Pseudosymmetric braidings, twines and twisted algebras

Pseudosymmetri c braidings, t wines a nd t wist ed algebras Florin P anaite ∗ Institute of Mathem atics of the Romanian Academ y PO-Bo x 1-764, R O-0147 00 Buc harest, Romania e-mail: Florin.P anaite@imar.ro Mihai D. Staic † Departmen t of Mathematics, Indiana Univers ity , Ra wles Hall, Blo omington, IN 4740 5, USA e-mail: mstaic@indiana.edu F r eddy V an Oystaey en Departmen t of Mathematics and Computer Science Univ ersit y of An t we rp, Middelheimlaan 1 B-2020 Ant w erp, Belgium e-mail: F ra ncine.Sc ho eters@ua.ac.b e Abstract A laycle is the categorica l analogue o f a lazy co cycle. Twines (as in tro duce d by Brug ui` eres) and str ong twines (as int r o duced b y the author s) are laycles sa tisfying s ome extra conditions. If c is a braiding, the do uble braiding c 2 is alwa ys a twine; w e prove that it is a strong twine if and o nly if c sa tisﬁes a sor t of mo diﬁed br aid relation (we call such c pseudosymmetric , a s any symmetric braiding satisﬁes this relation). It is known that symmetric Y e tter -Drinfeld categorie s are tr ivial; we prov e that the Y etter-Drinfeld category H Y D H ov er a Hopf algebra H is pseudosymmetric if and only if H is co mm utative and co comm utative. W e in tro duce as well the Hopf algebra ic counterpart of pseudosymmetric braidings under the name pseudo- triangular structur es and prove that all quasitriangula r structures on the 2 n +1 -dimensional po inted Hopf a lgebras E ( n ) are pseudotria ngular. W e o bserve that a laycle on a monoidal category induces a so-called pseudotwistor on every algebr a in the category , and we ob- tain some genera l r esults (and give some examples) concer ning pseudotwistors, inspired by prop erties of laycles and twines. ∗ Researc h carried out while the ﬁrst author was v isiting the Universit y of Antw erp, supp orted by a p ostdo ctoral fello wship oﬀered by FWO (Flemis h Scientiﬁc Researc h F oundation). This author w as also partially supported by the programme CEEX of the Romanian Ministry of Education and Research, con tract nr. 2-CEx06-11-20/2006 . † P ermanent address: Institute of Mathematics of the Romanian Academy , PO-Bo x 1-764, RO-01470 0 Buchare st, Romania. 1 In tro du ction The n otion of symmetric c ate gory is a classical concept in category theory . I t consists of a monoidal catego ry C equip p ed with a family of natural isomorph isms c X,Y : X ⊗ Y → Y ⊗ X satisfying natural “bilinearit y” conditions together with the symmetry relation c Y , X ◦ c X,Y = id X ⊗ Y , for all X, Y ∈ C . In 1985 Joy al and S treet w ere led by n atural consider ations to dr op this symmetry condition from the axioms, thus arriving at the concept of br aiding , which afterw ard s b ecame of cen tral imp ortance for the then emerging theory of quantum groups; for instance, if ( H, R ) is a qu asitriangular Hopf algebra as deﬁned b y Drinf eld, then the monoidal catego r y H M of left H -mo dules acquires a braiding deﬁn ed b y R , whic h is symmetric if and only if R is triangular, i.e. R 21 R = 1 ⊗ 1. There exist many examples of symmetric br aidin gs, as w ell as man y examples of br aidings whic h are not symmetric. Although some of the most basic examples of monoidal categories (suc h as the cate gory of v ector s paces) are sym metric, th e symmetry condition is a rather restrictiv e requirement , a claim whic h is pr ob ab ly b est illustrated by the follo wing result of Pa reigis (cf. [30]): if H is a Hopf algebra, then the Y etter-Drinfeld categ ory H Y D H is sym metric if and only if H is trivial (i.e. H = k ). Thus, the most basic examples of braided categories arising in Hopf algebra theory are virtually nev er symmetric. It app ears thus n atur al to look for br aidings satisfying some generalized (or wea kened) sym- metry cond itions. In a recen t pap er [15], Etingof and Gelaki prop osed the concept of quasisym- metric br aiding , as b eing a braiding w ith the prop erty that c Y , X ◦ c X,Y = id X ⊗ Y for all X , Y simple ob jects in the categ ory , and classiﬁed qu asisymmetric braided categories of exp on en - tial gro wth , generalizing Delig ne’s classiﬁcation of symmetric categories of exp onent ial gro wth. On the other hand, at the Hopf algebraic lev el, Liu and Z h u prop osed in [23] the concept of almost-triangula r Hopf alge b ra, as b eing a q u asitriangular Hopf alg ebr a ( H , R ) suc h that R 21 R is cen tral in H ⊗ H (ob viously , this concept generalizes the one of triangular Hopf algebra, but it is not clear whether it has a cat egorical coun terpart). The original aim of the present pap er wa s to con tin u e the stud y of some categorical concepts recen tly introd uced in [34 ], [5], [29] under the names pur e-br aide d structur e , twine and str ong twine . W e recall fr om [5] that a twine on a monoidal category C is a family of n atural isomor- phisms D X,Y : X ⊗ Y → X ⊗ Y in C satisfying a certain list of axioms chosen in su c h a wa y that, if c is a b raiding on C , th en the so-called double br aiding c 2 deﬁned b y c 2 X,Y = c Y , X ◦ c X,Y is a t wine (b y [29], the concept of t wine is equiv alen t to the concept of pu re-braided stru cture in tro d uced in [34]). Moreo v er, t wines are related to the p u re braid groups in the same w a y in whic h b raidings are