A Categorical Approach to Groupoid Frobenius Algebras

A CA TEGORICAL APPR O ACH TO GR OUPOID FR OBENIUS ALGEBRAS DA VID N. PHAM Abstra ct. In this pap er, w e sho w that G -F r ob enius algebras (for G a finite groupoid) correspond to a p articular class of F robenius ob jects in the represen tation category of D ( k [ G ]), where D ( k [ G ]) i s the Drinfeld double of t he quantum group oid k [ G ] [11]. 1. Introduction Group oid F rob enius algebras w ere in tro duced recen tly in [14] as a group oid v ersion of (non-pro jectiv e) G -F rob enius algebras ( G -F As) for G a fin ite group [15] [9 ]. As sho wn 1 in [1 5], G -F As are the a lgebraic structures whic h cla s- sify certain homotop y quan tum field theories (HQFTs). Ro ughly sp eaking, a ( d + 1)-dimensional HQFT is a top ologica l quan tum fi eld theory [1] for d -dimensional manifolds and ( d + 1)-dimensional co b ordisms endow ed with homotop y classes of maps in to a given space X . In the case when X is an Eilenberg-MacLane space of t yp e K ( G, 1), one fin ds that the asso ciated (1 + 1)-dimensional HQFTs are c lassified b y G - F As [15 ]. The author’s o riginal motiv atio n for generaliz ing G -F As to G -F As for G a fi nite group oid w as the app earance of certain “at ypical” G -F As in [10], whic h w ere co nstr ucted w ithin the framewo rk of stringy orbifold theory (cf [6] [7] [5]). T o get a basic idea of th is construction, let M b e a compact, almost complex m anifold with an action b y a finite group G wh ic h p reserv es the almost complex s tructure. Let I ( M ) denote the inertia manifold of M , that is, I ( M ) := G g ∈ G M g , (1) where M g := { m ∈ M | g · m = m } . Let H ( M , G ) := M g ∈ G H ev ( M g ) , (2) where H ev ( M g ) denotes the even part of the ordinary cohomology of M g with ratio nal co efficien ts. T hen H ( M , G ) can b e endo we d with a G -graded pro du ct, a G -action, an d a G -in v aria nt bilinear form whic h turns H ( M , G ) in to a G -F A; H ( M , G ) together with the aforemen tioned G -graded pro d u ct 1 F or an alternate approach to G -F As, see [9]. 1 2 DA VID N. PHAM is called the stringy c ohomolo gy ring of the G -manifold M . T o see ho w G - F As a rise from all this, w e lo ok to the inertial m anfiold I ( M ) which has a natural G -action giv en by h · ( g , m ) := ( hg h − 1 , hm ) . (3) If one tak es th e stringy cohomology of I ( M ) with its natural G -acti on, one obtains a G -F A with some additional structure; this additional struc- ture is p recisely that of a group oid F rob eniu s algebra. More sp ecifically , H ( I ( M ) , G ) tu rns out to b e a ∧ G -F A, wh er e ∧ G is th e lo op gr oup oid of the one ob ject group oid asso ciated with G . The existence of these at ypical G -F As motiv ated th e view that G -F As are actually a sp ecial case of some larger algebraic stru cture. Ultimately , it w as the transition from group to group oid that resulted in a framework that wa s capable of accommodating th ese at ypical G -F As. As it turned out, these early motiv ating examples were just the tip of the iceb erg. It was sho wn in [14] that by wo rking within th e G -F A f ramew ork, one could constru ct a tow er of increasingly complex G -F As, w here eac h G -F A in the to wer is deriv ed from some group oid F rob enius a lgebra. In addition to this, G -F As could a lso b e u sed to gain new insigh t on the problem of t wisting ordin ary G -F As. It w as sho wn in [8 ] that eve ry G -F A h as a twist b y an y elemen t of Z 2 ( G, k × ). Since these t wists apply to all G -F As, one can regard them as “univ ersal” G -F A twists. In an analogous manner, G -F As ha ve their own unive rsal t wists where the t wisting is no w by the elemen ts of Z 2 ( G , k × ) [14]. When one com bines this p oin t with the aforemen tioned to w er of “ G -F A i n- duced” G -F As, one obtains a significan t generaliza tion o f the G -F A t wisting result from [8]. Sp ecifically , for ev ery n ≥ 2, one ca n a lwa ys find a class of G -F As with t wists b y any elemen t in Z n ( G, k × ) [14]. While [14] illustrates the u tilit y of G -F As in add ressing and solving th ese problems, littl e w as done in [ 14] to motiv ate the choice of axio ms for a G - F A. The only motiv ation for the axioms came in the f orm of a short remark 2 whic h asserted that G -F As migh t actually corresp ond to certain kinds of F rob enius ob jects in Rep( D ( k [ G ])) (the represen tation categ ory of D ( k [ G ])), where D ( k [ G ]) is the Drinfeld double of the quan tum groupoid (w eak Hopf algebra) k [ G ] [11]. T his assertion is motiv ated by a recen t catego rical re- sult for G -F As [10] wh ic h sho we d that G -F As corresp ond to certain kinds of F rob enius o b j ects in Rep( D ( k [ G ])), where D ( k [ G ]) is the original Drinfeld double of the Hopf a lgebra k [ G ] [4]. Consequen tly , if the assertion pro v es true, the G -F A axioms o f [14 ] w ould e ssentia lly b e a consequence of gener- alizing D ( k [ G ]) to D ( k [ G ]). In other wo rd s, from this catego rical v an tage p oint , th e notion of a G -F A is a natural generalization of a G -F A for the case when G is replaced b y G . With the cu rren t pap er, w e sho w that the assertion of [14] is in deed true. 2 More sp ecifically , Remark 3.1 of [14]. A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 3 The rest of the pap er is organized as follo ws. In section 2, we giv e a brief review o f quantum group oids [2] [3 ] [13] and their represen tation category [11]. In section 3, we pro ve the assertion raised in [14]. W e conclude the pap er in sectio n 4 with some op en questions. 2. Preliminaries Throughout this pap er, we use th e follo wing notation. k is a field of c haracteristic 0. G = ( G 0 , G 1 ) d en otes a fi nite group oid whose set of ob jects is G 0 and whose set of morphisms is G 1 . The source and target maps from G 1 to G 0 are denoted a s s and t resp ectiv ely . F or x ∈ G 0 , e x denotes the iden tit y morphism asso ciated to x. F or x ∈ G 0 , Γ x is the group consisting of all g ∈ G 1 with s ( g ) = t ( g ) = x. 2.1. Quan tum Group oids. Quant um group oids or w eak Hopf algebras [2] [3] [13] generalize the notion of ordinary Hopf algebras by wea ke nin g the axioms concerning the copro du ct and counit in the follo wing w a y: 1. the copro d u ct is n ot necessarily unit-preserving; 2. the counit is not necessarily m ultiplicativ e. F ormally , a quantum group oid is defined as foll ows: Definition 2.1. A quantum group oid ov er a field k is a tuple ( H , · , 1 , ∆ , ε, S ) where (i) H is a finite d imensional un ital asso ciativ e algebra o v er k with pr o d- uct · and unit 1. (ii) H is a fin ite dimensional counital coasso ciativ e algebra o ver k with copro duct ∆ : H → H ⊗ k H and co un it ε : H → k . (iii) The algebra and coalgebra structure of H satisfy the follo wing com- patibilit y conditions. (a) Multiplicativit y of the copro du ct: for all x, y ∈ H , ∆( x · y ) = ∆( x ) · ∆( y ) (b) W eak m ultiplicativit y of the counit: for all x , y , z ∈ H , ε ( x · y · z ) = ε ( x · y (1) ) ε ( y (2) · z ) ε ( x · y · z ) = ε ( x · y (2) ) ε ( y (1) · z ) (c) W eak com ultiplicativit y of the unit: (∆ ⊗ id H ) ◦ ∆(1) = (∆(1) ⊗ 1) · (1 ⊗ ∆(1)) (∆ ⊗ id H ) ◦ ∆(1) = (1 ⊗ ∆(1)) · (∆(1) ⊗ 1) (iv) S : H → H is a k -linear map called the an tip o de whic h satisife s the follo w ing for all x ∈ H : 4 DA VID N. PHAM (a) x (1) · S ( x (2) ) = ε (1 (1) · x )1 (2) (b) S ( x (1) ) · x (2) = 1 (1) ε ( x · 1 (2) ) (c) S ( x (1) ) · x (2) · S ( x (3) ) = S ( x ) Remark 2.2. In Defin ition 2.1, Sweedler notation w as applied so that ∆( a ) is written as ∆( a ) = a (1) ⊗ a (2) . Remark 2.3. Its a straigh tforwa rd exercise to sho w the follo wing: 1. Ev ery Ho pf algebra is a quantum group oid. 2. F or a qu an tum group oid H , the follo win g statemen ts are equiv alen t: (i) H is a Hopf algebra (ii) ∆ (1) = 1 ⊗ 1 (iii) ε ( x · y ) = ε ( x ) ε ( y ) for all x, y ∈ H Example 2.4. Any finite g roup oid G defines a quan tum group oid ( k [ G ] , · , 1 , ∆ , ε, S ) where 1. k [ G ] := L g ∈G 1 k g as a v ector space o v er k 2. · is the multiplicat ion on k [ G ] in duced b y the c omp osition of mor- phisms, that is, for g , h ∈ G 1 , g · h = g h if s ( g ) = t ( h ) and g · h = 0 if s ( g ) 6 = t ( h ) 3. 