2 , b ) ¯ 0 ∈ Γ and a Kac simple V ( α, a ) ¯ p , ( 5.6 ) giv es M γ ,V = q − 2 αb id. W rite B := π b (Γ) ≤ Z . Hence V ( α, a ) ¯ p is lo cal iﬀ q − 2 αb = 1 for all b ∈ B . If B = d Z ( d > 0), this is equiv alen t to α ∈ r 2 d Z , while for B = 0 there is no restriction on α . So locality replaces the contin uous α -family by an arithmetic condition determined by B . In our gl (1 | 1) setting, C Γ con tains all one-dimensional simples ε ( nr 2 , b ) ¯ p . It con tains Kac simples V ( α, a ) ¯ p exactly for α ∈ r 2 d Z \ r 2 Z when B = d Z and d > 0, while for B = 0 there is no extra restriction b eyond α / ∈ r 2 Z . Th us locality remov es the con tin uous α -parameter exactly when π b ( L )  = 0. Finiteness of fg- b C loc A Γ is stronger, and will hold exactly when L has ﬁnite index in Z 2 . Prop osition 5.6. L et Γ ≤ Inv ( C ) satisfy q | Γ ≡ 1 , write Γ = L × { ¯ 0 } with L ≤ Z 2 , and let A Γ = L γ ∈ Γ γ . Then fg - b C loc A Γ is ﬁnite if and only if [ Z 2 : L ] < ∞ . Pr o of. By ( 5.5 ), ev ery one-dimensional simple ε ( nr 2 , b ) ¯ p cen tralizes Γ, hence b elongs to C Γ . The Γ-action on these simples is translation by L on ( n, b ) ∈ Z 2 , so their orbit set is ( Z 2 /L ) × ( Z / 2 Z ) . Therefore, if [ Z 2 : L ] = ∞ , then Irr( C Γ ) / Γ is already inﬁnite. Con v ersely , assume [ Z 2 : L ] < ∞ . Then N := π n ( L ) = n 0 Z and B := π b ( L ) = d Z with n 0 , d > 0. F or a Kac simple V ( α, a ) ¯ p , ( 5.6 ) giv es lo calit y if and only if q − 2 αb = 1 for all b ∈ B , equiv alen tly α ∈ r 2 d Z . Th us the lo cal Kac simples are parametrized by  r 2 d Z \ r 2 Z  × Z × Z / 2 Z . No w the Γ-action on Kac parameters is induced by ( n, b ) ∈ L 7→ ( α, a ) 7− →  α + nr 2 , a + b  . Let L ′ := n nr 2 , b  ∈ r 2 Z × Z    ( n, b ) ∈ L o ⊂ r 2 d Z × Z . Since L has ﬁnite index in Z 2 , it has rank 2, hence L ′ is a rank-2 sublattice of r 2 Z × Z . Moreo v er, h r 2 d Z × Z : r 2 Z × Z i = d < ∞ , so L ′ is also a ﬁnite-index sublattice of r 2 d Z × Z . Therefore  r 2 d Z × Z   L ′ 34 KENICHI SHIMIZU AND HARSHIT Y AD A V is ﬁnite. Since the excluded subset r 2 Z × Z is L ′ -stable, the same is true after restricting to  r 2 d Z \ r 2 Z  × Z . Hence the lo cal Kac simples con tribute only ﬁnitely man y Γ-orbits; the parit y parameter con- tributes only the ﬁnite factor Z / 2 Z . The one-dimensional part also contr ibutes ﬁnitely many Γ-orbits b ecause ( Z 2 /L ) × ( Z / 2 Z ) is ﬁnite. Hence Irr( C Γ ) / Γ is ﬁnite. No w apply Theorem 4.24 (c). □ In summary , ﬁnite-index subgroups L ≤ Z 2 yield explicit ﬁnite braided (and, when all n ( γ ) are ev en, ribb on) lo cal-mo dule categories in the gl (1 | 1) setting. Appendix A. Non-braided version of Etingof-Penneys’ lemma In App endix A , b y an ab elian monoidal c ate gory , we mean an ab elian category endow ed with a structure of a monoidal category such that the monoidal pro duct is additive and right exact in eac h v ariable. A.1. Etingof-P enneys’ Lemma. An ob ject X of an ab elian monoidal category C is said to b e left ﬂat if the endofunctor X ⊗ ( − ) on C is exact. Similarly , X is said to b e right ﬂat if ( − ) ⊗ X is exact. W e say that C has enough left (right) ﬂat ob jects if ev ery ob ject of C is a quotien t of a left (righ t) ﬂat ob ject of C . A ﬂat ob ject is a left and right ﬂat ob ject. Lemma A.1. L et C b e an ab elian monoidal c ate gory, and let 0 → Y i − → Z p − → X → 0 b e a short exact se quenc e in C . Assume that C has enough left ﬂat obje cts and enough right ﬂat obje cts. (a) If X and Y ar e left rigid and right ﬂat, then Z is left rigid. (b) If X and Z ar e left rigid, then Y is left rigid. (c) If Y and Z ar e left rigid and i ∗ is an epimorphism, then X is left rigid. This lemma w as established b y Etingof and P enneys under the assumption that C is braided [EP26, Lemma 4.2]. They also remark ed that the use of the braiding is not essen tial [EP26, Remark 4.4]. One of the aims of this App endix is to demonstrate that [EP26, Lemma 4.2] holds in the ab ov e form in the absence of a braiding b y tracing the original pro of carefully . As is w ell-kno wn, a left (righ t) rigid ob ject is left (righ t) ﬂat (see Lemma A.5 b elow). One tec hnical diﬃcult y in the non-braided case is that a left (right) rigid ob ject need not b e righ t (left) rigid. F or example, given a left rigid ob ject Y , a map b et w een Y oneda Ext 1 groups induced b y the functor Y ∗ ⊗ ( − ) is considered in the pro of of [EP26, Lemma 4.2]. Ho w ev er, since the functor Y ∗ ⊗ ( − ) is not necessarily exact in our non-braided setting, w e will face some tec hnical problems. F or completeness, in § A.4 , w e discuss maps b etw een Y oneda Ext 1 groups induced by an additiv e functor which is not necessarily exact. Another p oint to note is that techniq ues of the T or functor were used in the pro of of [EP26, Lemma 4.2]. Instead of justifying the use of the T or functor in our setting, w e ha ve chosen to assume that C has enough left ﬂat ob jects and enough right