r elated to th e braid groups. A strong twine, as deﬁn ed in [29], is also a family of natural isomorphisms D X,Y : X ⊗ Y → X ⊗ Y in C satisfying a list of (easier lo oking) axioms, which imply the axioms of a t wine. A double br aiding c 2 is not alw ays a stron g twine, so w e w ere led naturally to ask for wh at k in d of b raidings c is c 2 a strong twine. Th e ans wer is that this happ ens if and only if c satisﬁes the follo w ing condition: ( c Y , Z ⊗ id X ) ◦ ( id Y ⊗ c − 1 Z,X ) ◦ ( c X,Y ⊗ id Z ) = ( id Z ⊗ c X,Y ) ◦ ( c − 1 Z,X ⊗ id Y ) ◦ ( id X ⊗ c Y , Z ) for all X , Y , Z ∈ C . This is a sort of mo diﬁed braid relation, and it is obvious that if c is a symmetry then this condition b eco mes exactly the b raid relation satisﬁed by any braiding; th us, an y symmetric braiding s atisﬁes the ab o v e relation, so what we obtained is a generalized symmetry cond ition. A b raiding satisfying the ab o v e mo diﬁ ed braid relation will b e called pseudosymmetric . It should b e emphasized that, although we arr iv ed at this concept in an indirect w a y (via double braidings and str ong twines), the pseudosymm etry relation do es not 2 dep end on these concepts and could ha ve b een introd u ced directly . An ywa y , this concept is supp orted and fu rther ju stiﬁed b y our main r esult: if H is a Hopf algebra (with b ijectiv e an tip o de) then the canonical br aiding of the Y etter-Drinfeld catego ry H Y D H is pseudosymmetric if and only if H is comm utativ e and co comm utativ e. In view of P areigis’ result mentio ned ab o ve, this sh o ws that pseudosymmetries are far more n umerous than symmetries; and in the opp osite dir ection, it shows that not ev ery br aiding is p seudosymmetric (this w as not so obvio u s a priori). Note also that, incidentall y , our theorem provides a c haracterization of commutativ e and co comm utativ e Hopf alge b ras solely in terms of their Y etter-Drinfeld categ ories. W e in tro du ce the Hopf algebraic count erp art of p seudosymmetric b r aidings, und er the n ame pseudotriangular structur e , as b eing a quasitriangular stru cture R on a Hopf algebra H satis- fying the mo d iﬁed quan tum Y ang-Baxter equation R 12 R − 1 31 R 23 = R 23 R − 1 31 R 12 (from whic h it is visible that triangular implies pseudotriangular) or equiv alen tly th e elemen t F = R 21 R sat- isﬁes the condition F 12 F 23 = F 23 F 12 , whic h sho ws immediately that almost-triangular implies pseudotriangular. W e analyze in detail a class of quasitriangular Hopf algebras, namely the 2 n +1 -dimensional p oin ted Hopf algebras E ( n ) whose qu asitriangular stru ctures and cleft exten- sions h a v e b een classiﬁed in [27] and [28]: we pr ov e that all qu asitriangular structures of E ( n ) (whic h are in b ijection w ith n × n matrices) are pseudotriangular, and th e only almost-triangular structures of E ( n ) are the triangular ones (whic h are in b ijection with symmetric n × n matrices); in particular, this sho ws that pseudotriangular do es not imply almost-triangular. Apart from leading us to consider a certain class of b raidings (the p seudosymmetric ones), the stud y of t wines led u s also to consider certain classes of pseudotwistors , as in tro du ced in [24]. In ord er to explain this, we need to in tro duce ﬁrst some term in ology . A basic ob ject w e use all o v er the pap er is a monoidal structure of the iden tit y fu nctor on a m on oidal category (for instance, th is is part of the axioms for t win es and str ong t win es). W e needed to hav e a name for s u c h an ob ject, and in order to c ho ose it w e relied on the f act that these ob jects are the cate gorical analog u es of lazy c o cycles , a concept recen tly introduced in Hopf algebra theory and studied in a series of pap ers ([1], [7], [8], [9], [10], [33]). Th u s, w e ha ve c hosen the name la ycle , as deriv ed f rom la zy co c ycle . Th ese layc les ha v e some prop erties similar to those of lazy cocycles, for instance they act by conjugation on braidings and it is p ossib le to deﬁ ne for them an analogue of the Hopf lazy cohomol ogy . The concept of p seudot wistor (with particular cases called twistor and br aide d twistor ) was in tro d uced in [24] as an abstract and axiomatic device for “t wisting” the multiplicatio n of an algebra in a monoidal category in order to obtain a new algebra structur e (on the same ob j ect). More precisely , if ( A, µ, u ) is an algebra in a monoidal category C , a pseud ot wistor for A is a morphism T : A ⊗ A → A ⊗ A in C , for whic h there exist t wo morphisms ˜ T 1 , ˜ T 2 : A ⊗ A ⊗ A → A ⊗ A ⊗ A in C , call ed the companions of T , satisfying a list of axioms ensu ring that ( A, µ ◦ T , u ) is also an algebra in C . Examples of p seudot wistors are abund an t, cf. [24]. F or instance, if c is a braiding on C , then c 2 A,A is a p seudot wistor for ev ery alge b r a A in C . Since a dou b le b raiding is in particular a twine, this raises the n atural question whether an y t wine induces a pseu d ot wistor on eve ry algebra in the category . It tu rns out th at something m ore general h olds , namely that an y la ycle has this prop ert y . This seems to sho w that pseud otwistors are “lo cal” versions of la ycles (in the same sens e in w h ic h t wisting maps are “lo cal” versions of braidings, see [19] f or