1 := P x ∈G 0 e x 4. ∆ : k [ G ] → k [ G ] ⊗ k k [ G ] is the k -linear map indu ced by g 7→ g ⊗ g for all g ∈ G 1 5. ε : k [ G ] → k is the k -linear map indu ced b y g 7→ 1 k for al l g ∈ G 1 where 1 k is the unit elemen t of k 6. S : k [ G ] → k [ G ] is the k -linear map ind uced by g 7→ g − 1 for all g ∈ G 1 . W e conclude this section b y recalling a few things ab out quasitriangular quan tum group oids [11]; w e b egin with its defin ition. Definition 2.5. A quasitriangular quan tum group oid is a tuple ( H, · , 1 , ∆ , ε, S, R ) where (i) ( H , · , 1 , ∆ , ε, S ) i s a quantum group oid, an d (ii) R ∈ ∆ op (1)( H ⊗ k H )∆(1) satisfies the follo wing conditions for all h ∈ H : ∆ op ( h ) R = R ∆( h ) (4) ( id H ⊗ ∆)( R ) = R 13 · R 12 (5) (∆ ⊗ id H )( R ) = R 13 · R 23 (6) where ∆ op is the opp osite copro duct, R 12 = R ⊗ 1, R 23 = 1 ⊗ R , and R 13 = R (1) ⊗ 1 ⊗ R (2) . In add ition, there exists R ∈ ∆ (1)( H ⊗ k A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 5 H )∆ op (1) suc h that R · R = ∆ op (1) (7) R · R = ∆(1) . (8) Remark 2.6. In Definition 2.5, the R-matrix R w as written as R = R (1) ⊗ R (2) to simp lify notation. A Drinfeld double construction w as introd uced in [11] for generating qu a- sitriangular quan tum group oids fr om existing qu an tum group oids. When this construction is app lied to the qu antum group oid k [ G ], the result is the quasitriangular quantum group oid D ( k [ G ]) which is defin ed as follo ws: 1. As a v ector space o ver k , D ( k [ G ]) has basis { γ x g | g , x ∈ G 1 , s ( g ) = t ( g ) = t ( x ) } . (9) 2. F or γ x g , γ y h ∈ D ( k [ G ]), the m ultiplication la w is giv en b y γ x g · γ y h := δ x − 1 g x,h γ xy g . (10) (Note that when x − 1 g x = h , xy is defin ed since s ( x ) = s ( h ) = t ( y )). 3. The unit of D ( k [ G ]) is 1 = X x ∈G 0 1 x (11) where 1 x := X g ∈ Γ x γ e x g . (12) 4. The copro du ct of D ( k [ G ]) is defin ed as △ D ( γ x g ) := X { g 1 ,g 2 ∈ Γ t ( x ) | g 1 g 2 = g } γ x g 1 ⊗ γ x g 2 (13) 5. The counit of D ( k [ G ]) is defin ed as ε D ( γ x g ) = δ g ,xx − 1 . (14) 6. The ant ip o de of D ( k [ G ]) is d efined a s S ( γ x g ) = γ x − 1 x − 1 g − 1 x (15) 7. The R-matrix is R := X x ∈G 0 R x (16) where R x := X g ,h ∈ Γ x γ e x g ⊗ γ g h . (17) 6 DA VID N. PHAM Remark 2.7. Note that un less G has a single ob ject, ∆ D do es not p reserv e the un it since ∆ D (1 x ) = X g ∈ Γ x ∆ D ( γ e x g ) = X g ∈ Γ x X { g 1 ,g 2 ∈ Γ x | g 1 g 2 = g } γ e x g 1 ⊗ γ e x g 2 = 1 x ⊗ 1 x (18) and ∆ D (1) = X x ∈G 0 ∆ D (1 x ) = X x ∈G 0 1 x ⊗ 1 x 6 = X x , y ∈G 0 1 x ⊗ 1 y = 1 ⊗ 1 . (19) So b y Remark 2 .3, D ( k [ G ]) is a Hopf algebra only when G is a one-ob ject group oid (i.e., a group). In the case when G is the one-ob ject group oid whose set of morph isms is the group G , D ( k [ G ]) is exact ly D ( k [ G ]), the ordinary Drinfeld d ouble of the Hopf algebra k [ G ]. 2.2. Quan tum Group oids & Category Theory. It was sho wn in [12] that for a quantum group oid H , Rep( H ) 3 is a monoidal category . T o d efine the monoidal pro duct, let ( ρ 1 , A 1 ) and ( ρ 2 , A 2 ) b e ob jects of Rep( H ). Then ( ρ 1 , A 1 ) ⊗ ( ρ 2 , A 2 ) := ( ρ 12 , A 1 b ⊗ A 2 ) (20) where the H -action ρ 12 is indu ced by the copr o duct ∆ of H via ρ 12 ( h ) := [ ρ 1 ⊗ ρ 2 ] ◦ ∆( h ) , h ∈ H (21) and A 1 b ⊗ A 2 := { a ∈ A 1 ⊗ k A 2 | ρ 12 (1) a = a } = ρ 12 (1)( A 1 ⊗ k A 2 ) = [ ρ 1 (1 (1) ) ⊗ ρ 2 (1 (2) )] ( A 1 ⊗ k A 2 ) (22) where the second equalit y f ollo ws from the fact that ∆(1) · ∆(1) = ∆(1). The monoidal p r o duct of morp hisms is simp ly the restric tion of the usual tensor pr o duct of linea r maps. F or the unit ob ject, le t ε t : H → H b e d efined by ε t ( h ) := ε (1 (1) · h )1 (2) (23) where h ∈ H and ε is t he co un it of H . Then the unit ob ject o f Rep ( H ) is I = ( σ t , H t ) w here H t := ε t ( H ) , (24) and for h ∈ H and z ∈ H t , σ t ( h ) z := ε t ( h · z ) . (25) If ( ρ i , A i ) are ob jects of Rep( H ) for i = 1, 2, and 3, th en the asso ciator Φ 123 : ( A 1 b ⊗ A 2 ) b ⊗ A 3 ∼ − → A 1 b ⊗ ( A 2 b ⊗ A 3 ) (26) is the trivial one. 3 Rep( H ) is the category whose ob jects are finite dimensional left H -mo dules and whose morphisms are H -linear maps. A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 7 T o define the left \ righ t unit morph isms, let ( ρ, A ) b e an ob ject of Rep( H ). Then the left morph ism l A : H t b ⊗ A ∼ − → A (27) is defin ed b y l A  σ t (1 (1) ) z ⊗ ρ (1 (2) ) a  := ρ ( z ) a (28) where z ∈ H t and a ∈ A ; the righ t morphism r A : A b ⊗ H t ∼ − → A (29) is defin ed b y r A  ρ (1 (1) ) a ⊗ σ t (1 (2) ) z  := ρ ( S ( z )) a (30) where z ∈ H t , a ∈ A , and S is th e antipo d e of H . If H is also quasitriangular with R-matrix R , then Rep( H ) is also braided [11]. F or an y t wo ob jects ( ρ 1 , A 1 ) and ( ρ 2 , A 2 ) of Rep( H ), the br aiding c A 1 ,A 2 : A 1 b ⊗ A 2 → A 2 b ⊗ A 1 (31) is defin ed b y c A 1 ,A 2 ( x ) := ρ 2 ( R (2) ) x (2) ⊗ ρ 1 ( R (1) ) x (1) (32) where x = x (1) ⊗ x (2) ∈ A 1 b ⊗ A 2 . 2.2.1. F r ob enius O bje cts. Th roughout this section, ( C , ⊗ , I , Φ , l , r, c ) will de- note a b raided monoidal category where C is a small category , ⊗ is th e monoidal pro duct, I is the unit ob ject, Φ is the asso ciator, l and r are the left and righ t iden tit y maps, and c is the braidin g. Definition 2.8 . An alg ebra ob ject is a tup le ( A, m, µ ) where A is an ob ject of C , m : A ⊗ A → A is a morphism o f C c alled the prod uct, and µ : I → A is a morphism of C cal led the u nit whic h satisfy the follo wing t wo conditions: 1. m ◦ ( id A ⊗ m ) ◦ Φ A,A,A = m ◦ ( m ⊗ id A ) (asso ciativit y) 2. m ◦ ( µ ⊗ id A ) = l A , m ◦ ( id A ⊗ µ ) = r A (unit pr op ert y) ( A, m, µ ) is said to b e c ommutati ve if m ◦ c A,A = m . Definition 2.9 . A c oalgebra ob ject is a tuple ( A, ∆ , ε ) where A is an ob ject of C , ∆ : A → A ⊗ A is a morphism o f C ca lled the co pro d uct, and ε : A → I is a morphism of C called the counit whic h satisfy the follo wing t wo conditions: 1. ( id A ⊗ ∆) ◦ ∆ = Φ A,A,A ◦ (∆ ⊗ id A ) ◦ ∆ (c oasso ciativit y ) 2. l A ◦ ( ε ⊗ id A ) ◦ ∆ = id A = r A ◦ ( id A ⊗ ε ) ◦ ∆ (counit prop erty) ( A, ∆ , ε ) is said to b e co-comm utativ e if c A,A ◦ ∆ = ∆. 8 DA VID N. PHAM Definition 2.10. A F rob en ius ob ject is a tup le ( A, m, ∆ , µ, ε ) where ( A, m, µ ) is a commutati ve algebra ob ject and ( A, ∆ , ε ) is a co-comm utativ e coalg ebra ob ject wh ic h satisfies ∆ ◦ m = ( m ⊗ id A ) ◦ Φ − 1 A,A,A ◦ ( id A ⊗ ∆) (33) ∆ ◦ m = ( id A ⊗ m ) ◦ Φ A,A,A ◦ (∆ ⊗ id A ) . (34) Remark 2.11. T hroughout this pap er, we will disregard Φ fr om our expr es- sions since Rep( D ( k [ G ])) (our ca tegory of interest) has a trivial asso ciator. 3. G -F As & Frobenius Obj ects in Rep ( D ( k [ G ])) W e b egin this section by r ecalling the axiomatic definition of a G -F A [14]: Definition 3.1. A G -F rob enius Algebra ( G -F A) is giv en b y th e follo wing data < G , ( A, • , 1 A ) , η , ϕ > where (a) G = ( G 0 , G 1 ) is a finite groupoid. (b) ( A, • , 1 A ) is a finite dimensional asso ciativ e algebra o v er k with pro d - uct • and unit 1 A whic h splits as a d ir ect su m of algebras whic h are indexed by the ob jects of G : A = M x ∈G 0 A x . (35) (c) η : A × A → k is a bilinear f orm. (d) ϕ is a G -actio n wh ic h acts on A by al gebra homomorph ism s, that is, if (1) x ∈ G 1 with s ( x ) = x and t ( x ) = y, then ϕ ( x ) : A x → A y is an algebra isomorphism, (2) if g , h ∈ G 1 and s ( h ) = t ( g ), then ϕ ( h ) ◦ ϕ ( g ) = ϕ ( hg ), and (3) ϕ ( e x ) = id A x which satisfies the fol lowing for a l l x , y ∈ G 0 : (i) A x = L g ∈ Γ x A x g is a Γ x -graded a lgebra. (ii) if a x ∈ A x and b y ∈ A y , th en a x • b y = δ x , y a x • b y ∈ A x . (iii) η ( a • b, c ) = η ( a, b • c ) for all a, b, c ∈ A . (iv) η ( ϕ ( h ) a x , ϕ ( h ) b x ) = η ( a x , b x ) for all a x , b x ∈ A x and h ∈ G 1 satisfying s ( h ) = x. (v) ϕ ( x ) A x g ⊂ A y xg x − 1 for a morphism x ∈ G 1 satisfying s ( x ) = x and t ( x ) = y . (vi) η | A x g × A x h is nond egenerate for g h = e x and zero otherwise. (vii) a x g • a x h = ( ϕ ( g ) a x h ) • a x g ∈ A x g h for a x g ∈ A x g and a x h ∈ A x h . (viii) ϕ ( g ) | A x g = id A x g (ix) if g , h ∈ Γ x , c ∈ A x g hg − 1 h − 1 , and l c : A → A is the