ﬂat ob jects, and to v erify some necessary tec hnical lemmas on ﬂat ob jects in a direct wa y in § A.3 . The assumption on C do es not matter for our main application to the category of ﬁnitely generated modules ov er a cen tral RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 35 comm utativ e algebra in a braided tensor category in § 3 , since every ob ject of such a category is a quotien t of a ﬂat (in fact, rigid) ob ject. An imp ortant consequence of the ab ov e Lemma is: Theorem A.2. L et C b e an ab elian monoidal c ate gory in which every obje ct has ﬁnite length. Assume that C has enough left ﬂat obje cts and enough right ﬂat obje cts. If every simple obje ct of C is rigid, then C is rigid. Pr o of. It is ob vious that the zero ob ject of C is rigid. Let M be a non-zero ob ject of C . W e prov e the rigidit y of M by induction on the length of M . If M has length 1, then M is simple and hence rigid b y assumption. No w let M ha v e length n > 1 and assume that every ob ject of C of length < n is rigid. Cho ose a short exact sequence 0 → N → M → S → 0 with S simple. By induction, N is rigid, and b y assumption S is rigid. In particular, N and S are ﬂat (see Lemma A.5 below). Applying Lemma A.1 to this exact sequence shows that M is left rigid. Applying Lemma A.1 in the rev ersed monoidal category C rev sho ws that M is right rigid. Hence M is rigid. □ A.2. Lemmas on rigid ob jects. W e ﬁrst collect useful lemmas on rigid ob jects. The following lemma is found in the pro of of [EP26, Lemma 4.2]: Lemma A.3. L et C b e a monoidal c ate gory, let L and R b e obje cts of C , and let ε : L ⊗ R → 1 and η : 1 → R ⊗ L b e morphisms in C . Supp ose that ξ := ( ε ⊗ id L ) ◦ (id L ⊗ η ) and ζ := (id R ⊗ ε ) ◦ ( η ⊗ id R ) ar e invertible in C . Then we have ε ◦ ( ξ − 1 ⊗ id R ) = ε ◦ (id L ⊗ ζ − 1 ) , ( ζ − 1 ⊗ id L ) ◦ η = (id R ⊗ ξ − 1 ) ◦ η . L et ε ′ and η ′ b e b oth sides of the former and the latter e quation, r esp e ctively. Then the triples ( L, ε ′ , η ) and ( L, ε, η ′ ) ar e left dual obje cts of R . Pr o of. By the functorial prop ert y of ⊗ , it is easy to v erify ε ◦ ( ξ ⊗ id R ) = ( ε ⊗ ε ) ◦ (id L ⊗ η ⊗ id R ) = ε ◦ (id L ⊗ ζ ) . By comp osing ξ − 1 ⊗ ζ − 1 from the right, w e obtain the ﬁrst equation of the statement. The second one is pro ved in a similar wa y . No w we hav e ( ε ′ ⊗ id L ) ◦ (id L ⊗ η ) = ( ε ⊗ id L ) ◦ ( ξ − 1 ⊗ η ) = ξ ◦ ξ − 1 = id L , (id R ⊗ ε ′ ) ◦ ( η ⊗ id R ) = (id R ⊗ ε ) ◦ ( η ⊗ ζ − 1 ) = ζ ◦ ζ − 1 = id R , whic h means that ( L, ε ′ , η ) is a left dual ob ject of R . The case of ( L, ε, η ′ ) is pro ved in a similar w a y . □ Let A and B b e ob jects of C , and let f : A → B b e a morphism in C . T o discuss the rigidity of B from that of A or its conv erse, we note the follo wing lemma: Let A ′ and B ′ also b e ob jects of C , and let ε A : A ′ ⊗ A → 1 , η A : 1 → A ⊗ A ′ , ε B : B ′ ⊗ B → 1 , η B : 1 → B ⊗ B ′ and f ′ : B ′ → A ′ b e morphisms in C suc h that ε B ◦ (id B ′ ⊗ f ) = ε A ◦ ( f ′ ⊗ id A ) , ( f ⊗ id A ′ ) ◦ η A = (id B ⊗ f ′ ) ◦ η B . 36 KENICHI SHIMIZU AND HARSHIT Y AD A V Lemma A.4. In the ab ove situation, we have f ◦ (id A ⊗ ε A ) ◦ ( η A ⊗ id A ) = (id B ⊗ ε B ) ◦ ( η B ⊗ id B ) ◦ f , ( ε A ⊗ id A ′ ) ◦ (id A ′ ⊗ η A ) ◦ f ′ = f ′ ◦ ( ε B ⊗ id B ′ ) ◦ (id B ′ ⊗ η B ) . F or example, if ( A ′ , ε A , η A ) is a left dual ob ject of A , f is epic and f ′ is monic, then we can conclude that ( B ′ , ε B , η B ) is a left dual ob ject of B b y this lemma. Pr o of. The ﬁrst equation is prov ed as follo ws: f ◦ (id A ⊗ ε A ) ◦ ( η A ⊗ id A ) = (id B ⊗ ε A ⊗ id A ) ◦ ( f ⊗ id A ′ ⊗ id A ) ◦ ( η A ⊗ id A ) = (id B ⊗ ε A ⊗ id A ) ◦ (id A ⊗ f ′ ⊗ id A ) ◦ ( η B ⊗ id A ) = (id B ⊗ ε A ⊗ id A ) ◦ (id A ⊗ id B ′ ⊗ f ) ◦ ( η B ⊗ id A ) = (id B ⊗ ε B ) ◦ ( η B ⊗ id B ) ◦ f . The second one is prov ed by a straightforw ard computation in a similar w ay . □ Let X b e a left rigid ob ject of C . As is well-kno wn, the functor X ∗ ⊗ ( − ) is left adjoint to the functor X ⊗ ( − ). The unit and the counit of this adjunction are given by the coev aluation and the ev aluation, resp ectively . This observ ation yields the following well-kno wn fact: Lemma A.5. In an ab elian monoidal c ate gory, a left rigid obje ct is left ﬂat. Similarly, a right rigid obje ct is right ﬂat. Pr o of. The functor X ⊗ ( − ) is right exact by our deﬁnition of an ab elian monoidal category . As it has a left adjoin t X ∗ ⊗ ( − ), the functor X ⊗ ( − ) is also left exact. Thus X is left ﬂat. The righ t rigid case is similar. □ Let f : X → Y b e a morphism in a monoidal category C betw een left rigid ob jects X and Y . Then the left dual morphism of f is deﬁned and denoted by f ∗ := (ev Y ⊗ id X ∗ )(id Y ∗ ⊗ f ⊗ id X ∗ )(id Y ∗ ⊗ co ev X ) : Y ∗ → X ∗ . The zig-zag relations imply ev X ( f ∗ ⊗ id X ∗ ) = ev Y (id Y ∗ ⊗ f ) , (id Y ⊗ f ∗ )co ev Y = ( f ⊗ id X ∗ )co ev X . The follo wing fact was remarked in the pro of of [EP26, Lemma 4.2]: Lemma A.6. If f : X → Y is an epimorphism in an ab elian