the meaning of these concepts and references), bu t this is not quite tru e, b ecause f or instance a comp osition of la ycles is a la ycle while a comp osition of pseudot wistors is not in general a pseudot wistor. W e introdu ce th us the concept of str ong pseudotwistor , as a b etter candidate for b eing a lo cal version of la ycles (for instance, a comp osition of a strong p s eudot wistor with itself is again a strong p seudot wistor). W e also introd u ce a sort of lo cal version of t wines, und er 3 the name pur e pseudotwistor , as b eing a pseudotwistor whose companions satisfy the condition ( ˜ T 2 ⊗ id ) ◦ ( id ⊗ ˜ T 1 ) = ( id ⊗ ˜ T 1 ) ◦ ( ˜ T 2 ⊗ id ). Quite interestingly , it tu r ns out that virtually all the concrete examples of pseudot wistors we are aw are of are pure. What we discus s ed ab o v e are basically f acts ab out p seudot wistors inspired b y prop erties of la ycles and t wines. In the last section of the p ap er w e complete the p icture of the in terplay b e- t w een la ycles and t wines, on the one hand, and ps eu dot wistors, on the other hand, by presen ting a resu lt in the opp osite d irection. Namely , inspired by a result in [24] concerning pseud ot wistors and t wisting m ap s , we pro ve that, if C is a monoidal category , T a la ycle and d a b raiding on C related in a certain wa y , then the families d ′ X,Y = d X,Y ◦ T X,Y and d ′′ X,Y = T Y , X ◦ d X,Y are also br aidings on C . W e pr o v e also a sort of con v erse result, leading thus to a characte rization of generalized double braidings (i.e. twines of th e t yp e c ′ Y , X ◦ c X,Y , with c , c ′ braidings). 1 Preliminaries In th is section w e reca ll basic deﬁnitions and results and w e ﬁ x n otation to b e used throughout the pap er. All algebras, linear spaces, etc, will b e ov er a base ﬁeld k ; u nadorned ⊗ means ⊗ k . All monoidal categories are assumed to b e strict, with unit d enoted by I . F or a Hopf algebra H with comultiplica tion ∆ we denote ∆( h ) = h 1 ⊗ h 2 , for all h ∈ H . Unless otherwise stated, H will denote a Hopf algebra with bijectiv e antip o de S . F or terminology concerning Hopf algebras and monoidal catego ries we refer to [21], [26]. A linear map σ : H ⊗ H → k is called a left 2-co cycle if it satisﬁes the condition σ ( a 1 , b 1 ) σ ( a 2 b 2 , c ) = σ ( b 1 , c 1 ) σ ( a, b 2 c 2 ) , (1.1) for all a, b, c ∈ H , and it is called a right 2-co cycle if it satisﬁes the condition σ ( a 1 b 1 , c ) σ ( a 2 , b 2 ) = σ ( a, b 1 c 1 ) σ ( b 2 , c 2 ) . (1.2) Giv en a linear m ap σ : H ⊗ H → k , d eﬁ ne a pro d uct · σ on H by h · σ h ′ = σ ( h 1 , h ′ 1 ) h 2 h ′ 2 , for all h, h ′ ∈ H . Th en · σ is associativ e if and only if σ is a left 2-co cycle. If w e deﬁne · σ b y h · σ h ′ = h 1 h ′ 1 σ ( h 2 , h ′ 2 ), then · σ is asso ciativ e if and only if σ is a righ t 2-co cycle. In an y of the t w o cases, σ is normalized (i.e. σ (1 , h ) = σ ( h, 1) = ε ( h ) for all h ∈ H ) if and only if 1 H is the unit for · σ . If σ is a normalized left (resp ectiv ely righ t) 2-c o cycle, we d enote the algebra ( H, · σ ) b y σ H (resp ectiv ely H σ ). I t is well- kn own that σ H (resp ectiv ely H σ ) is a r igh t (r esp ectiv ely left) H -como du le alge br a via the comultiplica tion ∆ of H . If σ : H ⊗ H → k is normalized and con v olution in v ertible, then σ is a left 2-co cycle if and only if σ − 1 is a righ t 2-cocycle. If γ : H → k is linear, normalized (i.e. γ (1) = 1) and con vo lution in ve rtible, deﬁne D 1 ( γ ) : H ⊗ H → k , D 1 ( γ )( h, h ′ ) = γ ( h 1 ) γ ( h ′ 1 ) γ − 1 ( h 2 h ′ 2 ) , ∀ h, h ′ ∈ H . Then D 1 ( γ ) is a normalized and con v olution inv ertible left 2-cocycle. W e recall fr om [1] some facts ab out lazy co cycles and lazy cohomology . The set Reg 1 ( H ) (resp ectiv ely R eg 2 ( H )) consisting of normalized and con v olution in v ertible linear maps γ : H → k (resp ectiv ely σ : H ⊗ H → k ), is a group with r esp ect to the conv olution pro duct. An elemen t γ ∈ Reg 1 ( H ) is called lazy if γ ( h 1 ) h 2 = h 1 γ ( h 2 ), for all h ∈ H . The set of lazy elemen ts of Reg 1 ( H ), denoted by Reg 1 L ( H ), is a cen tral su bgroup of Reg 1 ( H ). An elemen t σ ∈ Reg 2 ( H ) is called lazy if σ ( h 1 , h ′ 1 ) h 2 h ′ 2 = h 1 h ′ 1 σ ( h 2 , h ′ 2 ) , ∀ h, h ′ ∈ H . (1.3) 4 The set of lazy elements of Reg 2 ( H ), denoted b y Reg 2 L ( H ), is a su bgroup of Reg 2 ( H ). W e denote by Z 2 ( H ) th e set of left 2-co cycles on H and by Z 2 L ( H ) the set Z 2 ( H ) ∩ R eg 2 L ( H ) of normalized and con v olution inv ertible lazy 2-cocycles. If σ ∈ Z 2 L ( H ), then the algebras σ H and H σ coincide and will b e denoted b y H ( σ ); moreo v er, H ( σ ) is an H -bicomodu le alge br a via ∆. It is w ell-kno wn that in general the s et Z 2 ( H ) of left 2-cocycles is not closed u nder con- v olution. One of the main features of lazy 2-cocycles is that th e set Z 2 L ( H ) is closed und er con v olution, and that the con vol u tion in v erse of an element σ ∈ Z 2 L ( H ) is again a lazy 2-cocycle, so Z 2 L ( H ) is a group under con vo lution. In particular, a lazy 2-co cycle is also a r igh t 2-co cycle. Consider now the map D 1 : Reg 1 ( H ) → Reg 2 ( H ), D 1 ( γ )( h, h ′ ) = γ ( h 1 ) γ ( h ′ 1 ) γ − 1 ( h 2 h ′ 2 ), for all h, h ′ ∈ H . T hen, b y [1], the map D 1 induces a group morphism Reg 1 L ( H ) → Z 2 L ( H ), with image con tained in the cen tre of Z 2 L ( H ); d enote b y B 2 L ( H ) th is cen tral sub group D 1 ( Reg 1 L ( H )) of Z 2 L ( H ) (its elemen ts are call ed lazy 2-cob oundaries ). Then d eﬁ ne the second lazy coho- mology group H 2 L ( H ) = Z 2 L ( H ) /B 2 L ( H ). Dually , an inv ertible elemen t T ∈ H ⊗ H is called a lazy twist if ( ε ⊗ id ) ( T ) = 1 = ( id ⊗ ε )( T ) , ( id ⊗ ∆)( T )(1 ⊗ T ) = (∆ ⊗ id )( T )( T ⊗ 1) , ∆( h ) T = T ∆ ( h ) , ∀ h ∈ H . As a consequence of these axioms we also ha v e (1 ⊗ T )( id ⊗ ∆)( T ) = ( T ⊗ 1)( ∆ ⊗ id )( T ). One can d eﬁ ne the analogues of Z 2 L ( H ), B 2 L ( H ) and H 2 L ( H ) with lazy twists instead of lazy co cycles; these will b e denoted resp ectiv ely by Z 2 LT ( H ), B 2 LT ( H ) an d H 2 LT ( H ). Remark 1.1 If C is a monoidal c ate gory and T X,Y : X ⊗ Y → X ⊗ Y is a family of natur al isomorph isms in C , the natur ality of T implies (for al l X, Y , Z ∈ C ): ( T X,Y ⊗ id Z ) ◦ T X ⊗ Y , Z = T X ⊗ Y , Z ◦ ( T X,Y ⊗ id Z ) , (1.4) ( id X ⊗ T Y , Z ) ◦ T X,Y ⊗ Z = T X,Y ⊗ Z ◦ ( id X ⊗ T Y , Z ) . (1.5) Deﬁnition 1.2 ([21]) L et C = ( C , ⊗ , I ) and D = ( D , ⊗ , I ) b e monoidal c ate gories. A monoidal functor fr om C to D is a triple ( F, ϕ 0 , ϕ 2 ) wher e F : C → D is a functor, ϕ 0 is an isomorphism in D fr om I to F ( I ) and ϕ 2 ( U, V ) : F ( U ) ⊗ F ( V ) → F ( U ⊗ V ) is a family of natur al isomorphisms in D indexe d by al l c ouples ( U, V ) of obje c ts in C such that, for al l U, V , W ∈ C : ϕ 2 ( U ⊗ V , W ) ◦ ( ϕ 2 ( U, V ) ⊗ id F ( W ) ) = ϕ 2 ( U, V ⊗ W ) ◦ ( id F ( U ) ⊗ ϕ 2 ( V , W )) , ϕ 2 ( I , U ) ◦ ( ϕ 0 ⊗ id F ( U ) ) = id F ( U ) , ϕ 2 ( U, I ) ◦ ( id F ( U ) ⊗ ϕ 0 ) = id F ( U ) . Deﬁnition 1.3 ([20]) L et C b e a monoidal c ate gory. A braiding on C c onsists of a family of natur al isomorphisms c X,Y : X ⊗ Y → Y ⊗ X in C such that, for al l X , Y , Z ∈ C : c X,Y ⊗ Z = ( id Y ⊗ c X,Z ) ◦ ( c X,Y ⊗ id Z ) , (1.6) c X ⊗ Y , Z = ( c X,Z ⊗ id Y ) ◦ ( id X ⊗ c Y , Z ) . (1.7) As c onse qu enc es of the axioms we also have c X,I = c I ,X = id X and the br aid r elation ( c Y , Z ⊗ id X ) ◦ ( id Y ⊗ c X,Z ) ◦ ( c X,Y ⊗ id Z ) = ( id Z ⊗ c X,Y ) ◦ ( c X,Z ⊗ id Y ) ◦ ( id X ⊗ c Y , Z ) . (1.8) If mor e over c satisﬁes c Y , X ◦ c X,Y = id X ⊗ Y , for al l X, Y ∈ C , then c is c al le d a symmetry . 5 Deﬁnition 1.4 ([12]) L et H b e a Hopf algebr a. An invertible element R ∈ H ⊗ H is c al le d a quasitriangular structure for H if (∆ ⊗ id )( R ) = R 13 R 23 , ( id ⊗ ∆)( R ) = R 13 R 12 , ( ε ⊗ id )( R ) = ( id ⊗ ε )( R ) = 1 , ∆ cop ( h ) R = R ∆( h ) , ∀ h ∈ H . If mor e over R satisﬁes R 21 R = 1 ⊗ 1 , then R is c al le d triangular . If R is a quasitriangular (r esp e ctively triangular) structur e for H , then the monoidal c ate gory H M of left H -mo dules b e c omes br aide d (r esp e ctively symmetric), with br aiding given by c M ,N : M ⊗ N → N ⊗ M , c M ,N ( m ⊗ n ) = R 2 · n ⊗ R 1 · m , for al l M , N ∈ H M , m ∈ M , n ∈ N . Deﬁnition 1.5 ([34]) L et C b e a monoidal c ate gory. A pure-braided structur e on C c onsists of two f amilies of natur al isomorphisms A U,V ,W : U ⊗ V ⊗ W → U ⊗ V ⊗ W and B U,V ,W : U ⊗ V ⊗ W → U ⊗ V ⊗ W in C such that (for al l U, V , W, X ∈ C ): A U ⊗ V ,W,X = A U,V ⊗ W,X ◦ ( id U ⊗ A V , W,X ) , (1.9) A U,V ,W ⊗ X = ( A U,V ,W ⊗ id X ) ◦ A U,V ⊗ W,X , (1.10) B U ⊗ V ,W,X = ( id U ⊗ B V , W,X ) ◦ B U,V ⊗ W,X , (1.11) B U,V ,W ⊗ X = B U,V ⊗ W,X ◦ ( B U,V ,W ⊗ id X ) , (1.12) ( A U,V ,W ⊗ id X ) ◦ ( id U ⊗ B V , W,X ) = ( id U ⊗ B V , W,X ) ◦ ( A U,V ,W ⊗ id X ) , (1.13) A U,I ,V = B U,I ,V . (1.14) A c ate g ory e quipp e d with a pur e-br aide d structur e is c al le d a pur e- br aide d c ate gory. Deﬁnition 1.6 ([5]) L et C b e a monoidal c ate gory. A t wine on C is a family of natur al iso- morphism s D X,Y : X ⊗ Y → X ⊗ Y in C satisfying the axioms (for al l X , Y , Z , W ∈ C ): D I ,I = id I , (1.15) ( D X,Y ⊗ id Z ) ◦ D X ⊗ Y , Z = ( id X ⊗ D Y , Z ) ◦ D X,Y ⊗ Z , (1.16) ( D X ⊗ Y , Z ⊗ id W ) ◦ ( id X ⊗ D − 1 Y , Z ⊗ id W ) ◦ ( id X ⊗ D Y , Z ⊗ W ) = ( id X ⊗ D Y , Z ⊗ W ) ◦ ( id X ⊗ D − 1 Y , Z ⊗ id W ) ◦ ( D X ⊗ Y , Z ⊗ id W ) . (1.17) A c ate g ory e quipp e d with a twine is c al le d an entwine d c ate gory. If ( C , D ) is entwine d then we also have D X,I = D I ,X = id X , for al l X ∈ C . By [29], these t w o concepts are equiv alen t in a certain (pr ecise) sense. Prop osition 1.7 ([5]) L et C b e a monoidal c ate gory and c , c ′ br aidings on C . Then the family T X,Y := c ′ Y , X ◦ c X,Y is a twine, c al le d a generalized double braiding ; if c = c ′ the family c Y , X ◦ c X,Y is c al le d a double braiding . Deﬁnition 1.8 ([29]) L et C b e a monoidal c ate gory and T X,Y : X ⊗ Y → X ⊗ Y a family of natur al isomorp hisms i n C . We say that T is a strong t wine (or ( C , T ) is str ongly entwine d) if for al l X, Y , Z ∈ C we have: T I ,I = id I , (1.18) ( T X,Y ⊗ id Z ) ◦ T X ⊗ Y , Z = ( id X ⊗ T Y , Z ) ◦ T X,Y ⊗ Z , (1.19) ( T X,Y ⊗ id Z ) ◦ ( id X ⊗ T Y , Z ) = ( id X ⊗ T Y , Z ) ◦ ( T X,Y ⊗ id Z ) . (1.20) 6 Prop osition 1.9 ([29]) If ( C , T ) is str ongly entwine d then ( C , T ) is entwine d. Prop osition 1.10 ([3], [4]) L et A b e an algebr a with multiplic ation denote d by µ A = µ and let T : A ⊗ A → A ⊗ A b e a line ar map satisfying the fol lowing c onditions: T (1 ⊗ a ) = 1 ⊗ a , T ( a ⊗ 1) = a ⊗ 1 , for al l a ∈ A , and µ 23 ◦ T 12 ◦ T 13 = T ◦ µ 23 : A ⊗ A ⊗ A → A ⊗ A, (1.21) µ 12 ◦ T 23 ◦ T 13 = T ◦ µ 12 : A ⊗ A ⊗ A → A ⊗ A, (1.22) T 12 ◦ T 13 ◦ T 23 = T 23 ◦ T 13 ◦ T 12 : A ⊗ A ⊗ A → A ⊗ A ⊗ A, (1.23) with standar d notation for µ ij and T ij . Then the map µ ◦ T : A ⊗ A → A deﬁnes an asso ciative algebr a structur e on A , with the same unit 1. The map T is c al le d an R -matrix for A . 