linear map induced b y left multiplica tion b y c , then T r  l c ◦ ϕ ( h ) | A x g : A x g → A x g  = T r  ϕ ( g − 1 ) ◦ l c | A x h : A x h → A x h  (36) where T r d enotes the trace. A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 9 Remark 3.2. The definition of group oid F r ob enius algebras giv en in [14] w as stated in terms of group F rob enius algebras. In an effort to mak e Defi- nition 3.1 self cont ained, we ha v e rew orded the original definition of [14] to a v oid an y r eference to group F rob en iu s algebras. Remark 3.3. In the sp ecial case when G is the o ne-ob ject group oid wh ose set of morph ism s is the group G , Definition 3.1 reduces to the definition of a G -F rob enius algebra. W e n o w stat e the main result of this pap er: Theorem 3.4. Every G -F A is derive d fr om a F r ob enius obje ct (( ρ, A ) , m, ∆ , µ, ε ) in R ep ( D ( k [ G ])) w hich satisfies (1) P x ∈G 0 P g ∈ Γ x ρ ( γ g g ) = id A (2) T r ( l c ◦ ρ ( γ h hg h − 1 )) = T r ( ρ ( γ g − 1 h ) ◦ l c ◦ ρ ( γ e x h )) ∀ x ∈ G 0 , g, h ∈ Γ x , and c ∈ ρ ( γ e x g hg − 1 h − 1 ) A wh er e T r denotes the tr ac e and l c is th e k -line ar map define d by l c ( v ) = m ( c ⊗ v ) for v ∈ ρ (1 x ) A . In addition, every F r ob enius obje ct in R ep ( D ( k [ G ])) which satisfies c ondi- tions (1) and (2) induc es a G -F A. W e n o w dedicate the r emainder of the pap er to the p ro of of Theorem 3.4. 3.1. G -F As via F rob enius ob jects. In this section, w e show th at ev ery F rob enius ob ject in Rep( D ( k [ G ])) w hic h s atisfies co nd itions (1) and (2) of Theorem 3.4 corresp onds to a particular G -F A. W e b egin b y sho w ing that every left D ( k [ G ])-mo du le has a ca nonical direct sum decomp osition and left G -action whic h r esem bles that of a G -F A. Prop osition 3.5. L et ( ρ, A ) b e a left D ( k [ G ]) -mo dule. Then (i) A has a dir e ct sum de c omp osition A = L x ∈G 0 A x which is indexe d by the obje cts of G w her e A x := ρ (1 x ) A . In p articular, ρ (1 y ) a x = δ x , y a x for a x ∈ A x . (ii) F or e ach x ∈ G 0 , A x has a dir e ct sum de c omp ostion A x = L g ∈ Γ x A x g wher e A x g := ρ ( γ e x g ) A x = ρ ( γ e x g ) A . In p articular, ρ ( γ e y h ) a x g = δ g ,h a x g for a x g ∈ A x g , y ∈ G 0 , and h ∈ Γ y . (iii) F or al l x , y ∈ G 0 , g ∈ Γ x , h ∈ Γ y , and x ∈ G 1 with t ( x ) = x , (a) ρ ( γ x g ) a y h = 0 for al l a y h ∈ A y h with h 6 = x − 1 g x , and (b) ρ ( γ x g ) is a ve ctor sp ac e isomo rphism fr om A s ( x ) x − 1 g x to A t ( x ) g . Pr o of. Since γ e x g · γ e y h = γ e y h · γ e x g = δ g ,h γ e x g (37) for g ∈ Γ x and h ∈ Γ y , it follo w s f rom (12 ) that 1 x · 1 y = X g ∈ Γ x X h ∈ Γ y δ g ,h γ e x g = δ x , y X g ∈ Γ x γ e x g = δ x , y 1 x . (38) 10 DA VID N. PHAM Hence, ρ (1 x ) ◦ ρ (1 y ) = δ x , y ρ (1 x ) . (39) It follo ws from (39) and the definition of A x that ρ (1 y ) a x = δ x , y a x (40) for a x ∈ A x . Since ρ (1 ) = id A , we also hav e A = ρ (1) A = X x ∈G 0 ρ (1 x ) A = X x ∈G 0 A x . (41) In addition, if P x ∈G 0 a x = 0 for a x ∈ A x , it follo w s f rom (40) that a y = ρ (1 y )   X x ∈G 0 a x   = 0 (42 ) for all y ∈ G 0 . (41) and (42) then sho w that A is a direct sum of the A x ’s. This completes the pro of of (i). F or (ii ), note that b y (40 ) A x = ρ (1 x ) A x = X g ∈ Γ x ρ ( γ e x g ) A x = X g ∈ Γ x A x g (43) and by (37) ρ ( γ e y h ) ◦ ρ ( γ e x g ) = δ g ,h ρ ( γ e x g ) . (44) It follo ws from (44) and the definition of A x g that ρ ( γ e y h ) a x g = δ g ,h a x g (45) for a x g ∈ A x g . Using (43) and (45) and applying an argumen t similar to the one used to pr o v e that A = ⊕ x ∈G 0 A x sho ws that A x itself is a direct sum of the A x g ’s. In addition, we also hav e A x g := ρ ( γ e x g ) A x = ρ ( γ e x g ) ( ρ (1 x ) A ) = ρ ( γ e x g ) A, (46) where the second equalit y follo ws f rom the definition of A x and the third equalit y follo ws from the fact that γ e x g · 1 x = γ e x g . This completes the pr o of of (ii). (iii-a) foll ows from (4 5) and the fact that ρ ( γ x g ) = ρ ( γ e t ( x ) g · γ x g · γ e s ( x ) x − 1 g x ) = ρ ( γ e t ( x ) g ) ◦ ρ ( γ x g ) ◦ ρ ( γ e s ( x ) x − 1 g x ) . (47) (47) also sh o ws that ρ ( γ x g ) A s ( x ) x − 1 g x ⊂ A t ( x ) g . T o see that ρ ( γ x g ) is also an isomorphism, note that ρ ( γ x g ) ◦ ρ ( γ x − 1 x − 1 g x ) | A t ( x ) g = ρ ( γ x g · γ x − 1 x − 1 g x ) | A t ( x ) g = ρ ( γ e t ( x ) g ) | A t ( x ) g = id A t ( x ) g (48) A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 11 and ρ ( γ x − 1 x − 1 g x ) ◦ ρ ( γ x g ) | A s ( x ) x − 1 gx = ρ ( γ x − 1 x − 1 g x · γ x g ) | A s ( x ) g = ρ ( γ e s ( x ) x − 1 g x ) | A s ( x ) x − 1 gx = id A s ( x ) x − 1 gx . (49) This completes the pro of of (iii-b).  Corollary 3.6. Ev ery left D ( k [ G ]) -mo dule ( ρ, A ) has a left G -action ϕ which acts as a k -line ar map on th e dir e ct sum de c omp osition A = ⊕ x ∈G 0 A x given by p art (i) of Pr op osition 3.5 wher e ϕ ( x ) := X g ∈ Γ t ( x ) ρ ( γ x g ) | A s ( x ) for x ∈ G 1 . (50) In addition, if x ∈ G 1 and A s ( x ) = ⊕ h ∈ Γ s ( x ) A s ( x ) h is the dir e c t sum de c omp o- sition given by p art (ii) of Pr op osition 3.5, t hen ϕ ( x ) A s ( x ) g ⊂ A t ( x ) xg x − 1 . Pr o of. If y ∈ G 1 with t ( y ) = s ( x ), then ϕ ( x ) ◦ ϕ ( y ) = ϕ ( xy ) since X g ∈ Γ t ( x ) γ x g · X h ∈ Γ t ( y ) γ y h = X g ∈ Γ t ( xy ) γ xy g . (51) It follo ws from the definition of ϕ an d part (iii) of Prop osition 3.5 that ϕ ( x ) is a linear map from A s ( x ) to A t ( x ) . Next we v erify that ϕ ( e x ) = id A x . T o do this, let a x ∈ A x . T h en a x can b e uniquely d ecomp osed as a x = P g ∈ Γ x a x g for a x g ∈ A x g . By part (ii) of Prop osition 3.5, w e ha ve ϕ ( e x ) a x = X g ∈ Γ x ρ ( γ e x g ) a x = X g ∈ Γ x ρ ( γ e x g ) a x g = X g ∈ Γ x a x g = a x . (52) T o complete the pro of that ϕ is a G -action, we only n eed to sho w that ϕ ( x ) : A s ( x ) → A t ( x ) is an isomorphism of vec tor sp aces and this follo ws from the previous calculation since ϕ ( x − 1 ) ◦ ϕ ( x ) = ϕ ( x − 1 x ) = ϕ ( e s ( x ) ) = id A s ( x ) (53) and ϕ ( x ) ◦ ϕ ( x − 1 ) = ϕ ( xx − 1 ) = ϕ ( e t ( x ) ) = id A t ( x ) . (54) Lastly , if a s ( x ) g ∈ A s ( x ) g , th en ϕ ( x ) a s ( x ) g = X h ∈ Γ t ( x ) ρ ( γ x h ) a s ( x ) g = ρ ( γ x xg x − 1 ) a s ( x ) g ∈ A t ( x ) xg x − 1 (55) b y part (iii) of Prop osition 3.5.  Notation 3.7. F or an ob ject ( ρ, A ) of R ep( D ( k [ G ])), we w ill often suppress the D ( k [ G ])-action ρ and simply w rite A for ( ρ, A ). The action of h ∈ D ( k [ G ]) on a ∈ A will b e denoted as h ⊲ a wh en ρ is omitted, that is, h ⊲ a := ρ ( h ) a . F urthermore, for x ∈ G 0 and g ∈ Γ x , A x will denote the direct summand of A give n by part (i) of Prop osition 3.5, and A x g will den ote the direct summand of A x giv en b y part (ii) of Prop osition 3.5. 12 DA VID N. PHAM W e no w look at the monoidal structur e of Rep( D ( k [ G ])), whic h is giv en by the next t w o le mmas. Lemma 3.8 . If A and B ar e obje cts of R ep ( D ( k [ G ])) , t hen th eir mo noidal pr o duct ( with D ( k [ G ]) -action ind uc e d by the c opr o duct ∆ D of D ( k [ G ])) is A b ⊗ B = M x ∈G 0 A x ⊗ k B x . (56) In add ition, for y ∈ G 0 ( A b ⊗ B ) y = A y ⊗ k B y . (57 ) Pr o of. By (22), A b ⊗ B := (1 (1) ⊲ A ) ⊗ k (1 (2) ⊲ B ). It follo ws from (19) and part (i) of Prop osition 3.5 that A b ⊗ B = X x ∈G 0 (1 x ⊲ A ) ⊗ k (1 x ⊲ B ) = X x ∈G 0 A x ⊗ k B x . (58) (56) then follo ws from th e fact that A = L x ∈G 0 A x and B = L x ∈G 0 B x . Lastly , note that since the D ( k [ G ])-actio n on A b ⊗ B is in duced by the copro duct of D ( k [ G ]) and ∆ D (1 y ) = 1 y ⊗ 1 y b y (18), (57) follo ws readily from part (i) of Prop osition 3.5 whic h implies that the image of A b ⊗ B under 1 y ⊗ 1 y is A y ⊗ k B y .  