monoidal c ate gory C b etwe en left rigid obje cts X and Y , then f ∗ is monic. Pr o of. W e ﬁx an ob ject T ∈ C and consider the commutativ e diagram Hom C ( T , Y ∗ ) Hom C ( T ⊗ Y , 1 ) Hom C ( T , X ∗ ) Hom C ( T ⊗ X , 1 ) . ∼ = Hom C ( T ,f ∗ ) Hom C (id T ⊗ f , 1 ) ∼ = Since f is an epimorphism, and since the monoidal product in C is assumed to be right exact, the morphism id T ⊗ f is an epimorphism. This implies that the right vertical arrow of the ab ov e diagram is injectiv e, and thus so is the left vertical arrow. The pro of is done. □ RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 37 A.3. Lemmas on ﬂat ob jects. Let C b e an ab elian monoidal category . W e pro vide some lemmas on ﬂat ob jects. Lemma A.7. L et 0 → Y → Z → X → 0 b e an exact se quenc e in C , and let A b e an obje ct of C . (a) If X is right ﬂat and A is a quotient of a left ﬂat obje ct, then 0 → A ⊗ Y → A ⊗ Z → A ⊗ X → 0 is exact. (b) If X is left ﬂat and A is a quotient of a right ﬂat obje ct, then 0 → Y ⊗ A → Z ⊗ A → X ⊗ A → 0 is exact. Pr o of. (a) W e c hoose an epimorphism p : F → A with F left ﬂat and let i : K → F b e the k ernel of p so that we ha v e an exact sequence 0 → K → F → A → 0. W e consider the following comm utativ e diagram: Ker( i ⊗ id Y ) Ker( i ⊗ id Z ) Ker( i ⊗ id X ) K ⊗ Y K ⊗ Z K ⊗ X 0 0 F ⊗ Y F ⊗ Z F ⊗ X 0 A ⊗ Y A ⊗ Z A ⊗ X 0 i ⊗ id Y i ⊗ id Z i ⊗ id X p ⊗ id Y p ⊗ id Z p ⊗ id X Here, for brevity , some of v ertical arro ws 0 → and → 0 are omitted, and the sym b ols  → and ↠ are used to mean a monomorphism and an epimorphism, resp ectively . The third ro w is exact since F is left ﬂat. The other ro ws and all columns are also exact by the right exactness of the tensor pro duct. Since X is right ﬂat, i ⊗ id X is monic. Th us the snak e lemma yields the exact sequence 0 → A ⊗ Y → A ⊗ Z → A ⊗ X → 0. (b) Apply (a) to C rev . □ Lemma A.8. L et 0 → Y → Z → X → 0 b e an exact se quenc e in C . (a) If C has enough left ﬂat obje cts and b oth X and Y ar e right ﬂat, then Z is right ﬂat. (b) If C has enough right ﬂat obje cts and b oth X and Y ar e left ﬂat, then Z is left ﬂat. Pr o of. (a) W e assume that C has enough left ﬂat ob jects and b oth X and Y are right ﬂat. Let 0 → B → C → A → 0 be an exact sequence in C . W e consider the follo wing comm utativ e diagram obtained b y tensoring 0 → B → C → A → 0 and 0 → Y → Z → X → 0: B ⊗ Y C ⊗ Y A ⊗ Y B ⊗ Z C ⊗ Z A ⊗ Z B ⊗ X C ⊗ X A ⊗ X The ﬁrst and the third ro w are exact since X and Y are right ﬂat. As C has enough left ﬂat ob jects and X is right ﬂat, by Lemma A.7 , the columns are exact. Thus, b y the nine lemma, the middle ro w is also exact. Therefore Z is right ﬂat. (b) Apply (a) to C rev . □ 38 KENICHI SHIMIZU AND HARSHIT Y AD A V A.4. Reminder on the Y oneda Ext. The Ext functor is deﬁned in an arbitrary ab elian cate- gory b y assigning the set of equiv alence classes of exact sequences of a particular form (so-called Y oneda Ext). Here w e give a reminder on basics on the Y oneda Ext functor, restricting ourselves to Ext 1 . A.4.1. Deﬁnition of the Y one da Ext functor. Let A b e an ab elian category . F or ob jects X and Y of A , the set Ext 1 A ( X , Y ) is deﬁned to b e the set of equiv alence classes of exact sequences of the form 0 → Y → E → X → 0 for some E ∈ A . When A is clear from the con text, we often write it as Ext 1 ( X , Y ). The construction of the set Ext 1 ( X , Y ) in fact extends to a functor from A op × A to the category Ab of ab elian groups. The functorial property and the ab elian group structure are explained as follo ws: (1) Given a morphism f : X ′ → X in A and an ob ject Y ∈ A , the map 2 f ⋆ : Ext 1 ( X , Y ) → Ext 1 ( X ′ , Y ) is deﬁned as follows: Let e b e an element of Ext 1 ( X , Y ) represented by an exact sequence 0 → Y → E → X → 0. W e consider the commutativ e diagram 0 Y E × X X ′ X ′ 0 0 Y E X 0 (PB) f (A.1) with exact rows, where (PB) means a pullback square. The element f ⋆ ( e ) is deﬁned to b e the elemen t represented by the ﬁrst row of the ab ov e diagram. (2) Given a morphism g : Y → Y ′ in A and an ob ject X ∈ A , the map g ⋆ : Ext 1 ( X , Y ) → Ext 1 ( X , Y ′ ) is deﬁned dually to (1): if e ∈ Ext 1 ( X , Y ) is represen ted b y 0 → Y → E → X → 0, consider the comm utative diagram 0 Y E X 0 0 Y ′ E ⨿ Y Y ′ X 0 g (PO) (A.2) with exact rows, where (PO) means a pushout square. The element g ⋆ ( e ) is deﬁned to b e the elemen t represented by the second row of the ab ov e diagram. (3) F or morphisms f and g as in (1) and (2), one can show that f ⋆ and g ⋆ comm ute. No w we deﬁne the map Ext 1 ( f , g ) by Ext 1 ( f , g ) := f ⋆ ◦ g ⋆ = g ⋆ ◦ f ⋆ : Ext 1 ( X , Y ) → Ext 1 ( X ′ , Y ′ ) . (4) F or e 1 , e 2 ∈ Ext 1 ( X , Y ), their sum is deﬁned by e 1 + e 2 := Ext 1 (diag X , sum Y )( e 1 ⊕ e 2 ) (the Baer sum) , where diag X : X → X ⊕ X and sum Y : Y ⊕ Y → Y are the diagonal morphism and the sum, resp ectively , and e 1 ⊕ e 2 ∈ Ext 1 ( X ⊕ X, Y ⊕ Y ) is the element represented by the 2 T o distinguish it