2 La ycles and quasi-braidings Deﬁnition 2.1 L et C b e a monoidal c ate gory and T X,Y : X ⊗ Y → X ⊗ Y a family of natur al isomorph isms in C . We say that T is a la ycle if for al l X , Y , Z ∈ C we have: T I ,I = id I , (2.1) ( T X,Y ⊗ id Z ) ◦ T X ⊗ Y , Z = ( id X ⊗ T Y , Z ) ◦ T X,Y ⊗ Z . (2.2) A c ate g ory e quipp e d with a laycle is c al le d a laycle d c ate gory. Remark 2.2 It T is a laycle on C then we also have T X,I = T I ,X = id X , for al l X ∈ C . Also , it is cle ar that if ( C , T ) is entwine d then ( C , T ) is laycle d. Remark 2.3 It i s obvious that T is a laycle if and only if ( id C , id I , ϕ 2 ( X, Y ) := T X,Y ) is a monoidal functor fr om C to i tself. So, dir e ctly fr om the pr op erties of monoidal fu nctors, it fol lows that the c omp osition of two laycles is a laycle and the inverse of a laycle is a laycle. Example 2.4 Let H b e a Hopf algebra, σ ∈ R eg 2 L ( H ) and C = M H , the category of right H -c omo d ules, with tensor pro d uct ( m ⊗ n ) (0) ⊗ ( m ⊗ n ) (1) = ( m (0) ⊗ n (0) ) ⊗ m (1) n (1) . Deﬁn e T M ,N ( m ⊗ n ) = m (0) ⊗ n (0) σ ( m (1) , n (1) ), for all M , N ∈ M H , m ∈ M , n ∈ N . Then σ is a lazy 2-cocycle on H if and only if T is a layc le on M H . Du ally , if F = F 1 ⊗ F 2 ∈ H ⊗ H is inv ertible and satisﬁes ( ε ⊗ id )( F ) = ( id ⊗ ε )( F ) = 1, consider the category H M of left H -mo d ules, with tensor pro duct giv en b y h · ( m ⊗ n ) = h 1 · m ⊗ h 2 · n , f or all M , N ∈ H M , m ∈ M , n ∈ N ; deﬁn e T M ,N ( m ⊗ n ) = F 1 · m ⊗ F 2 · n . T hen F is a lazy t wist if and only if T is a layc le on H M . If T is a la ycle on C , we deﬁne th e families T b X,Y ,Z , T f X,Y ,Z : X ⊗ Y ⊗ Z → X ⊗ Y ⊗ Z (notation as in [5]) of natural isomorphisms in C asso ciated to it, by T b X,Y ,Z := ( id X ⊗ T − 1 Y , Z ) ◦ T X ⊗ Y , Z = T X,Y ⊗ Z ◦ ( T − 1 X,Y ⊗ id Z ) , (2.3) T f X,Y ,Z := T X ⊗ Y , Z ◦ ( id X ⊗ T − 1 Y , Z ) = ( T − 1 X,Y ⊗ id Z ) ◦ T X,Y ⊗ Z . (2.4) 7 Prop osition 2.5 L et C b e a monoid al c ate gory. (i) If T is a laycle on C then for al l U, V , W ∈ C we have T f U ⊗ V ,W,X = T f U,V ⊗ W,X ◦ ( id U ⊗ T f V , W,X ) , (2.5) T f U,V ,W ⊗ X = ( T f U,V ,W ⊗ id X ) ◦ T f U,V ⊗ W,X . (2.6) Conversely, if A U,V ,W : U ⊗ V ⊗ W → U ⊗ V ⊗ W is a family of natur al isomorphisms such that (2.5) and (2.6) with A inste ad of T f hold, then T U,V := A U,I ,V is a laycle on C . (ii) If T is a laycle on C then for al l U, V , W ∈ C we have T b U ⊗ V ,W,X = ( id U ⊗ T b V , W,X ) ◦ T b U,V ⊗ W,X , (2.7 ) T b U,V ,W ⊗ X = T b U,V ⊗ W,X ◦ ( T b U,V ,W ⊗ id X ) . (2.8) Conversely, if B U,V ,W : U ⊗ V ⊗ W → U ⊗ V ⊗ W is a family of natur al isomorphisms such that (2.7) and (2.8) with B inste ad of T b hold, then T U,V := B U,I ,V is a laycle on C . Pr o of. W e p r o v e (i), while (ii) is similar and left to th e reader. W e compute: T f U ⊗ V ,W,X = T U ⊗ V ⊗ W,X ◦ ( id U ⊗ id V ⊗ T − 1 W,X ) = T U ⊗ V ⊗ W,X ◦ ( id U ⊗ T − 1 V ⊗ W,X ) ◦ ( id U ⊗ T V ⊗ W,X ) ◦ ( id U ⊗ id V ⊗ T − 1 W,X ) ( 2.2 ) = T f U,V ⊗ W,X ◦ ( id U ⊗ T f V , W,X ) , pro ving (2.5); the p r o of of (2.6) is similar and left to the r eader. Assume no w that A − , − , − is a family of n atural isomorph isms satisfying (2. 5 ) and (2.6); then ob viously the family T U,V = A U,I ,V consists also of n atur al isomorp hisms. If in (2.5) w e tak e V = W = X = I w e obtain T U,I = T U,I ◦ ( id U ⊗ T I ,I ), hence T I ,I = id I . If w e tak e W = I in (2.5) and V = I in (2.6) we obtain T U ⊗ V ,X = A U,V ,X ◦ ( id U ⊗ T V , X ) , T U,W ⊗ X = ( T U,W ⊗ id X ) ◦ A U,W,X , whic h together imply (2.2).  The cat egorical analogue of the op erator D 1 from the Preliminaries lo oks as follo ws: Prop osition 2.6 ([5]) L et C b e a monoidal c ate gory and R X : X → X a family of natur al isomorph isms in C such that R I = id I . Then the family D 1 ( R ) X,Y := ( R X ⊗ R Y ) ◦ R − 1 X ⊗ Y = R − 1 X ⊗ Y ◦ ( R X ⊗ R Y ) (2.9) is a laycle on C . The next result (whose pro of is s tr aigh tforw ard and will b e omitted) pr o vides the categoric al analogue of Hopf lazy cohomolo gy: Prop osition 2.7 L et C b e a smal l monoidal c ate gory. Then: (i) If we denote by Reg 1 L ( C ) the set of f amilies of natur al isomorphisms R X : X → X in C such that R I = id I , then Reg 1 L ( C ) is an ab elian gr oup under c omp osition. (ii) The set of laycles on C is a gr oup, denote d by Z 2 L ( C ) . (iii) The map D 1 : Reg 1 L ( C ) → Z 2 L ( C ) is a gr oup morphism with image (denote d by B 2 L ( C ) ) c ontaine d in the c entr e of Z 2 L ( C ) . We denote by H 2 L ( C ) the gr oup Z 2 L ( C ) /B 2 L ( C ) , and c al l it the lazy c ohomolo gy of C . 