Lemma 3.9. (i) The unit obje ct D ( k [ G ]) t of R ep ( D ( k [ G ])) is define d as fol lows: (a) A s a ve ctor sp ac e over k , D ( k [ G ]) t has b asis { 1 x } x ∈G 0 wher e 1 x ∈ D ( k [ G ]) is define d by (12). (b) The left D ( k [ G ]) -action on D ( k [ G ]) t is given by γ y h ⊲ 1 x = δ s ( y ) , x δ h,y y − 1 1 s ( h ) . (59) (ii) D ( k [ G ]) x t = k 1 x for x ∈ G 0 . In p articular, D ( k [ G ]) t b ⊗ A = M x ∈G 0 k 1 x ⊗ k A x (60) A b ⊗ D ( k [ G ]) t = M x ∈G 0 A x ⊗ k k 1 x (61) for an obje ct A of R ep ( D ( k [ G ])) . (iii) If l A : D ( k [ G ]) t b ⊗ A → A and r A : A b ⊗ D ( k [ G ]) t → A ar e the left \ right identity maps of R ep ( D ( k [ G ])) for an obje ct A of R ep ( D ( k [ G ])) , th en l A (1 x ⊗ a x ) = a x (62) r A ( a x ⊗ 1 x ) = a x (63) for a x ∈ A x . A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 13 Pr o of. Let ε D and ∆ D denote the coun it and copro d u ct of D ( k [ G ]) resp ec- tiv ely . B y defi n ition, D ( k [ G ]) t is the image of ε D t : D ( k [ G ]) → D ( k [ G ]) where ε D t is the map giv en b y (23 ). Sin ce ∆ D (1) = 1 (1) ⊗ 1 (2) = X x ∈G 0 1 x ⊗ 1 x (64) b y (1 9), w e ha ve ε D t ( h ) = ε D (1 (1) · h )1 (2) = X x ∈G 0 ε D (1 x · h )1 x (65) for h ∈ D ( k [ G ]). Hence, th e image of ε t is con tained in the su bspace sp anned b y { 1 x } x ∈G 0 . Since ε D (1 x · 1 z ) = δ x , z ε D (1 x ) = δ x , z X g ∈ Γ x ε D ( γ e x g ) = δ x , z ε D ( γ e x e x ) = δ x , z , (66) w e ha ve ε D t (1 z ) = 1 z . T his sho ws that D ( k [ G ]) t is precisely the sp ace spanned b y { 1 x } x ∈G 0 . It follo ws from (12) that the latter is also linearly indep end en t and this completes the pro of of (i -a). F or (i- b), we ha v e γ y h ⊲ 1 x := ε D t ( γ y h · 1 x ) = δ s ( y ) , x ε D t ( γ y h ) = δ s ( y ) , x X z ∈G 0 ε D (1 z · γ y h )1 z = δ s ( y ) , x ε D ( γ y h )1 s ( h ) = δ s ( y ) , x δ h,y y − 1 1 s ( h ) . F or (ii), note t hat if 1 x ∈ D ( k [ G ]) and 1 y ∈ D ( k [ G ]) t , then b y (i-b), w e ha v e 1 x ⊲ 1 y = X g ∈ Γ x γ e x g ⊲ 1 y = X g ∈ Γ x δ x , y δ g ,e x 1 s ( g ) = δ x , y 1 x . (67) Hence, it follo ws from this and (i-a) that D ( k [ G ]) x t := 1 x ⊲ D ( k [ G ]) t = k 1 x . (6 8) (60) and (6 1 ) then follo w from Lemma 3.8. F or (ii i), w e ha ve l A (1 x ⊗ a x ) = X y ∈G 0 l A (1 y ⊲ 1 x ⊗ 1 y ⊲ a x ) (69) = l A (1 (1) ⊲ 1 x ⊗ 1 (2) ⊲ a x ) (70) 14 DA VID N. PHAM and r A ( a x ⊗ 1 x ) = X y ∈G 0 r A (1 y ⊲ a x ⊗ 1 y ⊲ 1 x ) (71) = r A (1 (1) ⊲ a x ⊗ 1 (2) ⊲ 1 x ) (72) where (69) and (71) follo w from (40) and (67), and (70) and (72) follo w from (19). By (28) and (30 ), we ha v e l A (1 (1) ⊲ 1 x ⊗ 1 (2) ⊲ a x ) = 1 x ⊲ a x = a x (73) and r A (1 (1) ⊲ a x ⊗ 1 (2) ⊲ 1 x ) = S (1 x ) ⊲ a x = 1 x ⊲ a x = a x (74) whic h pro v es (ii i).  W e n o w sho w that a comm utativ e algebra ob ject in Rep( D ( k [ G ])) has the necessary structur e to en cod e axioms (i), (ii), and (vii) of Definition 3 .1. Prop osition 3.10 . Supp ose ( A , m, µ ) i s an algebr a obje ct in R ep ( D ( k [ G ])) and a x • b y :=  m ( a x ⊗ b y ) if x = y 0 if x 6 = y (75) for a x ∈ A x , b y ∈ A y and a • b := X x , y ∈G 0 a x • b y = X x ∈G 0 a x • b x (76) for a = P x ∈G 0 a x , b = P x ∈G 0 b x wher e a x , b x ∈ A x for al l x ∈ G 0 . Then (i) f or a x g ∈ A x g and b x h ∈ A x h , a x g • b x h ∈ A x g h . (77) In p articular, a x • b y = δ x , y a x • b y ∈ A x ; (ii) A is a unital asso ciative algebr a over k with multiplic ation • and unit 1 A := µ (1) = P x ∈G 0 µ (1 x ) ; (iii) if ( A, m, µ ) is a lso c ommutative, then a x g • b x h = ( ϕ ( g ) b x h ) • a x g (78) wher e ϕ is the G - action given by Cor ol lary 3.6. Pr o of. F or (77), w e ha ve a x g • b x h = m ( a x g ⊗ b x h ) (79) = m ( γ e x g ⊲ a x g ⊗ γ e x h ⊲ b x h ) (80) = m   X { g 1 ,h 1 ∈ Γ x | g 1 h 1 = g h } γ e x g 1 ⊲ a x g ⊗ γ e x h 1 ⊲ b x h   (81) = γ e x g h ⊲ m ( a x g ⊗ b x h ) (82) = γ e x g h ⊲ ( a x g • b x h ) ∈ A x g h (83) A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 15 where (80), (81), and (83) foll ow from statemen t (ii) o f Prop osition 3.5 and (82) follo ws from the fact that (1) m is a D ( k [ G ])-linear map from A b ⊗ A to A where A b ⊗ A = L y ∈G 0 A y ⊗ k A y b y Lemma 3.8; (2) the D ( k [ G ])-act ion on A b ⊗ A is indu ced by the copro duct of D ( k [ G ]); and (3) ∆( γ e x g h ) = X { g 1 ,h 1 ∈ Γ x | g 1 h 1 = g h } γ e x g 1 ⊗ γ e x h 1 . Since A x = L g ∈ Γ x A x g , it follo ws readily from (77) (and the defin ition of the prod u ct) that a x • b y = δ x , y a x • b y ∈ A x for a x ∈ A x , b y ∈ A y . This completes the pro of of (i) . F or (ii), note that since m is also k -linear, it follo ws that ( λ 1 a ) • ( λ 2 b ) = ( λ 1 λ 2 )( a • b ) for λ 1 , λ 2 ∈ k and a, b ∈ A . T o s ho w that 1 A is the u n it elemen t of A , we use the fact that m ◦ ( µ ⊗ id A ) = l A and m ◦ ( id A ⊗ µ ) = r A (where l A and r A denotes the left and righ t identit y maps of A ). The latter sho ws that m ( µ (1 x ) ⊗ a x ) = l A (1 x ⊗ a x ) = a x (84) m ( a x ⊗ µ (1 x )) = r A ( a x ⊗ 1 x ) = a x (85) for a x ∈ A x , where part (iii) of Lemma 3.9 h as b een applied in (84) and (85). No w for a ∈ A , a can b e u niquely written as P x ∈G 0 a x where a x ∈ A x . By (84) and (85), we hav e the follo wing: 1 A • a = X x ∈G 0 µ (1 x ) • a x = X x ∈G 0 m ( µ (1 x ) ⊗ a x ) = X x ∈G 0 a x = a (86) a • 1 A = X x ∈G 0 a x • µ (1 x ) = X x ∈G 0 m ( a x ⊗ µ (1 x )) = X x ∈G 0 a x = a. (87) F or associativit y , it suffices to c hec k that ( a x • b y ) • c z = a x • ( b y • c z ) (88) for a x ∈ A x , b y ∈ A y , and c z ∈ A z . F rom th e definition of th e pro duct, its easy to see that b oth sides of (8 8) are zero when x 6 = y or y 6 = z. F or the case when x = y = z, w e h a v e ( a x • b x ) • c x = m ( m ( a x ⊗ b x ) ⊗ c x ) (89 ) and a x • ( b x • c x ) = m ( a x ⊗ m ( b x ⊗ c x )) . (90) Since m ◦ ( m ⊗ id A ) = m ◦ ( id A ⊗ m ) (91) w e see that (89) and (9 0) are indeed equal. In addition, it fol lo ws r eadily from (75) and (76) th at a • ( b + c ) = a • b + a • c (92) ( b + c ) • a = b • a + c • a (93) 16 DA VID N. PHAM for a, b, c ∈ A . This completes the proof of (ii). Lastly for (iii), note that if ( A, m, µ ) is also commutat ive, we hav e a x g • b x h = m ( a x g ⊗ b x h ) (94) = m ◦ c A,A ( a x g ⊗ b x h ) (95) = m   X y ∈G 0 X l,m ∈ Γ y ( γ l m ⊲ b x h ) ⊗ ( γ e y l ⊲ a x g )   (96) = m  ( γ g g hg − 1 ⊲ b x h ) ⊗ ( γ e x g ⊲ a x g )  (97) = m  ( γ g g hg − 1 ⊲ b x h ) ⊗ a x g  (98) = ( γ g g hg − 1 ⊲ b x h ) • a x g (99) where the seco nd equalit y follo ws from the fact that m = m ◦ c A,A (since ( A, m, µ ) is a c ommutative algebra ob ject); the third equalit y follo ws from the defin ition of the braiding m orphism c , whic h is giv en b y (32), and t he definition of th e R-matrix of D ( k [ G ]), whic h is giv en by (1 6) and (17); and the f ourth and fifth equ alities follo w from parts (ii) and (iii) of Prop osition 3.5. Since γ g g hg − 1 ⊲ b x h = X l ∈ Γ x γ g l ⊲ b x h (100) (b y (iii-a) of Pr op osition 3.5) and the righ t side is just ϕ ( g ) b x h , we ha v e a x g • b x h = ( ϕ ( g ) b x h ) • a x g (101) and this completes the pro of of Pr op osition 3.10.  W e n o w sho w ho w any left D ( k [ G ])-mo du le ( ρ, A ) whic h is b oth an algebra and colagebra ob ject giv es rise to a bilinear form whic h satisfies all the axioms of a G -F A except p ossibly axiom (vi) of De fin ition 3.1. W e will see later in Pr op osition 3.13 that to ensure that the ind uced bilinear form is nondegenerate (i.e., satisfies axiom ( vi) of Definition 3.1), the algebra and coalge br a s tr ucture on ( ρ, A ) m ust s atisfy the F rob enius relations (equations (33) and (3 4 )). In other words, ( ρ, A ) must also b e a F r ob enius ob ject. W e b egin by examining the prop erties of a coalgebra ob j ect in Rep( D ( k [ G ])). Lemma 3.11. L et ( A, ∆ , ε ) b e a c o algebr a obje c t in R ep ( D ( k [ G ])) . Then ε : A → D ( k [ G ]) t and ∆ : A → A b ⊗ A satisfy the fol lowing: (i) ε ( a x g ) = δ g ,e x ε ( a x g ) ∈ k 1 x ⊂ D ( k [ G ]) t , and (ii) ∆ ( a x g ) ∈ M { g 1 ,g 2 ∈ Γ x | g 1 g 2 = g } A x g 1 ⊗ k A x g 2 for a x g ∈ A x g . Pr o of. By part (i-a) of Lemm a 3.9, ε ( a x g ) can b e written as ε ( a x g ) = X y ∈G 0 λ y 1 y , λ y ∈ k . (102) A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 17 Then ε ( a x g ) = ε ( γ e x g ⊲ a x g ) (103) = γ e x g ⊲ ε ( a x g ) (104) = X y ∈G 0 λ y γ e x g ⊲ 1 y (105) = X y ∈G 0 δ x , y δ g ,e x λ y 1 y (106) = δ g ,e x λ x 1 x (107) whic h pro v es part (i) of Lemma 3.11. In th e ab o v e calc ulation, the fourth equalit y follo ws from p art (i-b) of Lemma 3.9. F or part (ii), we hav e ∆( a x g ) = ∆( γ e x g ⊲ a x g ) (108) = X g 1 g 2 = g  γ e x g 1 ⊲ ( a x g ) (1)  ⊗  γ e x g 2 ⊲ ( a x g ) (2)  ∈ M g 1 g 2 = g A x g 1 ⊗ k A x g 2 (109) where the ab o v e calculati on follo ws fr om part (ii) of Pr op osition 3.5 an d the fact that γ e x g acts on ∆( a x g ) = ( a x g ) (1) ⊗ ( a x g ) (2) via the copro d u ct of D ( k [ G ]). (Note that the sum and direct sum in (109) are ov er all g 1 , g 2 ∈ Γ x suc h that g 1 g 2 = g .)  