from duals in a monoidal category , we use ⋆ to mean the maps b etw een Ext groups induced by a morphism. More sp eciﬁcally , ( − ) ⋆ is used in the contra v ariant case, while ( − ) ⋆ is used in the cov arian t case. RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 39 exact sequence obtained b y taking the direct sum of exact sequences representing e 1 and e 2 . (5) If e ∈ Ext 1 ( X , Y ) is represented by 0 → Y i → E → X → 0, then the in verse of e (with resp ect to the Baer sum) is represented by the sequence 0 Y E X 0 . − i The follo wing standard lemma is useful for computing f ⋆ and g ⋆ . Lemma A.9. L et 0 → Y → E → X → 0 b e an exact se quenc e in A , and let e b e an element of Ext 1 ( X , Y ) r epr esente d by this exact se quenc e. (a) If ther e is a c ommutative diagr am 0 Y F X ′ 0 0 Y E X 0 f (A.3) with exact r ows, then the top r ow r epr esents f ⋆ ( e ) . (b) If ther e is a c ommutative diagr am 0 Y E X 0 0 Y ′ G X 0 g (A.4) with exact r ows, then the b ottom r ow r epr esents g ⋆ ( e ) . Pr o of. W e only giv e a pro of of (a) since (b) can b e v eriﬁed in a similar wa y . W e assume that there is a commutativ e diagram as in (a), and lab el the arro ws in the diagrams ( A.1 ) and ( A.3 ) as follo ws: ( A.1 ): Y E × X X ′ X ′ Y E X a ′ pr 2 pr 1 f a b ( A.3 ): Y F X ′ Y E X a ′′ p 2 p 1 f a b By the universal prop erty of the pullbac k, there exists a unique morphism s : F → E × X X ′ suc h that p i = pr i ◦ s ( i = 1 , 2). Since pr 1 sa ′′ = p 1 a ′′ = a = pr 1 a ′ and pr 2 sa ′′ = p 2 a ′′ = 0 = pr 2 a ′ , w e ha v e sa ′′ = a ′ . Hence we obtain the commutativ e diagram 0 Y F X ′ 0 0 Y E × X X ′ X ′ 0 a ′′ p 2 s a ′ pr 2 with exact rows. The ﬁve lemma implies that s is an isomorphism. Since f ⋆ ( e ) is represented by the top ro w of ( A.1 ) by deﬁnition, f ⋆ ( e ) is also represen ted by the top row of ( A.3 ). □ It is known that the functor Ext 1 : A op × A → Ab is additiv e in each v ariable. Thus, in particular, there are canonical isomorphisms Ext 1 ( X ⊕ Y , Z ) ∼ = Ext 1 ( X , Z ) ⊕ Ext 1 ( Y , Z ) , Ext 1 ( X , Y ⊕ Z ) ∼ = Ext 1 ( X , Y ) ⊕ Ext 1 ( X , Z ) 40 KENICHI SHIMIZU AND HARSHIT Y AD A V for X , Y , Z ∈ A . W e note the follo wing well-kno wn relation b etw een the ab ov e isomorphisms and the Baer sum. Lemma A.10. The fol lowing diagr am is c ommutative: Ext 1 ( X , Y ) ⊕ Ext 1 ( X , Y ) Ext 1 ( X ⊕ X , Y ) Ext 1 ( X , Y ⊕ Y ) Ext 1 ( X , Y ) ∼ = ∼ = the Baer sum (diag X ) ⋆ (sum Y ) ⋆ A.4.2. Maps induc e d by functors. Let F : A → B b e an additive functor b etw een ab elian cat- egories. F or brevit y , w e deﬁne E F to b e the class of pairs ( X , Y ) consisting of ob jects of A suc h that F preserves extensions of X by Y , that is, for every exact sequence in A of the form 0 → Y → E → X → 0, the sequence 0 → F ( Y ) → F ( E ) → F ( X ) → 0 is exact. F or each pair ( X , Y ) ∈ E F , w e hav e an obvious map e F X,Y : Ext 1 A ( X , Y ) → Ext 1 B ( F ( X ) , F ( Y )) induced b y the functor F (when no confusion arises, w e often write e F X,Y ( e ) simply as F ( e )). Here, we exhibit prop erties of the map e F X,Y . All of the results should b e well-kno wn when the functor in question is exact. Since the exactness is to o strong for our intended applications, w e revisit the standard argument and explicitly demonstrate that the results con tin ue to hold under w eak er assumptions. First, w e discuss the naturality of e F X,Y . Lemma A.11. L et F b e as ab ove. (a) F or ﬁxe d X ∈ A , the map e F X,Y is natur al in Y ∈ A such that ( X, Y ) ∈ E F . (b) F or ﬁxe d Y ∈ A , the map e F X,Y is natur al in X ∈ A such t hat ( X, Y ) ∈ E F . Pr o of. W e only present the pro of of (a), since the pro of of (b) is analogous. W e ﬁx X ∈ A and let g : Y → Y ′ b e a morphism in A such that both ( X , Y ) and ( X , Y ′ ) belong to the class E F . Let e ∈ Ext 1 ( X , Y ) b e an elemen t represented by an exact sequence 0 → Y → E → X → 0. By applying F to the diagram ( A.2 ), we obtain a commutativ e diagram 0 F ( Y ) F ( E ) F ( X ) 0 0 F ( Y ′ ) F ( E ⨿ Y Y ′ ) F ( X ) 0 F ( g ) whose rows are exact since ( X, Y ) , ( X , Y ′ ) ∈ E F . The second ro w represents F ( g ⋆ ( e )). Since the ﬁrst row represents F ( e ), the second ro w also represents F ( g ) ⋆ ( F ( e )) by Lemma A.9 . Therefore w e ha ve F ( g ⋆ ( e )) = F ( g ) ⋆ ( F ( e )). This means that e F X,Y is natural as stated. □ The additivit y of e F X,Y app ears to require a somewhat subtle condition. Lemma A.12. L et F b e as ab ove, and let X , Y ∈ A . If b oth ( X , Y ) and ( X , Y ⊕ Y ) b elong to the class E F , then the map e F X,Y is additive. RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 41 Pr o of. Let e i ( i = 1 , 2) b e an element of Ext 1 ( X , Y ) represented by an exact sequence 0 → Y → E i → X → 0. W e consider the following commutativ e diagram with exact ro ws: 0 Y ⊕ Y E 1 ⊕ E 2 X ⊕ X 0 0 Y ⊕ Y • X 0 0 Y • X 0 sum Y (PB) diag X (PO) W e call this diagram D and consider the diagram F ( D ) obtained