8 A b asic prop ert y of lazy co cycles on Hopf algebras (see [1]) is that they act on co qu asitrian- gular structures. This prop ert y extends to the categorica l setting: Prop osition 2.8 L et C b e a monoidal c ate gory, T a laycle and c a b r aiding on C . Then the family c T X,Y := T Y , X ◦ c X,Y ◦ T − 1 X,Y is also a br aiding on C . Pr o of. The naturalit y of c with resp ect to th e morphism s id X and T − 1 Y , Z together with (1.6) imply ( T − 1 Y , Z ⊗ id X ) ◦ ( id Y ⊗ c X,Z ) ◦ ( c X,Y ⊗ id Z ) = ( id Y ⊗ c X,Z ) ◦ ( c X,Y ⊗ id Z ) ◦ ( id X ⊗ T − 1 Y , Z ) . (2.10) The naturalit y of c with resp ect to the morp h isms T − 1 X,Y and id Z together with (1.7) imply ( id Z ⊗ T − 1 X,Y ) ◦ ( c X,Z ⊗ id Y ) ◦ ( id X ⊗ c Y , Z ) = ( c X,Z ⊗ id Y ) ◦ ( id X ⊗ c Y , Z ) ◦ ( T − 1 X,Y ⊗ id Z ) . (2.11) W e c hec k (1.6) for c T ; w e compute: c T X,Y ⊗ Z = T Y ⊗ Z ,X ◦ c X,Y ⊗ Z ◦ T − 1 X,Y ⊗ Z ( 1.6 ) , ( 2.2 ) = ( id Y ⊗ T Z,X ) ◦ T Y , Z ⊗ X ◦ ( T − 1 Y , Z ⊗ id X ) ◦ ( id Y ⊗ c X,Z ) ◦ ( c X,Y ⊗ id Z ) ◦ ( id X ⊗ T Y , Z ) ◦ T − 1 X ⊗ Y , Z ◦ ( T − 1 X,Y ⊗ id Z ) ( 2.10 ) = ( id Y ⊗ T Z,X ) ◦ T Y , Z ⊗ X ◦ ( id Y ⊗ c X,Z ) ◦ ( c X,Y ⊗ id Z ) ◦ T − 1 X ⊗ Y , Z ◦ ( T − 1 X,Y ⊗ id Z ) natural ity of T = ( id Y ⊗ T Z,X ) ◦ ( id Y ⊗ c X,Z ) ◦ T Y , X ⊗ Z ◦ T − 1 Y ⊗ X,Z ◦ ( c X,Y ⊗ id Z ) ◦ ( T − 1 X,Y ⊗ id Z ) ( 2.2 ) = ( id Y ⊗ T Z,X ) ◦ ( id Y ⊗ c X,Z ) ◦ ( id Y ⊗ T − 1 X,Z ) ◦ ( T Y , X ⊗ id Z ) ◦ ( c X,Y ⊗ id Z ) ◦ ( T − 1 X,Y ⊗ id Z ) = ( id Y ⊗ c T X,Z ) ◦ ( c T X,Y ⊗ id Z ) , q.e.d. Similarly , w e c hec k (1.7) for c T : c T X ⊗ Y , Z = T Z,X ⊗ Y ◦ c X ⊗ Y , Z ◦ T − 1 X ⊗ Y , Z ( 1.7 ) , ( 2.2 ) = ( T Z,X ⊗ id Y ) ◦ T Z ⊗ X,Y ◦ ( id Z ⊗ T − 1 X,Y ) ◦ ( c X,Z ⊗ id Y ) ◦ ( id X ⊗ c Y , Z ) ◦ ( T X,Y ⊗ id Z ) ◦ T − 1 X,Y ⊗ Z ◦ ( id X ⊗ T − 1 Y , Z ) ( 2.11 ) = ( T Z,X ⊗ id Y ) ◦ T Z ⊗ X,Y ◦ ( c X,Z ⊗ id Y ) ◦ ( id X ⊗ c Y , Z ) ◦ T − 1 X,Y ⊗ Z ◦ ( id X ⊗ T − 1 Y , Z ) natural ity of T = ( T Z,X ⊗ id Y ) ◦ ( c X,Z ⊗ id Y ) ◦ T X ⊗ Z,Y ◦ T − 1 X,Z ⊗ Y ◦ ( id X ⊗ c Y , Z ) ◦ ( id X ⊗ T − 1 Y , Z ) ( 2.2 ) = ( T Z,X ⊗ id Y ) ◦ ( c X,Z ⊗ id Y ) ◦ ( T − 1 X,Z ⊗ id Y ) ◦ ( id X ⊗ T Z,Y ) ◦ ( id X ⊗ c Y , Z ) ◦ ( id X ⊗ T − 1 Y , Z ) = ( c T X,Z ⊗ id Y ) ◦ ( id X ⊗ c T Y , Z ) , 9 ﬁnishing the pro of.  Prop osition 2.9 L et C b e a monoidal c ate gory, c a br aiding on C and R X : X → X a family of natur al isomorphisms in C such that R I = id I . Then c D 1 ( R ) = c , wher e D 1 ( R ) is the laycle given b y (2.9). Pr o of. F ollo ws immediately by usin g the naturality of c and R .  Corollary 2.10 If C is a smal l monoidal c ate gory, then the gr oup H 2 L ( C ) acts on the set of br aidings of C . Prop osition 2.11 In the hyp otheses of P r op osition 2.8 , the br aide d monoidal c ate gories ( C , c ) and ( C , c T ) ar e e quiv alent (as br aide d monoida l c ate gories). Pr o of. W e d eﬁ ne th e monoidal functor ( F , ϕ 0 , ϕ 2 ) : C → C b y F = id C , ϕ 0 = id I and ϕ 2 ( X, Y ) : X ⊗ Y → X ⊗ Y , ϕ 2 ( X, Y ) := T − 1 X,Y , whic h is obviously a monoidal equiv alence. Moreo ver, the form ula c T X,Y = T Y , X ◦ c X,Y ◦ T − 1 X,Y expresses exactly the fact that ( F , ϕ 0 , ϕ 2 ) is a b raided functor from ( C , c ) to ( C , c T )  If C is a b raided monoidal catego ry with br aidin g c , w e denote b y B r ( C , c ) its Brauer group as introdu ced in [35]. Thus, as a consequence of Prop osition 2.11, we obtain the follo wing generalizat ion of [6], Prop ositio n 3.1: Corollary 2.12 In the hyp otheses of Pr op osition 2.8, the Br auer gr oups B r ( C , c ) and B r ( C , c T ) ar e isomorphic. Deﬁnition 2.13 L et C b e a monoidal c ate gory. A quasi-braiding on C is a family of natur al isomorph isms q X,Y : X ⊗ Y → Y ⊗ X in C satisfying the fol lowing axioms (for al l X , Y , Z ∈ C ): q I ,I = id I , (2.12) q X,Z ⊗ Y ◦ ( id X ⊗ q Y , Z ) = q Y ⊗ X ,Z ◦ ( q X,Y ⊗ id Z ) . (2.13) If q Y , X ◦ q X,Y = id X ⊗ Y for al l X , Y ∈ C , then ( C , q ) is what Drinfeld c al ls a cob oundary category i n [13]. Remark 2.14 If q is a q u asi-br aiding on C then we also have q X,I = q I ,X = id X and ( q Y , Z ⊗ id X ) ◦ q X,Y ⊗ Z = q X,Z ⊗ Y ◦ ( id X ⊗ q Y , Z ) = q Y ⊗ X,Z ◦ ( q X,Y ⊗ id Z ) = ( id Z ⊗ q X,Y ) ◦ q X ⊗ Y , Z . (2.14) Conse quently, the family p X,Y := q − 1 Y , X is also a quasi-br aiding. The concept of qu asi-braiding w as considered (with a diﬀerent name) b y L. M. Ionescu in [18], as follo ws. Deﬁne a monoidal category C op , wh ich is the same as C as a category , h as the same unit I , and rev ersed tensor p r o duct: X ⊗ op Y = Y ⊗ X . Then, a family q X,Y : X ⊗ Y → Y ⊗ X is a quasi-braidin g on C if and only if ( id C , id I , ϕ 2 ( X, Y ) := q X,Y ) is a monoidal functor from C op to C , or equiv alent ly ( id C , id I , ϕ 2 ( X, Y ) := q Y , X ) is a monoidal fun ctor from C to C op . As noted in [18], an y braiding is a quasi-braiding (this follo ws easily by (1.6), (1.7) and (1.8)), and quasi-braidings are related to Drinfeld’s cob oundary Hopf algebras: 10 Deﬁnition 2.15 ([12]) A c ob oundary Hopf algebr a is a p air ( H , R ) , wher e H is a Hopf algebr a and R ∈ H ⊗ H is an invertible element such that: R 12 (∆ ⊗ id )( R ) = R 23 ( id ⊗ ∆)( R ) , (2.15) ( ε ⊗ id )( R ) = ( id ⊗ ε )( R ) = 1 , (2.16) ∆ cop ( h ) R = R ∆( h ) , ∀ h ∈ H , (2.17) R 21 R = 1 ⊗ 1 . (2.18) If R do es not ne c essarily satisfy (2.18), we c al l it a quasi-cob oundary . Prop osition 2.16 ([18]) L et H b e a Hopf algebr a and R = R 1 ⊗ R 2 ∈ H ⊗ H an invertible element. If U, V ar e left H -mo dules, deﬁne q U,V : U ⊗ V → V ⊗ U by q U,V ( u ⊗ v ) = R 2 · v ⊗ R 1 · u . Then q is a quasi-br aiding on H M if and only if R is a quasi-c ob oundary on