Prop osition 3.12. Supp ose ( A, m, µ ) and ( A, ∆ , ε ) ar e r esp e ctively algebr a and c olagebr a obje cts in Rep ( D ( k [ G ])) . L et ε ′ : A → k b e the k - line ar map define d by 4 ε ( a ) = X x ∈G 0 ε ′ ( a x ) 1 x (110) for a = P x ∈G 0 a x with a x ∈ A x and let η : A × A → k b e the map define d by η ( a, b ) = ε ′ ( a • b ) (111) wher e a • b is the pr o duct given in Pr op osition 3.10. Then (i) η is a k - b iline ar map; (ii) η ( a • b, c ) = η ( a, b • c ) for a, b, c ∈ A (iii) η ( a x g , b x h ) = 0 for g h 6 = e x wher e a x g ∈ A x g and b x h ∈ A x h ; and (iv) η ( a x , b x ) = η ( ϕ ( x ) a x , ϕ ( x ) b x ) wher e ϕ is the G - action define d in Cor ol lary 3.6, a x , b x ∈ A x , and x ∈ G 1 with s ( x ) = x . Pr o of. (i) follo ws easily from the d efinition of η and (ii) follo ws f rom the fact that ( a • b ) • c = a • ( b • c ) by Prop osition 3.10. F or (ii i), note that since a x g • b x h ∈ A x g h (b y P rop osition 3 .10), w e h a v e ε ( a x g • b x h ) = ε ′ ( a x g • b x h )1 x . (112) 4 Note th at by part (i) of Lemma 3.11 the coefficient of 1 x in (110) d epen ds only up on a x , the x-comp onent of a . 18 DA VID N. PHAM With a x g • b x h ∈ A x g h , it follo ws fr om p art (i) of Lemma 3.11 that ε ( a x g • b x h ) = 0 for g h 6 = e x . By (1 12 ), ε ′ ( a x g • b x h ) is also zero for g h 6 = e x and th is pro ves (iii). F or (iv), it s uffices to consider the case when a x = a x g ∈ A x g and b x = b x h ∈ A x h . By Corollary 3.6, ϕ ( x ) a x g ∈ A y xg x − 1 and ϕ ( x ) b x h ∈ A y xhx − 1 where we ha ve set y = t ( x ). If gh 6 = e x , then η ( a x g , b x h ) = η ( ϕ ( x ) a x g , ϕ ( x ) b x h ) = 0 (113) b y part (iii) of Prop osition 3.12. F or the case wh en g h = e x , we ha v e η ( ϕ ( x ) a x g , ϕ ( x ) b x g − 1 ) = ε ′  ( ϕ ( x ) a x g ) • ( ϕ ( x ) b x g − 1 )  = ε ′  ( γ x xg x − 1 ⊲ a x g ) • ( γ x xg − 1 x − 1 ⊲ b x g − 1 )  = ε ′  m (( γ x xg x − 1 ⊲ a x g ) ⊗ ( γ x xg − 1 x − 1 ⊲ b x g − 1 ))  = ε ′   m   X uv = e y , u,v ∈ Γ y ( γ x u ⊲ a x g ) ⊗ ( γ x v ⊲ b x g − 1 )     = ε ′  γ x e y ⊲ m ( a x g ⊗ b x g − 1 )  = ε ′  γ x e y ⊲ ( a x g • b x g − 1 )  (114) where the second equalit y follo ws fr om the defin ition of ϕ and part (iii-a) of Prop osition 3.5; the f ou r th equalit y also f ollo ws from part (iii-a ) of Prop osi- tion 3.5; and th e fifth equalit y follo ws f rom the f act that m is D ( k [ G ])-linear. In addition, ε  γ x e y ⊲ ( a x g • b x g − 1 )  = γ x e y ⊲ ε  a x g • b x g − 1  (115) = ε ′  a x g • b x g − 1  γ x e y ⊲ 1 x (116) = ε ′  a x g • b x g − 1  1 y (117) where the third equalit y follo ws from part (i-b) of Lemma 3.9. Since γ x e y ⊲ ( a x g • b x g − 1 ) ∈ A y e y , we also hav e ε  γ x e y ⊲ ( a x g • b x g − 1 )  = ε ′  γ x e y ⊲ ( a x g • b x g − 1 )  1 y . (118) Hence, ε ′  γ x e y ⊲ ( a x g • b x g − 1 )  = ε ′  a x g • b x g − 1  . (119) It follo ws from this as w ell as the definition of η and (114) th at η ( ϕ ( x ) a x g , ϕ ( x ) b x g − 1 ) = η ( a x g , b x g − 1 ) (120) and this completes the pro of of (iv).  A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 19 Prop osition 3.13. L et ( A, m, △ , µ, ε ) b e a F r ob enius obje ct in R ep ( D ( k [ G ])) and let η : A × A → k b e the biline ar form given by Pr op osition 3.12. Then η | A x g × A x h is nonde gener ate for al l x ∈ G 0 and g, h ∈ Γ x satisfying g h = e x . Pr o of. T o start, set 1 x A := µ (1 x ) ∈ A x . (121) Then from the pro of o f Prop osition 3.10, w e h a v e 1 x A • a x = a x • 1 x A = a x (122) for a x ∈ A x . F urthermore, (7 7 ) of Prop osition 3.10 implies that 1 x A ∈ A x e x . By (ii) of Lemma 3. 11, ∆(1 x A ) ∈ M h ∈ Γ x A x h ⊗ k A x h − 1 . (123) No w let v 1 , . . . , v n b e a basis for A x with v i ∈ A x h i for some h i ∈ Γ x . Th en ∆(1 x A ) = n X i =1 u i ⊗ v i (124) for some u i ∈ A x h − 1 i . If a x ∈ A x , th en a x = l A ◦ ( ε ⊗ id A ) ◦ ∆( a x ) (125) = l A ◦ ( ε ⊗ id A ) ◦ ∆( a x • 1 x A ) (126) = l A ◦ ( ε ⊗ id A ) ◦ ∆ ◦ m ( a x ⊗ 1 x A ) (127) = l A ◦ ( ε ⊗ id A ) ◦ ( m ⊗ id A ) ◦ ( id A ⊗ ∆)( a x ⊗ 1 x A ) (128) = n X i =1 ε ′ ( a x • u i ) v i , (129) where the first equalit y is just th e counit prop erty of a coalgebra ob ject; the fourth equalit y foll o ws from (33); and the fifth equalit y emplo ys the linear map ε ′ : A → k that w as d efined in Prop osition 3 .12. Setting a x = v j and using the fact that th e v i ’s a re linearly indep enden t giv es η ( v j , u i ) := ε ′ ( v j • u i ) = δ j i . (130) Using (34), a similar ca lculation sho ws that a x = n X i =1 ε ′ ( v i • a x ) u i . (131) Since a x is arbitrary and the dimension of A x is n , (131) sho ws that { u i } n i =1 is also a basis of A x . (130) com b ined with the fact th at { v i } n i =1 and { u i } n i =1 are b oth b ases of A x (where v i ∈ A x h i and u i ∈ A x h − 1 i ) sho ws that η | A x g × A x h is nondegenerate for g h = e x .  The next result establishes the second h alf of Theorem 3.4. 20 DA VID N. PHAM Prop osition 3.14. If (( A, ρ ) , m, ∆ , µ, ε ) is a F r ob enius obje ct i n R ep ( D ( k [ G ])) which satisfies c onditions (1) and (2) of The or em 3.4, then < G , ( A, • , 1 A ) , η , ϕ > (132) is a G - F A wher e (i) • and 1 A ar e r esp e ctively the pr o duct and multiplic ative u ni t giv en by Pr op osition 3.10; (ii) η is the biline ar form given by Pr op osition 3.12; and (iii) ϕ is the G -action given by Cor ol lary 3.6. Pr o of. Axioms (b), (i), an d (ii) of Definition 3.1 are satisfied b y parts (i) and (ii) of Prop osition 3.5 and p arts (i) a nd (ii) of Prop osition 3.10. Axioms (v) and (vii) of Definition 3.1 are satisfied by Corollary 3.6 and part (iii) of Prop osition 3.10 resp ectiv ely . F or axiom (d ), we only need to v erify that ϕ ( x ) : A x → A y is a n algebra isomorphism for x ∈ G 1 where s ( x ) = x and t ( x ) = y . By Corollary 3.6, ϕ ( x ) is already an isomorphism of v ector spaces. Hence, we only need to c hec k that ϕ ( x )( a x • b x ) = ( ϕ ( x ) a x ) • ( ϕ ( x ) b x ) . (133) It su ffices to verify this for the case when a x = a x g and b x = b x h . In this case, w e h a v e ϕ ( x )( a x g • b x h ) = γ x xg hx − 1 ⊲ ( a x g • b x h ) (134) = γ x xg hx − 1 ⊲ m ( a x g ⊗ b x h ) (135) = m  ( γ x xg x − 1 ⊲ a x g ) ⊗ ( γ x xhx − 1 ⊲ b x h )  (136) = m  ( ϕ ( x ) a x g ) ⊗ ( ϕ ( x ) b x h )  (137) = ( ϕ ( x ) a x g ) • ( ϕ ( x ) b x h ) . (138) Throughout the ab o v e calculation w e ha ve made use of part (i ii) of Prop o- sition 3.5, and in the third equ alit y , we ha v e made use of the fact m is D ( k [ G ])-linear. Axioms (c), (iii), and (iv) of Definition 3.1 are satisfied by Prop osition 3.12; axiom (vi) of Definition 3.1 is satisfied by part (iii) of Prop osition 3.12 and by Prop osition 3 .13 . All that remains left to do is to sh o w that axioms (viii) and (ix) of Def- inition 3.1 a re also sati sfi ed. W e will no w show that axioms (viii) and(ix) follo w resp ective ly from cond itions (1) and (2) of Theorem 3.4. F or axi om (viii) of Definition 3.1, let a x g ∈ A x g . Th en a x g = X y ∈G 0 X h ∈ Γ y γ h h ⊲ a x g (139) = γ g g ⊲ a x g (140) = ϕ ( g ) a x g (141) A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 21 where the fi rst equalit y follo ws from condition (1) of Theorem 3.4 and the second a nd third equalitie s follo w from part (iii-a) of Prop osition 3.5. T his sho ws that axiom (viii) of Definition 3.1 is sat isfied . F or axiom (ix) of De fin ition 3.1, let g, h ∈ A x and for c ∈ A x g hg − 1 h − 1 , let l c : A x → A x b e the linear map defin ed by l c ( a x ) := c • a x . Then by p art (iii) of Prop osition 3.5 and by part (i) of Pr op osition 3.10, we hav e the follo wing: T r  l c ◦ ρ ( γ h hg h − 1 )  = T r  l c ◦ ρ ( γ h hg h − 1 ) | A x g : A x g → A x g  (142) T r  ρ ( γ g − 1 h ) ◦ l c ◦ ρ ( γ e x h )  = T r  ρ ( γ g − 1 h ) ◦ l c | A x h : A x h → A x h  . (143) Condition (2) of Theorem 3.4 then giv es T r  l c ◦ ρ ( γ h hg h − 1 ) | A x g : A x g → A x g  = T r  ρ ( γ g − 1 h ) ◦ l c | A x h : A x h → A x h  . (144) Since l c ◦ ρ ( γ h hg h − 1 ) | A x g = l c ◦ ϕ ( h ) | A x g (145) and ρ ( γ g − 1 h ) ◦ l c | A x h = ϕ ( g − 1 ) ◦ l c | A x h (146) b y p art (iii) of Prop osition 3.5 and the definition of ϕ , (144) sho ws that axiom (ix) is satisfied and this complete s the pro of of Prop osition 3.14.  