from D by applying F . Since e 1 + e 2 is represented by the third row of D , the third row of F ( D ) represents F ( e 1 + e 2 ). No w we compute the third row of F ( D ) in a diﬀerent w a y . Since F is additive, w e may identify F ( V ⊕ W ) = F ( V ) ⊕ F ( W ) for V , W ∈ A . Under this iden tiﬁcation, w e ha v e F (diag X ) = diag F ( X ) and F (sum Y ) = sum F ( Y ) . It is straigh tforw ard to verify that the ﬁrst row of F ( D ) is exact and represen ts F ( e 1 ) ⊕ F ( e 2 ). By the assumption that ( X , Y ) , ( X , Y ⊕ Y ) ∈ E F , the second and the third rows of F ( D ) are also exact. Thus, by using Lemma A.9 , we see that the third row of F ( D ) represen ts F ( e 1 ) + F ( e 2 ). The pro of is done. □ This lemma pro vides an application that is directly related to our ob jective: Lemma A.13. L et C b e an ab elian monoidal c ate gory, and let X, Y , A ∈ C . If X is right ﬂat and A is a quotient of a left ﬂat obje ct, then the functor A ⊗ ( − ) induc es a homomorphism Ext 1 ( X , Y ) → Ext 1 ( A ⊗ X, A ⊗ Y ) of ab elian gr oups. Pr o of. W e consider the functor F = A ⊗ ( − ). Lemma A.7 implies that F preserv es extensions of X b y arbitrary ob ject. Thus the functor F induces a map b et w een Ext 1 -groups as stated, and this map is in fact additive by Lemma A.12 . □ A.4.3. The adjunction isomorphism. As b efore, F : A → B is an additiv e functor b etw een abelian categories. W e assume that F has a righ t adjoint G : B → A . Let η : id A → GF and ε : F G → id B b e the unit and the counit of the adjunction, resp ectiv ely . Under a suitable condition for X ∈ A and Y ∈ B , the following lemma giv es an isomorphism Ext 1 A ( X , G ( Y )) ∼ = Ext 1 B ( F ( X ) , Y ) giv en b y a formula similar to the adjunction isomorphism. Lemma A.14. If ( X , G ( Y )) ∈ E F and ( F ( X ) , Y ) ∈ E G , then the map Φ : Ext 1 A ( X , G ( Y )) Ext 1 B ( F ( X ) , F G ( Y )) Ext 1 B ( F ( X ) , Y ) e F ( ε Y ) ⋆ is invertible with the inverse Ψ : Ext 1 B ( F ( X ) , Y ) Ext 1 A ( GF ( X ) , G ( Y )) Ext 1 A ( X , G ( Y )) . e G ( η X ) ⋆ Pr o of. W e sho w that Ψ ◦ Φ = id; the pro of that Φ ◦ Ψ = id is analogous. Let e ∈ Ext 1 A ( X , G ( Y )) b e an element represen ted by a short exact sequence 0 → G ( Y ) a − → E b − → X → 0. By deﬁnition, 42 KENICHI SHIMIZU AND HARSHIT Y AD A V Φ( e ) is represen ted by the second row of the following commutativ e diagram with exact rows: 0 F G ( Y ) F ( E ) F ( X ) 0 0 Y E ′ F ( X ) 0 F ( a ) (PO) ε Y j F ( b ) a ′ b ′ Ψ(Φ( e )) is represen ted by the ﬁrst row of the following commutativ e diagram with exact rows: 0 G ( Y ) E ′′ X 0 0 G ( Y ) G ( E ′ ) GF ( X ) 0 a ′′ (PB) v u η X G ( a ′ ) G ( b ′ ) Consider the comp osite t := G ( j ) ◦ η E : E → G ( E ′ ). By naturality of η , w e ha v e G ( b ′ ) ◦ t = G ( b ′ ) ◦ G ( j ) ◦ η E = GF ( b ) ◦ η E = η X ◦ b. Hence, b y the univ ersal prop erty of the pullbac k, there is a unique morphism s : E → E ′′ suc h that u ◦ s = t and v ◦ s = b . Using naturality of η , we get u ◦ s ◦ a = t ◦ a = G ( j ) ◦ η E ◦ a = G ( j ) ◦ GF ( a ) ◦ η G ( Y ) = G ( a ′ ) ◦ G ( ε Y ) ◦ η G ( Y ) = G ( a ′ ) = u ◦ a ′′ , where the last equality is the triangle identit y for the unit and the counit. Since also v ◦ s ◦ a = b ◦ a = 0 = v ◦ a ′′ , w e hav e s ◦ a = a ′′ . Thus the map s is a morphism of short exact sequences 0 G ( Y ) E X 0 0 G ( Y ) E ′′ X 0 . a b s a ′′ v By the ﬁv e lemma, s is an isomorphism, hence e and Ψ(Φ( e )) are equiv alen t extensions. Thus they deﬁne the same class in Ext 1 A ( X , G ( Y )). □ A.5. Exercise in homological algebra. Let A be an abelian category . Throughout this sub- section, w e consider the commutativ e diagram A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 j 1 i 1 j 2 p 1 j 3 q 1 i 2 q 2 p 2 q 3 i 3 p 3 (A.5) in A with exact ro ws and exact columns. Here, w e establish technical lemmas concerning this diagram. Lemma A.15. Ther e is an exact se quenc e 0 A 11 Ker( p 3 q 2 ) A 13 ⊕ A 31 0 , ι π (A.6) RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 43 wher e ι is the r estriction of i 2 j 1 : A 11 → A 22 and π is the unique morphism making the fol lowing diagr am c ommutative: A 22 Ker( p 3 q 2 ) A 22 A 23 A 13 A 13 ⊕ A 31 A 31 A 32 p 2 π q 2 j 3 i 3 Pr o of. W e use Mitchell’s embedding theorem to treat ob jects of A as if they were ab elian groups. W e ﬁrst deﬁne π . Since the diagram ( A.5 ) comm utes, we hav e q 3 p 2 = p 3 q 2 : A 22 → A 33 . Th us, for z ∈ Ker( p 3 q 2 ) w e hav e q 3 p 2 ( z ) = 0, so p 2 ( z ) ∈ Ker( q 3 ) = Im( j 3 ). Since j 3 is monic, there is a unique morphism π 13 : Ker( p 3 q 2 ) → A 13 suc h that j 3 ◦ π 13 = p 2 | Ker( p 3 q 2 ) . Similarly , if z ∈ Ker( p 3 q 2 ) then p 3 q 2 ( z ) = 0, so q 2 ( z ) ∈ Ker( p 3 ) = Im( i 3 ). Since i 3 is monic, there is a unique morphism π 31 : Ker( p 3 q 2 ) → A 31 suc h that i 3 ◦ π 31 = q 2 | Ker( p 3 q 2 ) . W e set π = ( π 13 , π 31 ) : Ker( p 3 q 2 ) → A 13 ⊕ A 31 . W e prov e that the sequence ( A.6 ) is exact. Injectivit y of ι