H . Remark 2.17 If T is a laycle on a monoidal c ate gory C , then the family T X,Y := T Y , X is a laycle on C op . F rom the description of layc les and q u asi-braidings as monoidal stru ctures for some iden tit y functors and the fact that a composition of mon oidal fu nctors is monoidal, w e obtain: Prop osition 2.18 L et C b e a monoidal c ate gory, T a laycle and p , q two quasi-br aidings on C . Then the family D X,Y := p Y , X ◦ q X,Y is a laycle on C and the families q ′ X,Y := T Y , X ◦ q X,Y and q ′′ X,Y := q X,Y ◦ T X,Y ar e quasi-b r aidings on C . Corollary 2.19 L et C b e a monoidal c ate gory, T a laycle and q a qu asi-br aiding on C . Then the f amily q T X,Y := T Y , X ◦ q X,Y ◦ T − 1 X,Y is also a quasi-br aiding on C . Remark 2.20 Pr op osition 2.9 is also true with q u asi-br aidings inste ad of br aidings, so we obtain also an action of H 2 L ( C ) on the set of quasi-br aidings of C . Prop osition 2.21 L et C b e a monoidal c ate gory, c a br aiding and q a quasi-br aiding on C . Then the family c q X,Y := q − 1 Y , X ◦ c Y , X ◦ q X,Y is also a br aiding on C . Mor e over, the br aide d c ate gories ( C , c ) and ( C , c q ) ar e br aide d e q u ivalent. Pr o of. F ollo ws immediately from Prop osition 2.8, since c q = c T , wher e T is the la ycle T X,Y = q − 1 X,Y ◦ c X,Y .  Remark 2.22 F or the p articular c ase when q itself is a br aiding, we wil l obtain an alternative pr o of in P r op osition 5.3. Let now C b e a small monoidal category . W e denote by Z 2 ( C ) the s et of all natural isomor- phisms in C th at are la ycles or qu asi-braidings. Then, with notation as in Prop ositio n 2.7, w e ha ve : Prop osition 2.23 (i) Z 2 ( C ) is a gr oup. (ii) Z 2 L ( C ) is an index 2 sub gr oup in Z 2 ( C ) . (iii) B 2 L ( C ) is a c e ntr al sub g r oup in Z 2 ( C ) . We deﬁne the “c ohomolo gy gr oup” H 2 ( C ) := Z 2 ( C ) /B 2 L ( C ) . 11 Pr o of. W e giv e ﬁ rst the explicit descrip tion of th e m ultiplication in Z 2 ( C ). T ake R and P quasi-braidings, S and T la ycles. W e ha ve , for all U, V ∈ C : ( S T ) U,V = S U,V ◦ T U,V , ( T R ) U,V = T V , U ◦ R U,V , ( RT ) U,V = R U,V ◦ T U,V , ( RP ) U,V = R V , U ◦ P U,V . No w (i) and (ii) follo w fr om Prop osition 2.18, while (iii) is just an easy computatio n.  Similarly , if H is a Hopf algebra, we may consider the group Z 2 ( H ) consisting of the elemen ts in H ⊗ H that are lazy t wists or quasi-cob oundaries, its cen tral s u bgroup B 2 LT ( H ) and the “cohomolog y group” H 2 ( H ) = Z 2 ( H ) /B 2 LT ( H ). Example 2.24 Let k b e a ﬁeld with char ( k ) 6 = 2 and H = k [ C 2 ], the group algebra of the cyclic group with tw o elements C 2 (denote its generato r by g ). One can see th at the lazy t wists on H are given b y the formula T a = 3+ a 4 (1 ⊗ 1) + 1 − a 4 (1 ⊗ g ) + 1 − a 4 ( g ⊗ 1) − 1 − a 4 ( g ⊗ g ), with a ∈ k ∗ . It is interesting to note that T 0 is not inv ertible b ut has all the other prop erties in the deﬁnition of a lazy t wist. Consider the ele ment θ α = 1+ g 2 + α 1 − g 2 ∈ H , with α ∈ k . One can see that θ α is in vertible if and only if α 6 = 0. Also it is easy to see that T α − 2 = ∆( θ α )( θ − 1 α ⊗ θ − 1 α ) and so T a is trivial in H 2 ( H ) if and only if a ∈ ( k ∗ ) 2 . O ne can also note that T a T b = T ab . Since H is comm utativ e and co commutativ e, one can see that the q u asi-cob oundaries for H are giv en b y the form ula R a = 3+ a 4 (1 ⊗ 1) + 1 − a 4 (1 ⊗ g ) + 1 − a 4 ( g ⊗ 1) − 1 − a 4 ( g ⊗ g ), with a ∈ k ∗ . Among these, only R 1 and R − 1 are qu asitriangular. If w e put ev erything together we obtain H 2 ( H ) = k ∗ / ( k ∗ ) 2 × C 2 . 3 Strong t wines and pseudosymmetric b raidings A k ey result for this section is the follo w ing charac terization of strong t wines: Prop osition 3.1 L et C b e a monoid al c ate gory and T a laycle on C . Then T is a str ong twine if and only if the families T b and T f given b y (2.3) and (2.4) c oincide. Pr o of. Let X, Y , Z ∈ C and assume that T is a strong t wine; then we h a v e: T b X,Y ,Z = ( id X ⊗ T − 1 Y , Z ) ◦ T X ⊗ Y , Z = ( T − 1 X,Y ⊗ id Z ) ◦ ( T X,Y ⊗ id Z ) ◦ ( id X ⊗ T − 1 Y , Z ) ◦ T X ⊗ Y , Z ( 1.20 ) = ( T − 1 X,Y ⊗ id Z ) ◦ ( id X ⊗ T − 1 Y , Z ) ◦ ( T X,Y ⊗ id Z ) ◦ T X ⊗ Y , Z ( 1.19 ) = ( T − 1 X,Y ⊗ id Z ) ◦ ( id X ⊗ T − 1 Y , Z ) ◦ ( id X ⊗ T Y , Z ) ◦ T X,Y ⊗ Z = ( T − 1 X,Y ⊗ id Z ) ◦ T X,Y ⊗ Z = T f X,Y ,Z . Con versely , assume that T b = T f . By u sing (2.3), (2.2) and (2.4) it is easy to see that T b X,Y ,Z ◦ ( T X,Y ⊗ id Z ) ◦ ( id X ⊗ T Y , Z ) = T f X,Y ,Z ◦ ( id X ⊗ T Y , Z ) ◦ ( T X,Y ⊗ id Z ), and since T b = T f it follo ws that (1.20) holds.  12 Deﬁnition 3.2 ([2]) L et C b e a monoidal c ate g ory. A D- structur e on C c onsists of a family of natur al morphisms R X : X → X in C , such that R I = id I and (for al l X, Y , Z ∈ C ): ( R X ⊗ Y ⊗ id Z )( id X ⊗ R Y ⊗ Z ) = ( id X ⊗ R Y ⊗ Z )( R X ⊗ Y ⊗ id Z ) . (3.1) It w as pr o v ed in [29] that if R is a D -structure consisting of isomorphism s then the family D 1 ( R ) giv en by (2.9) is a strong twine. Using Prop ositio n 3.1 w e can pro ve th e con v erse: Prop osition 3.3 L et C b e a monoidal c ate gory and R X : X → X a family of natur al isomor- phisms in C with R I = id I . Then D 1 ( R ) is a str ong twine if and only if R is a D -structur e. Pr o of. W e compu te: D 1 ( R ) b X,Y ,Z = ( id X ⊗ R Y ⊗ Z ) ◦ ( id X ⊗ R − 1 Y ⊗ R − 1 Z ) ◦ ( R X ⊗ Y ⊗ R Z ) ◦ R − 1 X ⊗ Y ⊗ Z = ( id X ⊗ R Y ⊗ Z ) ◦ ( id X ⊗ R − 1 Y ⊗ R − 1 Z ) ◦ ( id X ⊗ R Y ⊗ R Z ) ◦ ( id X ⊗ R − 1 Y ⊗ id Z ) ◦ ( R X ⊗ Y ⊗ id Z ) ◦ R − 1 X ⊗ Y ⊗ Z = ( id X ⊗ R Y ⊗ Z ) ◦ ( id X ⊗ R − 1 Y ⊗ id Z ) ◦ ( R X ⊗ Y ⊗ id Z ) ◦ R − 1 X ⊗ Y ⊗ Z natural ity of R = ( id X ⊗ R − 1 Y ⊗ id Z ) ◦ ( id X ⊗ R Y ⊗ Z ) ◦ ( R X ⊗ Y ⊗ id Z ) ◦ R − 1 X ⊗ Y ⊗ Z , and similarly one can see that D 1 ( R ) f X,Y ,Z = ( id X ⊗ R − 1 Y ⊗ id Z ) ◦ ( R X ⊗ Y ⊗ id Z ) ◦ ( id X ⊗ R Y ⊗ Z ) ◦ R − 1 X ⊗ Y ⊗ Z , and it is clear that D 1 ( R ) b = D 1 ( R ) f (i.e. D 1 ( R ) is a strong twine) if and on ly if (3.1) holds.  