3.2. F rob enius ob jects via G -F As. In this section, we mo ve in the op- p osite direction and s ho w that ev ery G -F A is also a F rob enius ob ject in Rep( D ( k [ G ])) whic h satisfies cond itions (1) and (2) of T heorem 3.4. W e b egin w ith the follo wing result: Prop osition 3.15. Eve ry G -F A is a left D ( k [ G ]) -mo dule. If A is a G -F A with G - action ϕ , then its D ( k [ G ]) -action is the line ar map define d b y ρ ( γ x g ) a y h := δ h,x − 1 g x ϕ ( x ) a y h (147) for γ x g ∈ D ( k [ G ]) and a y h ∈ A y h . Pr o of. T o start, let a ∈ A and decomp ose it as a = P x ∈G 0 P g ∈ Γ x a x g . T o sho w that (147) do es in deed define a D ( k [ G ])-actio n, we need to v erify that (i) ρ (1) = id A , and (ii) ρ ( γ x 1 g 1 ) ◦ ρ ( γ x 2 g 2 ) = ρ ( γ x 1 g 1 · γ x 2 g 2 ). F or (i) , w e h a v e ρ (1) a = X x ∈G 0 X g ∈ Γ x ρ ( γ e x g ) a = X x ∈G 0 X g ∈ Γ x ϕ ( e x ) a x g = X x ∈G 0 X g ∈ Γ x a x g = a. F or (ii ), w e h a v e ρ ( γ x 1 g 1 ) ◦ ρ ( γ x 2 g 2 ) a = ρ ( γ x 1 g 1 ) ◦ ϕ ( x 2 ) a s ( x 2 ) x − 1 2 g 2 x 2 = δ x − 1 1 g 1 x 1 ,g 2 ϕ ( x 1 ) ◦ ϕ ( x 2 ) a s ( x 2 ) x − 1 2 g 2 x 2 . (148) 22 DA VID N. PHAM Since γ x 1 g 1 · γ x 2 g 2 = δ x − 1 1 g 1 x 1 ,g 2 γ x 1 x 2 g 1 , we see that (ii) is satisfied for the case when x − 1 1 g 1 x 1 6 = g 2 . F or the c ase when x − 1 1 g 1 x 1 = g 2 , (148) reduces to ϕ ( x 1 x 2 ) a s ( x 2 ) x − 1 2 g 2 x 2 = ρ ( γ x 1 x 2 x 1 g 2 x − 1 1 ) a s ( x 2 ) x − 1 2 g 2 x 2 (149) = ρ ( γ x 1 x 2 g 1 ) a s ( x 2 ) x − 1 2 g 2 x 2 (150) = ρ ( γ x 1 x 2 g 1 ) a (151) = ρ ( γ x 1 g 1 · γ x 2 g 2 ) a (152) and this completes the pro of of (ii).  Prop osition 3.15 will b e applied implicit ly throughout this sectio n. Remark 3.16. Note that if one applies (i) and (ii) of Prop osition 3.5 to the left D ( k [ G ])-mo du le give n b y Pr op osition 3.15, the r esulting direct sum decomp osition is exactl y the one fr om the original G -F A. Hence, if ( ρ, A ) is the left D ( k [ G ])-mo dule of Prop osition 3.15 and A = L x ∈G 0 A x is the d ir ect sum decomp osition of the orig inal G -F A, then the monoidal pro d uct A b ⊗ A of ( ρ, A ) with itself is L x ∈G 0 A x ⊗ k A x b y Lemma 3.8. Prop osition 3.17. If < G , ( A, • , 1 A ) , η , ϕ > is a G -F A, th en (( ρ, A ) , m, µ ) is a c ommutative algebr a o bje ct in R ep ( D ( k [ G ])) wher e (i) m : A b ⊗ A → A is given by m ( a x ⊗ b x ) := a x • b x ∈ A x , and (ii) µ : D ( k [ G ]) t → A is given by 1 x 7→ 1 x A := ρ (1 x ) 1 A ∈ A x e x . Pr o of. Its clear from the asso ciativit y of the G -F A pro du ct and the fact that 1 A is the m u ltiplicativ e unit that m and µ satisfy 1 and 2 of Definition 2.8. Next, w e v erify that m ◦ c A,A = m . Without loss of generalit y , tak e a = a x g ∈ A x g and b = b x h ∈ A x h . Th en m ( a x g ⊗ b x h ) = a x g • b x h (153) = ( ϕ ( g ) b x h ) • a x g (154) = m ( ϕ ( g ) b x h ⊗ a x g ) (155) = m ( ρ ( γ g g hg − 1 ) b x h ⊗ a x g ) (156) = m ( ρ ( γ g g hg − 1 ) b x h ⊗ ρ ( γ e x g ) a x g ) (157) = m   X y ∈G 0 X l,m ∈ Γ y ρ ( γ l m ) b x h ⊗ ρ ( γ e y l ) a x g   (158) = m ◦ c A,A ( a x g ⊗ b x h ) (159) where the second equalit y follo ws from axiom (vii) of Definition 3.1. A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 23 The only thing that r emains to b e d one is to sho w that m and µ are D ( k [ G ])-linear. In the case of m , for x ∈ G 1 with s ( x ) = x, we hav e ρ ( γ x xg hx − 1 ) m ( a x g ⊗ b x h ) = ϕ ( x )( a x g • b x h ) (160) = ( ϕ ( x ) a x g ) • ( ϕ ( x ) b x h ) (161) = m  ( ρ ( γ x xg x − 1 ) a x g ) ⊗ ( ρ ( γ x xhx − 1 ) b x h )  (162) = X g 1 h 1 = g h m  ( ρ ( γ x xg 1 x − 1 ) a x g ) ⊗ ( ρ ( γ x xh 1 x − 1 ) b x h )  (163) (where the su m in the last equalit y is o ve r all g 1 , h 1 ∈ Γ x satisfying g 1 h 1 = g h ). Since the D ( k [ G ])-acti on on A b ⊗ A is induced by the copro duct of D ( k [ G ]), the ab o v e calculatio n sho ws that m is D ( k [ G ])-linear. In th e case of µ , it suffices to show that µ ( γ y h ⊲ 1 x ) = ρ ( γ y h ) µ (1 x ) . (16 4) By (i-b) of Lemma 3.9 , the left side is µ ( γ y h ⊲ 1 x ) = δ s ( y ) , x δ h,y y − 1 µ (1 s ( h ) ) = δ s ( y ) , x δ h,y y − 1 1 s ( h ) A , (165) and the righ t sid e is ρ ( γ y h ) µ (1 x ) = ρ ( γ y h ) ρ (1 x ) 1 A = δ s ( y ) , x ρ ( γ y h ) 1 A . (166) Since 1 A = X z ∈G 0 1 z A , it follo ws easily from axioms (i) and (ii) o f Definition 3.1 and the definition of ρ that 1 z A is the unit elemen t of A z and 1 z A ∈ A z e z . Hence, ρ ( γ y h ) 1 A = X z ∈G 0 ρ ( γ y h ) 1 z A = X z ∈G 0 δ e z ,y − 1 hy ϕ ( y )1 z A = δ h,y y − 1 ϕ ( y ) 1 s ( y ) A = δ h,y y − 1 1 t ( y ) A (167) where the last equalit y follo ws from the fact th at ϕ ( y ) : A s ( y ) → A t ( y ) is an isomorphism of algebras and m ust therefore map the unit of A s ( y ) to that of A t ( y ) . By substituting (167 ) into (166) and u sing the fact that s ( h ) = t ( y ), w e see that the righ t side and left side o f (164) are indeed equal and this completes the pro of.  Notation 3.18. As in the pro of of Prop osition 3.17, w e will use 1 x A to denote the unit elemen t of A x . 24 DA VID N. PHAM Notation 3.19. F or a v ector space V , let V ∗ denote the du al sp ace of V , and for a linear m ap f : V → U , let f ∗ : U ∗ → V ∗ denote the dual of f . The next lemma is a te c hn ical r esult whic h will b e used shortly to ind uce a copro duct on the G -F A. Lemma 3.20. Supp ose < G , ( A, • , 1 A ) , η , ϕ > is a G - F A and ψ : A → A ∗ is the k -line ar map define d by ψ ( a )( b ) := η ( a, b ) wher e a, b ∈ A a nd ψ ( a ) ∈ A ∗ . Then (i) ψ | A x g is a ve ctor sp ac e isomorphism fr om A x g to ( A x g − 1 ) ∗ , wher e an element f in ( A x g − 1 ) ∗ is also r e gar de d as an element in A ∗ via f ( b y h ) = δ g − 1 ,h f ( b y h ) for b y h ∈ A y h . (ii) ψ : A → A ∗ is a ve ctor sp ac e isomorphism; a nd (iii) ψ  ρ ( γ x g ) a x x − 1 g x  = ρ ( γ x − 1 x − 1 g − 1 x ) ∗ ψ ( a x x − 1 g x ) . Pr o of. F or (i), the isomorphism from A x g to ( A x g − 1 ) ∗ follo w s d irectly from axiom (vi) of Definition 3.1. The same axiom also imp lies that ψ ( a x g )( b y h ) = 0 when y = x and h 6 = g − 1 . F or y 6 = x, w e h a v e ψ ( a x g )( b y h ) = η ( a x g , b y h ) = η ( a x g • b y h , 1 A ) = 0 where the second and third equalit y f ollo w f rom axioms (iii) and (ii) of Definition 3.1 resp ectiv ely . In other w ords, ψ ( a x g )( b y h ) = δ g − 1 ,h ψ ( a x g )( b y h ) . (168) (ii) is a consequence of part (i) of Lemma 3.20 and the fact that A d e- comp oses as A = L x ∈G 0 L g ∈ Γ x A x g . F or (ii i), let b y h ∈ A y h . Then w e n eed to sho w that ψ  ρ ( γ x g ) a x x − 1 g x  ( b y h ) = ψ ( a x x − 1 g x )  ρ ( γ x − 1 x − 1 g − 1 x ) b y h  . (16 9) Both sides of (169) are zero for the case wh en h 6 = g − 1 b y (168) and the definition of ρ . F or the case when h = g − 1 , we ha v e y = t ( x ) and ψ  ρ ( γ x g ) a x x − 1 g x  ( b y g − 1 ) = η ( ρ ( γ x g ) a x x − 1 g x , b y g − 1 ) = η ( ϕ ( x ) a x x − 1 g x , b y g − 1 ) = η ( ϕ ( x − 1 ) ϕ ( x ) a x x − 1 g x , ϕ ( x − 1 ) b y g − 1 ) = η ( a x x − 1 g x , ϕ ( x − 1 ) b y g − 1 ) = η ( a x x − 1 g x , ρ ( γ x − 1 x − 1 g − 1 x ) b y g − 1 ) = ψ ( a x x − 1 g x )  ρ ( γ x − 1 x − 1 g − 1 x ) b y g − 1  A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 25 where the third equalit y follo ws from axiom (iv) of Definition 3.1.  