is clear since i 2 j 1 is monic. Moreo v er, π ◦ ι = 0 b ecause p 2 i 2 = 0 and q 2 i 2 = i 3 q 1 . Th us Im( ι ) ⊆ Ker( π ). T o show that π is epic, let ( x, y ) ∈ A 13 ⊕ A 31 . Cho ose z 0 ∈ A 22 with q 2 ( z 0 ) = i 3 ( y ) (p ossible since q 2 is epic). Then p 3 q 2 ( z 0 ) = 0, hence z 0 ∈ Ker( p 3 q 2 ). Since q 3 p 2 ( z 0 ) = p 3 q 2 ( z 0 ) = 0, w e ha v e p 2 ( z 0 ) ∈ Ker( q 3 ) = Im( j 3 ), so there is x 0 ∈ A 13 with p 2 ( z 0 ) = j 3 ( x 0 ). Let d := x 0 − x ∈ A 13 . Cho ose w ∈ A 12 with p 1 ( w ) = d (p ossible since p 1 is epic). Then q 2 ( z 0 − j 2 ( w )) = q 2 ( z 0 ) and p 2 ( z 0 − j 2 ( w )) = p 2 ( z 0 ) − p 2 j 2 ( w ) = j 3 ( x 0 ) − j 3 p 1 ( w ) = j 3 ( x ) . Th us z := z 0 − j 2 ( w ) lies in Ker( p 3 q 2 ) and satisﬁes π ( z ) = ( x, y ). So π is epic. Finally , let z ∈ Ker( π ). Then p 2 ( z ) = j 3 π 13 ( z ) = 0 and q 2 ( z ) = i 3 π 31 ( z ) = 0. Hence z ∈ Ker( p 2 ) ∩ Ker( q 2 ) = Im( i 2 ) ∩ Im( j 2 ). Choose u ∈ A 21 and v ∈ A 12 with i 2 ( u ) = j 2 ( v ) = z . Applying q 2 and using comm utativity gives 0 = q 2 ( z ) = q 2 i 2 ( u ) = i 3 q 1 ( u ) , so q 1 ( u ) = 0 since i 3 is monic. Thus u ∈ Ker( q 1 ) = Im( j 1 ), so u = j 1 ( a ) for some a ∈ A 11 . Then z = i 2 ( u ) = i 2 j 1 ( a ) = j 2 i 1 ( a ) , and since j 2 is monic, w e get v = i 1 ( a ). Therefore z = ι ( a ), i.e. Ker( π ) = Im( ι ). □ By the additivit y of Ext 1 , w e hav e an isomorphism Ext 1 ( A 13 ⊕ A 31 , A 11 ) ∼ = Ext 1 ( A 13 , A 11 ) ⊕ Ext 1 ( A 31 , A 11 ) (A.7) of ab elian groups. The exact sequence ( A.6 ) deﬁnes an elemen t of the left hand side of ( A.7 ). The elemen t of the right hand side of ( A.7 ) corresp onding to ( A.6 ) is describ ed as follows: Lemma A.16. L et α 1 and β 1 b e the elements of the Ext gr oup c orr esp onding to the ﬁrst r ow and the ﬁrst c olumn of ( A.5 ) , r esp e ctively. Then the element r epr esente d by the exact se quenc e ( A.6 ) c orr esp onds to the element ( α 1 , β 1 ) of the right hand side of ( A.7 ) . 44 KENICHI SHIMIZU AND HARSHIT Y AD A V Pr o of. Under the canonical isomorphism Ext 1 ( A 13 ⊕ A 31 , A 11 ) ∼ = Ext 1 ( A 13 , A 11 ) ⊕ Ext 1 ( A 31 , A 11 ), the extension class is sent to the pair of its pullbacks along the inclusions A 13  → A 13 ⊕ A 31 and A 31  → A 13 ⊕ A 31 . Let e b e the extension class represen ted by ( A.6 ). The pullbac k of ( A.6 ) along A 13  → A 13 ⊕ A 31 has middle term Ker( π 31 ), where π 31 is the comp osite Ker( p 3 q 2 ) π − → A 13 ⊕ A 31 → A 31 . By construction of π in Lemma A.15 , we ha v e i 3 ◦ π 31 = q 2 | Ker( p 3 q 2 ) , hence Ker( π 31 ) = Ker( q 2 ) ∼ = A 12 . Moreo v er, the induced map Ker( π 31 ) → A 13 is identiﬁed with p 1 b y the comm utativit y p 2 j 2 = j 3 p 1 and the deﬁning prop erty j 3 ◦ π 13 = p 2 | Ker( p 3 q 2 ) . Th us this pullbac k extension is isomorphic to the ﬁrst ro w of ( A.5 ), hence equals α 1 . Similarly , the pullback along A 31  → A 13 ⊕ A 31 has middle term Ker( π 13 ). Since j 3 ◦ π 13 = p 2 | Ker( p 3 q 2 ) , w e hav e Ker( π 13 ) = Ker( p 2 ) ∼ = A 21 . The induced map Ker( π 13 ) → A 31 is identiﬁed with q 1 using comm utativit y q 2 i 2 = i 3 q 1 and the deﬁning prop ert y i 3 ◦ π 31 = q 2 | Ker( p 3 q 2 ) . Hence this pullbac k extension is isomorphic to the ﬁrst column of ( A.5 ), so it equals β 1 . Therefore e corresp onds to ( α 1 , β 1 ). □ In the opp osite category A op , the diagram ( A.5 ) lo oks like: A 33 A 32 A 31 A 23 A 22 A 21 A 13 A 12 A 11 . p 3 q 3 p 2 i 3 p 1 j 3 q 2 j 2 i 2 j 1 q 1 i 1 W e apply the ab ov e lemmas to this diagram and then interpret the result in A to obtain: Lemma A.17. Ther e is an exact se quenc e 0 A 13 ⊕ A 31 Cok er( i 2 j 1 ) A 33 0 , ι π (A.8) wher e π is induc e d by p 3 q 2 : A 22 → A 33 and ι is the unique morphism making the fol lowing diagr am c ommutative: A 12 A 13 A 13 ⊕ A 31 A 31 A 21 A 22 Cok er( i 2 j 1 ) A 22 p 1 j 2 ι q 1 i 2 Under the c anonic al isomorphism Ext 1 ( A 33 , A 13 ⊕ A 31 ) ∼ = Ext 1 ( A 33 , A 13 ) ⊕ Ext 1 ( A 33 , A 31 ) (A.9) of ab elian gr oups, the element of the left hand side r epr esente d by ( A.8 ) c orr esp onds to the element ( α 3 , β 3 ) of the right hand side, wher e α 3 and β 3 ar e elements r epr esente d by the thir d c olumn and the thir d r ow of ( A.5 ) , r esp e ctively. F or the rest of this section, w e ﬁx a short exact sequence in an ab elian monoidal category C with enough left ﬂat and enough right ﬂat ob jects: 0 Y Z X 0 . i p (A.10) RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 45 A.6. Pro of of Lemma A.1 (1). W e give a pro of of Lemma A.1 (1). Assume that X and Y are righ t ﬂat and left rigid. Lemma A.5 implies that X and Y are also left ﬂat. A.6.1. Construction of the dual obje ct. W e ﬁrst consider the following diagram: Ext 1 ( X , Y ) Ext 1 ( X ⊗ X ∗ , Y ⊗ X ∗ ) Ext 1 ( 1 , Y ⊗ X ∗ ) Ext 1 ( Y ∗ ⊗ X , Y ∗ ⊗ Y ) Ext 1 ( Y ∗ ⊗ X ⊗ X ∗ , Y ∗ ⊗ Y ⊗ X ∗ ) Ext 1 ( Y ∗ , Y ∗ ⊗ Y ⊗ X ∗ ) Ext 1 ( Y ∗ ⊗ X , 1 ) Ext 1 ( Y ∗ ⊗ X ⊗ X ∗ , X ∗ ) Ext 1 ( Y ∗ , X ∗ ) , ^ −⊗ X ∗ ^ Y ∗ ⊗− (coev X ) ⋆ ^ Y ∗ ⊗− ^ Y ∗ ⊗− ^ −⊗ X ∗ (ev Y ) ⋆ (id ⊗ coev X ) ⋆ (ev Y ⊗ id) ⋆ (ev Y ⊗ id) ⋆ ^ −⊗ X ∗ (id ⊗ coev X ) ⋆ where ^ − ⊗ X ∗ and ^ Y ∗ ⊗ − are the maps induced by the functors − ⊗ X ∗ and Y ∗ ⊗ − , respec- tiv ely . Since X ∗ is righ t ﬂat by Lemma A.5 , the maps ^ − ⊗ X ∗ in the diagram are w ell-deﬁned homomorphisms of ab elian groups. The ob jects X , X ⊗ X ∗ and 1 are right ﬂat. Thus, by Lemma A.13 , the maps ^ Y ∗ ⊗ − in the diagram are also well-deﬁned homomorphisms of abelian groups. The top-left square comm utes since the functors − ⊗ X ∗ and Y ∗ ⊗ − commute. By the basic prop ert y of the Y oneda Ext, the b ottom-right square also commutes. The remaining squares also comm ute by Lemma A.11 . The rows and the columns of the ab o v e diagram are bijections induced b y adjunctions discussed in Lemma A.14 . Hence the ab o v e diagram shrinks to the follo wing comm utative diagram of isomorphisms of ab elian groups: Ext 1 ( X , Y ) Ext 1 ( 1 , Y ⊗ X ∗ ) Ext 1 ( Y ∗ ⊗ X , 1 ) Ext 1 ( Y ∗ , X ∗ ) ∼ = ∼ = ∼ = ∼ = (A.11) W e denote b y α ∈ Ext 1 ( X , Y ) the element represented by ( A.10 ), and let β ∈ Ext 1 ( Y ∗ , X ∗ ) b e the elemen t corresp onding to − α . W e choose an exact sequence 0 X ∗ Z ∨ Y ∗ 0 j q (A.12) represen ting β . W e will show that the middle term, Z ∨ , is a left dual ob ject of Z . A.6.2. Construction of the c o evaluation. By taking the tensor pro duct of exact sequences ( A.10 ) and ( A.12 ), w e obtain the following commutativ e diagram: Y ⊗ X ∗ Z ⊗ X ∗ X ⊗ X ∗ Y ⊗ Z ∨ Z ⊗ Z ∨ X ⊗ Z ∨ Y ⊗ Y ∗ Z ⊗ Y ∗ X ⊗ Y ∗ i ⊗ id id ⊗ j p ⊗ id id ⊗ j id ⊗ j i ⊗ id id ⊗ q p ⊗ id id ⊗ q id ⊗ q i ⊗ id p ⊗ id (A.13) Since X and Y are left ﬂat, Z is also left ﬂat by Lemma A.8 . Th us all columns of this diagram are exact. Since X ∗ and Y ∗ are right ﬂat, Z ∨ is also right ﬂat b y Lemma A.8 , and th us all rows of this diagram are exact. Hence, by Lemma A.15 , we obtain an exact sequence 0 X ⊗ Y ∗ Ker( p ⊗ q ) ( X ⊗ X ∗ ) ⊕ ( Y ⊗ Y ∗ ) 0 , i ⊗ j π (A.14) 46 KENICHI SHIMIZU AND HARSHIT Y AD A V where π is a morphism in C such that the following diagram is commutativ e: Z ⊗ Z ∨ Ker( p ⊗ q ) Z ⊗ Z ∨ X ⊗ Z ∨ X ⊗ X ∗ ( X ⊗ X ∗ ) ⊕ ( Y ⊗ Y ∗ ) Y ⊗ Y ∗ Z ⊗ Y ∗ , p ⊗ id π id ⊗ q id ⊗ j pr X pr Y i ⊗ id (A.15) where pr X and pr Y are resp ective pro jections. W e compute the pullback of the exact sequence ( A.14 ) along the morphism co ev X,Y := (co ev X ⊕ co ev Y ) ◦ diag 1 . By Lemma A.10 and the naturalit y of the Baer sum, we hav e the following comm utativ e diagram: Ext 1 ( X ⊗ X ∗ , X ⊗ Y ∗ ) ⊕ Ext 1 ( Y ⊗ Y ∗ , X ⊗ Y ∗ ) Ext 1 (( X ⊗ X ∗ ) ⊕ ( Y ⊗ Y ∗ ) , X ⊗ Y ∗ ) Ext 1 ( 1 , X ⊗ Y ∗ ) ⊕ Ext 1 ( 1 , X ⊗ Y ∗ ) Ext 1 ( 1 , X ⊗ Y ∗ ) ∼ = (coev X ) ⋆ ⊕ (coev Y ) ⋆ (coev X,Y ) ⋆ the Baer sum By Lemma A.16 , the elemen t ( A.14 ) of the upper righ t corner corresp onds to the elemen t ( α ⊗ X ∗ , Y ⊗ β ) of the upp er left corner. Thus, by the commutativit y of the diagram, w e ha v e (co ev X,Y ) ⋆ ( A.14 ) = (co ev X ) ⋆ ( α ⊗ X ∗ ) + (coev Y ) ⋆ ( Y ⊗ β ) in Ext 1 ( 1 , X ⊗ Y ∗ ) The ﬁrst and the second term of the righ t hand side corresp ond to α ∈ Ext 1 ( X , Y ) and β ∈ Ext 1 ( Y ∗ , X ∗ ) in the diagram ( A.11 ), resp ectively . Since β corresp onds to − α ∈ Ext 1 ( X , Y ) in the diagram ( A.11 ) by deﬁnition, the right hand side of the ab o v e equation is in fact zero. This means that the pullback of ( A.14 ) along co ev X,Y splits. Summarizing our discussion so far, w e hav e the comm utativ e diagram 0 X ⊗ Y ∗ Ker( p ⊗ q ) ( X ⊗ X ∗ ) ⊕ ( Y ⊗ Y ∗ ) 0 0 X ⊗ Y ∗ E 1 0 i ⊗ j π r u (PB) coev X,Y s (A.16) with exact ro ws, where (PB) is a pullback and s is a section of r . W e deﬁne η Z : 1 s − → E u − → Ker( p ⊗ q )  → Z ⊗ Z ∨ , (A.17) whic h, in fact, b ecomes the co ev aluation after suitable mo diﬁcation. Before pro ceeding further, w e remark that the morphism η Z mak es the following diagram commute: Y ⊗ Y ∗ 1 X ⊗ X ∗ Z ⊗ Y ∗ Z ⊗ Z ∨ X ⊗ Z ∨ i ⊗ id coev Y coev X η Z id ⊗ j id ⊗ q p ⊗ id (A.18) Indeed, b y the commutativ e diagrams ( A.15 ) and ( A.16 ), w e ha ve (id Z ⊗ q ) ◦ η Z = ( i ⊗ id Y ∗ ) ◦ pr Y ◦ π ◦ u ◦ s = ( i ⊗ id Y ∗ ) ◦ pr Y ◦ co ev X,Y = ( i ⊗ id Y ∗ ) ◦ coev Y and, in a similar w ay , ( p ⊗ id Z ∨ ) ◦ η Z = (id X ⊗ j ) ◦ co ev X . RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 47 A.6.3. Construction of the evaluation. By taking the tensor pro duct of exact sequences ( A.10 ) and ( A.12 ) in the diﬀeren t order as b efore, we obtain the following commutativ e diagram: X ∗ ⊗ Y X ∗ ⊗ Z X ∗ ⊗ X Z ∨ ⊗ Y Z ∨ ⊗ Z Z ∨ ⊗ X Y ∗ ⊗ Y Y ∗ ⊗ Z Y ∗ ⊗ X id ⊗ i j ⊗ id id ⊗ p j ⊗ id j ⊗ id id ⊗ i q ⊗ id id ⊗ p q ⊗ id q ⊗ id id ⊗ i id ⊗ p (A.19) By Lemma A.8 and the ﬂatness of X and Y , the columns of the diagram ( A.19 ) are exact. By Lemma A.7 applied to A = X ∗ , Z ∨ , Y ∗ , w e see that the rows of ( A.19 ) are exact. W e compute the pullback of the exact sequence 0 → ( X ∗ ⊗ X ) ⊕ ( Y ∗ ⊗ Y ) → Cok er( j ⊗ i ) → Y ∗ ⊗ X → 0 obtained by applying Lemma A.17 to the diagram ( A.19 ) along the morphism ev X,Y := sum 1 ◦ (ev X ⊕ ev Y ). By an argumen t dual to one when w e ha v e obtained the comm utativ e diagram ( A.16 ), w e obtain the following commutativ e diagram with exact ro ws: 0 ( X ∗ ⊗ X ) ⊕ ( Y ∗ ⊗ Y ) Cok er( j ⊗ i ) Y ∗ ⊗ X 0 0 1 E ′ Y ∗ ⊗ X 0 ι ev X,Y u ′ (PO) r ′ s ′ Here, s ′ is a section of r ′ . Now we set ε Z : Z ∨ ⊗ Z ↠ Cok er( j ⊗ i ) u ′ − → E ′ s ′ − → 1 . (A.20) Again b y the dual argument, we see that the following diagram is commutativ e: Z ∨ ⊗ Y Z ∨ ⊗ Z X ∗ ⊗ Z Y ∗ ⊗ Y 1 X ∗ ⊗ X id ⊗ i q ⊗ id ε Z j ⊗ id id ⊗ p ev Y ev X (A.21) A.6.4. V erifying the zig-zag r elation. W e ha v e deﬁned the morphisms η Z : 1 → Z ⊗ Z ∨ and ε Z : Z ∨ ⊗ Z → 1 b y ( A.17 ) and ( A.20 ), resp ectiv ely . W e set ξ = ( ε Z ⊗ id Z ∨ )(id Z ∨ ⊗ η Z ) and ζ = (id Z ⊗ ε Z )( η Z ⊗ id Z ) , and aim to show that ξ and ζ are inv ertible. T o show that ξ is in vertible, w e consider the diagram of Figure 4 . The cells in the diagram lab eled ‘(nat.)’ is comm utativ e by the naturalit y of the monoidal pro duct. Thus the diagram is comm utativ e and it shrinks to the following comm utative diagram with exact ro ws: 0 X ∗ Z ∨ Y 0 0 X ∗ Z ∨ Y 0 j q ξ j q 48 KENICHI SHIMIZU AND HARSHIT Y AD A V X ∗ X ∗ ⊗ X ⊗ X ∗ X ∗ X ∗ ⊗ Z ⊗ Z ∨ X ∗ ⊗ X ⊗ Z ∨ Z ∨ Z ∨ ⊗ Z ⊗ Z ∨ Z ∨ Z ∨ ⊗ Y ⊗ Y ∗ Z ∨ ⊗ Z ⊗ Y ∗ Y ∗ Y ∗ ⊗ Y ⊗ Y ∗ Y ∗ j id ⊗ coev X id ⊗ η Z ( A.18 ) ev X ⊗ id id ⊗ id ⊗ j j (nat.) id ⊗ p ⊗ id j ⊗ id ⊗ id (nat.) ( A.21 ) ev X ⊗ id id ⊗ η Z id ⊗ coev Y q ε Z ⊗ id id ⊗ id ⊗ q ( A.18 ) q (nat.) id ⊗ i ⊗ id q ⊗ id ⊗ id ε Z ⊗ id (nat.) ( A.21 ) id ⊗ coev Y ev Y ⊗ id Figure 4. Pro of of the zig-zag equation By the ﬁv e lemma, ξ is inv ertible. A similar argument sho ws that ζ is also inv ertible (the detail of this part is written in [EP26]). Hence, by Lemma A.3 , Z is left rigid. The pro of is done. A.7. Pro of of Lemma A.1 (2). W e giv e a pro of of Lemma A.1 (2). Assume that X and Z are left rigid. W e deﬁne Y ∨ = Coker( p ∗ : X ∗ → Z ∗ ) and denote b y π : Z ∗ → Y ∨ the quotien t morphism. Since X is left ﬂat, we hav e an exact sequence 0 Y ⊗ Y ∨ Z ⊗ Y ∨ X ⊗ Y ∨ 0 i ⊗ id p ⊗ id b y Lemma A.7 . W e set ξ = (id Z ⊗ π )coev Z . Since π p ∗ = 0, w e hav e ( p ⊗ id Y ∨ ) ξ = ( p ⊗ π )coev Z = (id X ⊗ π p ∗ )co ev X = 0 . Th us ξ factors through Ker( p ⊗ id Y ∨ ), which is equal to Im( i ⊗ id Y ∨ ). Since i ⊗ id Y ∨ is monic, there exists a morphism η Y : 1 → Y ⊗ Y ∨ in C suc h that ( i ⊗ id Y ∨ ) ◦ η Y = (id Z ⊗ π ) ◦ co ev Z . (A.22) There is also an exact sequence X ∗ ⊗ Y Z ∗ ⊗ Y Y ∨ ⊗ Y 0 . p ∗ ⊗ id π ⊗ id W e set ζ = ev Z (id Z ∗ ⊗ i ). Since ζ ( p ∗ ⊗ id Y ) = ev X (id X ∗ ⊗ p ◦ i ) = 0, the kernel of ζ con tains the image of p ∗ ⊗ id Y , which is equal to the kernel of π ⊗ id Y . Hence there exists a morphism ε Y : Y ∨ ⊗ Y → 1 in C suc h that ε Y ◦ ( π ⊗ id Y ) = ev Z ◦ (id Z ∗ ⊗ i ) . (A.23) By ( A.22 ), ( A.23 ) and Lemma A.4 , we hav e i ◦ (id Y ⊗ ε Y ) ◦ ( η Y ⊗ id Y ) = i, ( ε Y ⊗ id Y ∨ ) ◦ (id Y ∨ ⊗ η Y ) ◦ π = π . Since i is monic and π is epic, we conclude that ( Y ∨ , ε Y , η Y ) is a left dual ob ject of Y . The pro of is done. RIBBON CA TEGORIES FROM IND-EXACT ALGEBRAS: SIMPLE CURRENT CASE 49 A.8. Pro of of Lemma A.1 (3). W e giv e a pro of of Lemma A.1 (3). Assume that Y and Z are left rigid, and that i ∗ : Z ∗ → Y ∗ is epic in C . W e deﬁne X ∨ = Ker( i ∗ ) and denote b y ι : X ∨ → Z ∗ the inclusion morphism. Since i ∗ is epic, w e hav e an exact sequence 0 → X ∨ ι − → Z ∗ i ∗ − → Y ∗ → 0. W e note that Y ∗ is right ﬂat since it has a righ t dual ob ject Y . Thus, by Lemma A.7 , w e ha ve the following exact sequence: 0 X ⊗ X ∨ X ⊗ Z ∗ X ⊗ Y ∗ 0 . id X ⊗ ι id X ⊗ i ∗ Since (id X ⊗ i ∗ )( p ⊗ id Z ∗ )co ev Z = 0, the morphism ( p ⊗ id Z ∗ )co ev Z factors through the kernel of id X ⊗ i ∗ , which is equal to the image of id X ⊗ ι . Since id X ⊗ ι is monic, we hav e a morphism η X : 1 → X ⊗ X ∨ suc h that (id X ⊗ ι ) ◦ η X = ( p ⊗ id Z ∗ ) ◦ coev Z . (A.24) There is also an exact sequence X ∨ ⊗ Y → X ∨ ⊗ Z → X ∨ ⊗ X → 0. By the same argumen t as the pro of of Part (2), we hav e a morphism ε X : X ∨ ⊗ X → 1 in C satisfying the follo wing equation: ε X ◦ (id X ∨ ⊗ p ) = ev Z ◦ ( ι ⊗ id Z ) . (A.25) By ( A.24 ), ( A.25 ) and Lemma A.4 , we conclude that ( X ∨ , ε X , η X ) is a left dual ob ject of X . The pro of is done. References [A CE15] Nicol´ as Andruskiewitsch, Juan Cuadra Carre ˜ no, and Pa vel Etingof. On tw o ﬁniteness conditions for Hopf algebras with nonzero in tegral. A nnali del la Scuola Normale Sup erior e di Pisa. 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Dep ar tment of Ma thema tical Sciences, Shiba ura Institute of Technology, 307 Fukasaku, Minuma- ku, Sait ama-shi, Sait ama 337-8570, Jap an Email address : kshimizu@shibaura-it.ac.jp Dep ar tment of Ma thema tics, University of Alber t a, Edmonton, AB, Canada T6G 2G1 Email address : hyadav3@ualberta.ca

Ribbon categories from ind-exact algebras: simple current case

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