In the n ext t w o lemmas, w e construct the counit and copro duct maps wh ic h will give ev ery G -F A the str u cture o f a co-co mmutativ e coal gebra ob ject in Rep( D ( k [ G ])). Lemma 3.21. Supp ose < G , ( A, • , 1 A ) , η , ϕ > is a G -F A and ε : A → D ( k [ G ]) t is the k -line ar map define d by ε ( a ) := X x ∈G 0 η ( a x , 1 A )1 x (170) for a = P x ∈G 0 a x . Then ε is D ( k [ G ]) -line ar. Pr o of. It suffices to sho w that ε  ρ ( γ x g ) a y h  = γ x g ⊲ ε ( a y h ) (171) for a y h ∈ A y h . F rom the definition of ρ , w e see th at the left side of (171) is zero when y 6 = s ( x ). Like wise, the right side is also zero when y 6 = s ( x ) since γ x g ⊲ ε ( a y h ) = η ( a y h , 1 A ) γ x g ⊲ 1 y = δ s ( x ) , y δ g ,xx − 1 η ( a y h , 1 A )1 t ( x ) = δ s ( x ) , y δ g ,xx − 1 η ( a y h • 1 y A , 1 A )1 t ( x ) = δ s ( x ) , y δ g ,xx − 1 η ( a y h , 1 y A )1 t ( x ) (172) where the second equalit y follo ws fr om part (i-b) of L emma 3.9 and the last equalit y follo ws from axiom (iii) of De fin ition 3.1 . F or the case wh en y = s ( x ), the left side of (171) can b e rewritten as ε  ρ ( γ x g ) a s ( x ) h  = η ( ρ ( γ x g ) a s ( x ) h , 1 A )1 t ( x ) = δ x − 1 g x,h η ( ϕ ( x ) a s ( x ) h , 1 A )1 t ( x ) = δ x − 1 g x,h η (( ϕ ( x ) a s ( x ) h ) • 1 t ( x ) A , 1 A )1 t ( x ) = δ x − 1 g x,h η ( ϕ ( x ) a s ( x ) h , 1 t ( x ) A )1 t ( x ) = δ x − 1 g x,h η ( ϕ ( x ) a s ( x ) h , ϕ ( x ) 1 s ( x ) A )1 t ( x ) = δ x − 1 g x,h η ( a s ( x ) h , 1 s ( x ) A )1 t ( x ) = δ g ,xx − 1 η ( a s ( x ) h , 1 s ( x ) A )1 t ( x ) (173) where the first equalit y follo ws from the fact that ρ ( γ x g ) a s ( x ) h ∈ A t ( x ) ; the sixth equalit y follo ws fr om axiom (iv) of Definition 3.1; and the sev en th equalit y follo ws from axiom (vi) of Defin ition 3.1 and the fact that 1 s ( x ) A ∈ A s ( x ) e s ( x ) . By comparin g (173) with (172), w e see that (171) is also satisfied when y = s ( x ).  26 DA VID N. PHAM Lemma 3.2 2. Su pp ose < G , ( A, • , 1 A ) , η , ϕ > is a G -F A. L et ψ b e the map given in L emma 3.20, m op : A b ⊗ A → A b e the k -line ar map given by m op ( a x ⊗ b x ) := b x • a x , and let ∆ : A → A b ⊗ A b e the k -line ar map given by ∆ := ( ψ − 1 ⊗ ψ − 1 ) ◦ ( m op ) ∗ ◦ ψ . Then (i) ∆ ( a x g ) ∈ L g 1 g 2 = g A x g 1 ⊗ k A x g 2 for al l a x g ∈ A x g , and (ii) ∆ is D ( k [ G ]) -line ar (wher e the dir e c t sum in (i) is over al l g 1 , g 2 ∈ Γ x satisfying g 1 g 2 = g ). Pr o of. F or (i), n ote that by part (i) of Lemma 3.20, ( m op ) ∗ ◦ ψ ( a x g )( b y h ⊗ c y l ) = 0 for all b y h ∈ A y h and c y l ∈ A y l satisfying lh 6 = g − 1 . Th is implies that ( m op ) ∗ ◦ ψ ( a x g ) ∈ M g 1 g 2 = g ( A x g − 1 1 ) ∗ ⊗ ( A x g − 1 2 ) ∗ . (174) P art (i) of Lemma 3.22 then follo ws from part (i) of Lemma 3.20. F or (ii ), it su ffices to sho w that ∆( ρ ( γ x g ) a y h ) = X g 1 g 2 = g [ ρ ( γ x g 1 ) ⊗ ρ ( γ x g 2 )]∆( a y h ) (175) for a y h ∈ A y h (where the sum is o ve r all g 1 , g 2 ∈ Γ t ( x ) satisfying g 1 g 2 = g ). F rom the definition of ρ , the left sid e is zero f or h 6 = x − 1 g x . By (i) of L emm a 3.22, the righ t side is also zero for h 6 = x − 1 g x . Let x = s ( x ). F or the case when h = x − 1 g x , w e ha v e ( m op ) ∗ ◦ ψ ( ρ ( γ x g ) a x x − 1 g x ) = ( m op ) ∗ ◦ ρ ( γ x − 1 x − 1 g x ) ∗ ◦ ψ ( a x x − 1 g x ) = " X g 1 g 2 = g ρ ( γ x − 1 x − 1 g − 1 1 x ) ∗ ⊗ ρ ( γ x − 1 x − 1 g − 1 2 x ) ∗ # ◦ ( m op ) ∗ ◦ ψ ( a x x − 1 g x ) (176) where the firs t equalit y follo ws from part (iii) of Lemma 3.20 and the second equalit y follo ws the defin ition of ρ and the fact t hat ϕ ( x − 1 ) is an alge br a homomorphism. Since ( m op ) ∗ ◦ ψ ( a x x − 1 g x ) ∈ M g 1 g 2 = g ( A x x − 1 g − 1 1 x ) ∗ ⊗ k ( A x x − 1 g − 1 2 x ) ∗ b y (1 74 ), it foll ows fr om (i) of Lemma 3.20 that ( m op ) ∗ ◦ ψ ( a x x − 1 g x ) = X g 1 g 2 = g n [ g 1 ] X i =1 ψ ( u x x − 1 g 1 x,i ) ⊗ ψ ( v x x − 1 g 2 x,i ) (177) for some u x x − 1 g 1 x,i ∈ A x x − 1 g 1 x and v x x − 1 g 2 x,i ∈ A x x − 1 g 2 x . In particular, ∆( a x x − 1 g x ) = X g 1 g 2 = g n [ g 1 ] X i =1 u x x − 1 g 1 x,i ⊗ v x x − 1 g 2 x,i . (17 8) A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 27 Substituting (177) in to the r igh t sid e of (176) and applying (iii) of Lemma 3.20 as w ell as the fact that ρ ( γ y s ) ∗ ψ ( c z t ) = 0 for s 6 = t − 1 giv es ( m op ) ∗ ◦ ψ ( ρ ( γ x g ) a x x − 1 g x ) = X g 1 g 2 = g n [ g 1 ] X i =1 ψ ( ρ ( γ x g 1 ) u x x − 1 g 1 x,i ) ⊗ ψ ( ρ ( γ x g 2 ) v x x − 1 g 2 x,i ) . (179) Applying ψ − 1 ⊗ ψ − 1 to both sides of (1 79 ) (and using the definition o f ρ ) yields ∆( ρ ( γ x g ) a x x − 1 g x ) = X g 1 g 2 = g [ ρ ( γ x g 1 ) ⊗ ρ ( γ x g 2 )]∆( a x x − 1 g x ) whic h co mp letes the pro of.  Prop osition 3.23. Supp ose < G , ( A, • , 1 A ) , η , ϕ > is a G -F A and ε and ∆ ar e the maps gi v en in L emmas 3.21 and 3.22 r esp e ctively. Then (( ρ, A ) , ∆ , ε ) is a c o-c ommutative c o algebr a obje ct in R ep ( D ( k [ G ])) . Pr o of. By Lemmas 3.21 an d 3.22, ε and ∆ are D ( k [ G ])-linear. W e no w v erify that ∆ and ε satisfy the axioms of a co -comm utativ e c oalgebra. F or the coasso ciativit y o f ∆, we ha v e (∆ ⊗ id A ) ◦ ∆ =  ( ψ − 1 ⊗ ψ − 1 ) ◦ ( m op ) ∗  ⊗ ψ − 1  ◦ ( m op ) ∗ ◦ ψ (180) =  ψ − 1 ⊗ ψ − 1 ⊗ ψ − 1  ◦  ( m op ) ∗ ⊗ id A ∗  ◦ ( m op ) ∗  ◦ ψ (181) =  ψ − 1 ⊗ ψ − 1 ⊗ ψ − 1  ◦  id A ∗ ⊗ ( m op ) ∗  ◦ ( m op ) ∗  ◦ ψ (182) =  ψ − 1 ⊗  ( ψ − 1 ⊗ ψ − 1 ) ◦ ( m op ) ∗  ◦ ( m op ) ∗ ◦ ψ (183) = ( id A ⊗ ∆) ◦ ∆ (184) where the third equalit y is a consequence of the fact th at the opp osite mul- tiplication map m op of Lemma 3.22 is asso ciativ e. F or the counit prop ert y , we need to sho w that l A ◦ ( ε ⊗ id A ) ◦ ∆( a ) = a = r A ◦ ( id A ⊗ ε ) ◦ ∆( a ) (185) for all a ∈ A . By lin earit y , it suffices to prov e (185 ) for th e case when a = a x g ∈ A x g . If a x g is zero, there is not hin g to pro v e. S o assume then that a x g 6 = 0 and let { u j } n j = 1 b e a basis for A x g − 1 and let { v i } m i =1 b e a basis for A x e x where v 1 is t ake n to b e the pro jection of 1 A on to A x . (As was shown in pr o of of Prop osition 3.17, v 1 is indeed an elemen t of A x e x and is also the un it elemen t of A x .) F urthermore, let { u ∗ j } n j = 1 and { v ∗ i } m i =1 denote the dual basis of { u j } n j = 1 and { v i } m i =1 resp ectiv ely (where an elemen t f in ( A y h ) ∗ is also regarded as an element of A ∗ b y extending the d efinition of f via f ( a z l ) = δ h,l f ( a z l )). 28 DA VID N. PHAM By p art (i) o f Lemma 3.20, w e hav e ψ ( a x g ) = n X j = 1 α j u ∗ j (186) where α j = ψ ( a x g )( u j ). In addition, b y part (i) of Lemm as 3.20 and 3.22 we can also express ( m op ) ∗ ◦ ψ ( a x g ) as ( m op ) ∗ ◦ ψ ( a x g ) = X i,j α ij v ∗ i ⊗ u ∗ j + ω ∈ M g 1 g 2 = g ( A x g − 1 1 ) ∗ ⊗ k ( A x g − 1 2 ) ∗ (187) where α ij = ψ ( a x g )( u j • v i ) and ω ∈ M g 1 g 2 = g , g 1 6 = e x ( A x g − 1 1 ) ∗ ⊗ k ( A x g − 1 2 ) ∗ . In particular, note that α j = α 1 j . Next, note that for f ∈ ( A x h ) ∗ , we hav e ε ◦ ψ − 1 ( f ) = η ( ψ − 1 ( f ) , 1 A ) 1 x = ψ ( ψ − 1 ( f ))( 1 A ) 1 x = f ( 1 A ) 1 x = δ h,e x f ( 1 A ) 1 x . (188) Applying (187) and (188) to the fi rst half of (185) g ive s l A ◦ ( ε ⊗ id A ) ◦ ∆( a x g ) = l A ◦ ( ε ◦ ψ − 1 ⊗ ψ − 1 ) ◦ ( m op ) ∗ ◦ ψ ( a x g ) (189 ) = X i,j α ij v ∗ i ( 1 A ) ψ − 1 ( u ∗ j ) (190) = X i,j α ij v ∗ i ( v 1 ) ψ − 1 ( u ∗ j ) (191) = X i,j α 1 j ψ − 1 ( u ∗ j ) (19 2) = X i,j α j ψ − 1 ( u ∗ j ) (193) = a x g . (194) The pro of of the other half of (185) is entirely similar. Lastly , for co-comm utativit y , we need to sho w that c A,A ◦ ∆( a ) = ∆( a ) ∀ a ∈ A. (195) Again, by linearit y , it suffi ces to pro v e (195) for the case when a = a x g ∈ A x g . T o start, note that b y applying ψ − 1 to b oth sides of part (iii) of Lemma 3.20 (and using the definition of ρ ), it follo ws that ρ ( γ x g ) ◦ ψ − 1 = ψ − 1 ◦ ρ ( γ x − 1 x − 1 g − 1 x ) ∗ . (196) A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 29 Next, note that if b x g − 1 1 ∈ A x g − 1 1 and c x g − 1 2 ∈ A x g − 1 2 with g 1 g 2 = g , then ( m op ) ∗ ◦ ψ ( a x g )( b x g − 1 1 ⊗ c x g − 1 2 ) = ψ ( a x g )( c x g − 1 2 • b x g − 1 1 ) = ψ ( a x g )(( ϕ ( g − 1 2 ) b x g − 1 1 ) • c x g − 1 2 ) = ψ ( a x g )(( ρ ( γ g − 1 2 g − 1 2 g − 1 1 g 2 ) b x g − 1 1 ) • ( ρ ( γ e x g − 1 2 ) c x g − 1 2 )) (197) where the second equalit y follo ws from axiom (vii) of Definition 3.1 and the third equalit y follo ws d irectly from t he definition of ρ . (197) then implies that ( m op ) ∗ ◦ ψ ( a x g ) = " X g 1 g 2 = g ρ ( γ g − 1 1 g − 1 2 ) ∗ ⊗ ρ ( γ e x g − 1 1 ) ∗ # ◦ m ∗ ◦ ψ ( a x g ) . (198) No w let τ : A b ⊗ A → A b ⊗ A b e the k -linear map defined by τ ( a y ⊗ b y ) := b y ⊗ a y . The proof of (195) then follo w s from (196) and (198): c A,A ◦ ∆( a x g ) = τ ◦     X y ∈G 0 X h,l ∈ Γ y ρ ( γ e y h ) ◦ ψ − 1 ⊗ ρ ( γ h l ) ◦ ψ − 1   ◦ ( m op ) ∗ ◦ ψ ( a x g )   = τ ◦ " X g 1 g 2 = g ρ ( γ e x g 1 ) ◦ ψ − 1 ⊗ ρ ( γ g 1 g 1 g 2 g − 1 1 ) ◦ ψ − 1 ! ◦ ( m op ) ∗ ◦ ψ ( a x g ) # = τ ◦ " X g 1 g 2 = g ψ − 1 ◦ ρ ( γ e x g − 1 1 ) ∗ ⊗ ψ − 1 ◦ ρ ( γ g − 1 1 g − 1 2 ) ∗ ! ◦ ( m op ) ∗ ◦ ψ ( a x g ) # = ( ψ − 1 ⊗ ψ − 1 ) ◦ " X g 1 g 2 = g ρ ( γ g − 1 1 g − 1 2 ) ∗ ⊗ ρ ( γ e x g − 1 1 ) ∗ ! ◦ m ∗ ◦ ψ ( a x g ) # = ( ψ − 1 ⊗ ψ − 1 ) ◦ ( m op ) ∗ ◦ ψ ( a x g ) = ∆ ( a x g ) where the third equ alit y follo ws from (196) and the fifth equalit y f ollo ws from (198).  The next t w o lemmas will b e used to show that the algebra and coalge br a ob jects giv en b y Propositions 3. 17 and 3. 23 sati sfy the F rob enius relations (equations (33) and (34)). Lemma 3.24. Supp ose < G , ( A, • , 1 A ) , η , ϕ > is a G - F A . Then (i) ϕ ( g − 1 ) | A x g = id A x g , (ii) a x g • b x g − 1 = b x g − 1 • a x g for al l a x g ∈ A x g , b x g − 1 ∈ A x g − 1 , and (iii) η is symmetric for al l x ∈ G 0 , g ∈ Γ x . 30 DA VID N. PHAM Pr o of. P art (i) f ollo ws immediately from axiom (viii) of Definition 3.1 and the fa ct that ϕ ( e x ) = id A x . Part (ii) then follo ws from part (i ) of Lemma 3.24 and axio m (vii) of Definition 3.1. F or (iii), w e ha ve η ( a x g , b x h ) = η ( a x g • b x h , 1 A ) = η (( ϕ ( g ) b x h ) • a x g , 1 A ) = η ( ϕ ( g ) b x h , a x g ) = η ( b x h , ϕ ( g − 1 ) a x g ) = η ( b x h , a x g ) where the fourth equalit y follo ws from axiom (iv) of Definition 3.1 and the last equalit y follo ws from part (i) of Lemma 3.24.  Lemma 3.25. Supp ose < G , ( A, • , 1 A ) , η , ϕ > is a G -F A and { u i } is any b asis of A x wher e u i ∈ A x g i for some g i ∈ Γ x . L et b u i := ψ − 1 ( u ∗ i ) (199) wher e { u ∗ i } is the dual b asis of { u i } . Then (i) b u i ∈ A x g − 1 i ; (ii) { b u i } is a b asis of A x ; and (iii) ψ ( u i ) = b u ∗ i . Pr o of. P arts (i) and (ii) a re both i mmed iate consequences of L emma 3.20. By part (iii) of Lemma 3.24, we also ha ve ψ ( u i )( b u j ) = η ( u i , b u j ) = η ( b u j , u i ) = ψ ( b u j )( u i ) = u ∗ j ( u i ) = δ ij , whic h pro v es (ii i).  The last tw o results of this section w ill b e u sed shortly to establish the fir st half of Theorem 3.4. Prop osition 3.26. Supp ose < G , ( A, • , 1 A ) , η , ϕ > is a G - F A and (( ρ, A ) , m, µ ) and (( ρ, A ) , ∆ , ε ) ar e th e algebr a and c o algebr a obje c ts given r esp e c tively in Pr op ositions 3.17 and 3.23. Then (( ρ, A ) , m, ∆ , µ, ε ) is a F r ob enius obje ct in R ep ( D ( k [ G ])) . Pr o of. The only thing w e ha ve left to c heck are the F rob enius relations: ∆ ◦ m = ( m ⊗ id A ) ◦ ( id A ⊗ ∆) (200) ∆ ◦ m = ( id A ⊗ m ) ◦ (∆ ⊗ id A ) (201) A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 31 T o start, let { u i } b e an y basis of A x where u i ∈ A x g i for some g i ∈ Γ x and let { b u i } b e the basis g ive n b y Lemma 3.25. T hen u i • u j = X t C t ij u t (202) b u l • b u m = X t b C t lm b u t (203) for some C t ij , b C t lm ∈ k where we note that C t ij = 0 if g t 6 = g i g j and b C t lm = 0 if g t 6 = g m g l . Next, note that b u l • u i = X s C l is b u s . (204) (204) follo ws from the fact that if α s ∈ k is the scalar m ultiplying b u s then α s = b u ∗ s ( b u l • u i ) = ψ ( u s )( b u l • u i ) = η ( u s , b u l • u i ) = η ( b u l • u i , u s ) = η ( b u l , u i • u s ) = X t C t is η ( b u l , u t ) = X t C t is ψ ( b u l )( u t ) = X t C t is u ∗ l ( u t ) = C l is . W e no w pro ve (2 00 ). (The pro of of (201 ) is s im ilar.) T o do this, it suffices to sh ow that ∆( u i • u j ) = ( m ⊗ id A ) ◦ ( id A ⊗ ∆)( u i ⊗ u j ) . (205) It follo ws from the definition of ∆ as well as that of { u i } and { b u i } that ∆( a x ) = X l,m η ( a x , b u m • b u l ) u l ⊗ u m . (206) Hence, the righ t side of (205) is X l,m η ( u j , b u m • b u l ) ( u i • u l ) ⊗ u m . (207) 32 DA VID N. PHAM Computing the left side o f (205) giv es ∆( u i • u j ) = X l,m η ( u i • u j , b u m • b u l ) u l ⊗ u m = X l,m η ( b u m • b u l , u i • u j ) u l ⊗ u m = X l,m η ( b u m • ( b u l • u i ) , u j ) u l ⊗ u m = X l,m X s C l is η ( b u m • b u s , u j ) u l ⊗ u m = X m,s η ( b u m • b u s , u j ) X l C l is u l ! ⊗ u m = X m,s η ( b u m • b u s , u j ) ( u i • u s ) ⊗ u m . = X m,s η ( u j , b u m • b u s ) ( u i • u s ) ⊗ u m . Comparing the last line of the ab o v e calculation with (207) sh ows that the left and righ t sides of (205) are indeed equal.  Prop osition 3.27. Supp ose < G , ( A, • , 1 A ) , η , ϕ > is a G - F A and (( ρ, A ) , m, ∆ , µ, ε ) is the F r ob enius obje ct of Pr op osition 3.26. Then (( ρ, A ) , m, ∆ , µ, ε ) sat isfies c onditions (1) and (2) of The or em 3.4. Pr o of. F or cond ition (1), let a = P x ∈G 0 P g ∈ Γ x a x g . Th en X x ∈G 0 X g ∈ Γ x ρ ( γ g g ) a = X x ∈G 0 X g ∈ Γ x ϕ ( g ) a x g = X x ∈G 0 X g ∈ Γ x a x g = a where the first equalit y follo w s fr om the d efinition of ρ and the second equal- it y foll ows from axio m (viii) of Definition 3.1. F or co nd ition (2), note that T r  l c ◦ ρ ( γ h hg h − 1 )  = T r  l c ◦ ϕ ( h ) | A x g : A x g → A x g  (208) and T r  ρ ( γ g − 1 h ) ◦ l c ◦ ρ ( γ e x h )  = T r  ϕ ( g − 1 ) ◦ l c | A x h : A x h → A x h  . (209) Condition (2) then follo ws from axiom (ix) of Definition 3.1.  A CA TEGORICAL APPR OA CH TO GR OUPOID FROBENIUS ALGEBRAS 33 3.3. Proof of T heorem 3.4. T he proof of T heorem 3 .4 now follo ws from Prop ositions 3.14 and 3.26. Sp ecifically , Prop osition 3.14 sho ws that ev ery F rob enius o b j ect in Rep( D ( k [ G ])) satisfying the tw o conditions of Theorem 3.4 induces a G -F A. This pr o v es th e second h alf of Theorem 3.4 . In addition, ev ery G -F A is d eriv ed f rom a F r ob enius ob j ect in Rep( D ( k [ G ])) whic h satis- fies conditions (1) and (2) of Theorem 3.4. T o see this, let A b e any G -F A and use Prop osition 3 .26 to represent A a s a F rob eniu s ob ject in Rep( D ( k [ G ])). By Prop osition 3. 27 , this F rob eniu s o b ject satisfies conditions (1) and (2) of T heorem 3.4. Its easy to c heck that if Prop osition 3.14 is applied to the aforemen tioned F r ob enius ob ject, the resulting G -F A is exactly A and this pro ve s the first part of Theorem 3.4. 4. Concl usions & Direct ions for Fu ture W ork In th is p ap er, w e ha ve shown that G -F As corresp ond to a certain typ e of F rob enius ob ject in the representa tion category of D ( k [ G ]). This result generalizes an earlier result for group F rob enius algebras [10], and, in the pro cess, p ro vides a ca tegory-theoretic “deriv ation” of the original G -F A ax- ioms in tro duced in [14]. F urthermore, when one compares the original G -F A definition (wh ic h is quite length y) with the cate gory-theoretic statement of Theorem 3.4 (whic h is quite concise), one c an certainly mak e the case that the natural setting for G -F As is c ate goric al in n ature. W e conclude the paper with the f ollo wing op en questions 5 : 1. Is there a relationship b et wee n G -F As and HQFT (b ey ond the sp ecial case when G is a fin ite group)? 2. Does the notion of a G -F A mak e sense if G is replaced by a c ate gory fib er e d in g r oup oids (e.g., De ligne-Mumford stac ks)? These qu estions will b e explored in a future wo rk. Referen ces [1] M. Atiyah, T op ological q uantum field th eory , Public ations Mathematiques de l’IHES (1988), 175-186. [2] G. Bohm, K. 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PHAM [9] R.M. Kaufmann, Orbifolding F rob enius algebras, Int. J. of Math. , 14 (2003), 573619. [10] R. M. Kaufmann, D. Pham, The Drinfeld double a nd t wisting in stringy orbifold theory , I nt. J. of Math. 20 (2009), 623-657. [11] D. Nikshyc h, V. T u raev, and L. V ainerman, Q uantum group oids and in v ariants of knots an d 3-manifolds, math .QA/000607 8 (2000). [12] D. Nikshyc h, L. V ainerman, Finite quantum group oids and their applications, math.QA/0006057v2. [13] F. Nill, Axioms of W eak Bialgebras, math.QA/9805104 (1998). [14] D. Pham, Group oid F rob enius Algebras, Comm. in Contemp. M ath. , V ol 12, No. 6, 2010 pp. 939-952. [15] V. T uraev, Homotop y Quantum field th eory in dimension 2 and group algebras, math.QA/9910010. Dep ar tment o f Ma thema tics, Mar ymount Man ha tt a n College, 221 E 71st Street, NY , NY 10021, email: dpham90@gmai l.com, dph am@mmm.edu