W e recall that a (generalized) d ouble braiding is alw a ys a t wine; it is natural to ask und er what conditions is it a strong t wine. The answ er is provided by our n ext result: Theorem 3.4 L et C b e a monoid al c ate gory, c and d br aidings on C and T X,Y = d Y , X ◦ c X,Y . Then T is a str ong twine if and only if the fol lowing r elation holds, for al l X , Y , Z ∈ C : ( d Z,X ⊗ id Y ) ◦ ( id Z ⊗ c − 1 X,Y ) ◦ ( c Y , Z ⊗ id X ) ◦ ( d Z,Y ⊗ id X ) ◦ ( id Z ⊗ c X,Y ) ◦ ( c X,Z ⊗ id Y ) = ( id X ⊗ c Y , Z ) ◦ ( d − 1 X,Y ⊗ id Z ) ◦ ( id Y ⊗ d Z,X ) ◦ ( id Y ⊗ c X,Z ) ◦ ( d X,Y ⊗ id Z ) ◦ ( id X ⊗ d Z,Y ) . (3.2) Pr o of. W e compu te the f amilies T b and T f : T b X,Y ,Z = ( id X ⊗ T − 1 Y , Z ) ◦ T X ⊗ Y , Z ( 1.6 ) = ( id X ⊗ c − 1 Y , Z ) ◦ ( id X ⊗ d − 1 Z,Y ) ◦ ( id X ⊗ d Z,Y ) ◦ ( d Z,X ⊗ id Y ) ◦ c X ⊗ Y , Z = ( id X ⊗ c − 1 Y , Z ) ◦ ( d Z,X ⊗ id Y ) ◦ c X ⊗ Y , Z , T f X,Y ,Z = T X ⊗ Y , Z ◦ ( id X ⊗ T − 1 Y , Z ) ( 1.7 ) = d Z,X ⊗ Y ◦ ( c X,Z ⊗ id Y ) ◦ ( id X ⊗ c Y , Z ) ◦ ( id X ⊗ c − 1 Y , Z ) ◦ ( id X ⊗ d − 1 Z,Y ) = d Z,X ⊗ Y ◦ ( c X,Z ⊗ id Y ) ◦ ( id X ⊗ d − 1 Z,Y ) . 13 By Prop osition 3.1, T is a strong twine if and only if T b = T f , and this holds if and only if ( d Z,X ⊗ id Y ) ◦ c X ⊗ Y , Z ◦ ( id X ⊗ d Z,Y ) = ( id X ⊗ c Y , Z ) ◦ d Z,X ⊗ Y ◦ ( c X,Z ⊗ id Y ) . (3.3) Th u s, it is enough to p ro ve that the left h and sides of equations (3.2) and (3.3) coincide, and the same for the righ t hand sides. W e compute: ( d Z,X ⊗ id Y ) ◦ c X ⊗ Y , Z ◦ ( id X ⊗ d Z,Y ) = ( d Z,X ⊗ id Y ) ◦ ( id Z ⊗ c − 1 X,Y ) ◦ ( c Y , Z ⊗ id X ) ◦ c X,Y ⊗ Z ◦ ( id X ⊗ d Z,Y ) = ( d Z,X ⊗ id Y ) ◦ ( id Z ⊗ c − 1 X,Y ) ◦ ( c Y , Z ⊗ id X ) ◦ ( d Z,Y ⊗ id X ) ◦ c X,Z ⊗ Y = ( d Z,X ⊗ id Y ) ◦ ( id Z ⊗ c − 1 X,Y ) ◦ ( c Y , Z ⊗ id X ) ◦ ( d Z,Y ⊗ id X ) ◦ ( id Z ⊗ c X,Y ) ◦ ( c X,Z ⊗ id Y ) (for the ﬁrst equalit y w e used (2.14), for the s econd the naturalit y of c and for th e third (1.6)), ( id X ⊗ c Y , Z ) ◦ d Z,X ⊗ Y ◦ ( c X,Z ⊗ id Y ) = ( id X ⊗ c Y , Z ) ◦ ( d − 1 X,Y ⊗ id Z ) ◦ ( id Y ⊗ d Z,X ) ◦ d Z ⊗ X,Y ◦ ( c X,Z ⊗ id Y ) = ( id X ⊗ c Y , Z ) ◦ ( d − 1 X,Y ⊗ id Z ) ◦ ( id Y ⊗ d Z,X ) ◦ ( id Y ⊗ c X,Z ) ◦ d X ⊗ Z,Y = ( id X ⊗ c Y , Z ) ◦ ( d − 1 X,Y ⊗ id Z ) ◦ ( id Y ⊗ d Z,X ) ◦ ( id Y ⊗ c X,Z ) ◦ ( d X,Y ⊗ id Z ) ◦ ( id X ⊗ d Z,Y ) (for the ﬁ rst equalit y w e used (2.1 4 ), for the second the naturalit y of d and for the third (1.7)), ﬁnishing the pro of.  Deﬁnition 3.5 L et C b e a monoidal c ate gory and c a br aiding on C . We wil l say that c is a pseudosymmetry if the fol lowing c ondition holds, for al l X, Y , Z ∈ C : ( c Y , Z ⊗ id X ) ◦ ( id Y ⊗ c − 1 Z,X ) ◦ ( c X,Y ⊗ id Z ) = ( id Z ⊗ c X,Y ) ◦ ( c − 1 Z,X ⊗ id Y ) ◦ ( id X ⊗ c Y , Z ) . (3.4) In this c ase we wil l say that C i s a pseudosymmetric braided category . If c is a sym m etry , i.e. c − 1 Z,X = c X,Z , then ob viously c is a pseudosymmetry , b y (1.8). Theorem 3.6 L et C b e a monoidal c ate gory and c a br aiding on C . Then the double br aiding T X,Y = c Y , X ◦ c X,Y is a str ong twine if and only if c i s a pseudosymmetry. Pr o of. In (3.2) written for c = d w e ha ve, b y (1.8), ( c Z,Y ⊗ id X ) ◦ ( id Z ⊗ c X,Y ) ◦ ( c X,Z ⊗ id Y ) = ( id Y ⊗ c X,Z ) ◦ ( c X,Y ⊗ id Z ) ◦ ( id X ⊗ c Z,Y ) , so (3.2) reduces in this case to (3.4).  Let H b e a Hopf algebra. Consid er the category H Y D H of left-right Y etter-D rin feld mod ules o v er H , whose ob jects are ve ctor spaces M that are left H -mo dules (denote the action by h ⊗ m 7→ h · m ) and righ t H -comodu les (denote the coac tion b y m 7→ m (0) ⊗ m (1) ∈ M ⊗ H ) satisfying the compatibilit y condition ( h · m ) (0) ⊗ ( h · m ) (1) = h 2 · m (0) ⊗ h 3 m (1) S − 1 ( h 1 ) , ∀ h ∈ H , m ∈ M . (3.5) 14 It is a monoidal categ ory , with tensor p ro du ct giv en b y h · ( m ⊗ n ) = h 1 · m ⊗ h 2 · n, ( m ⊗ n ) (0) ⊗ ( m ⊗ n ) (1) = m (0) ⊗ n (0) ⊗ n (1) m (1) . Moreo v er, it has a (canonica l) braiding giv en by c M ,N : M ⊗ N → N ⊗ M , c M ,N ( m ⊗ n ) = n (0) ⊗ n (1) · m, c − 1 M ,N : N ⊗ M → M ⊗ N , c − 1 M ,N ( n ⊗ m ) = S ( n (1) ) · m ⊗ n (0) . It is k n o wn (cf. [30]) that this braiding is a symmetry only in the degenerate case H = k . Theorem 3.7 The c anonic al br aiding of H Y D H is pseudosym metric if and only if H is c om- mutative and c o c ommutative. Pr o of. Assume ﬁrst that H is comm utativ e and co comm utativ e; in this case, the compatibilit y condition (3.5) b ecomes the Long condition ( h · m ) (0) ⊗ ( h · m ) (1) = h · m (0) ⊗ m (1) , ∀ h ∈ H , m ∈ M . (3.6) F or all X , Y , Z ∈ H Y D H , x ∈ X , y ∈ Y , z ∈ Z we compu te: ( c Y , Z ⊗ id X ) ◦ ( id Y ⊗ c − 1 Z,X ) ◦ ( c X,Y ⊗ id Z )( x ⊗ y ⊗ z ) = ( c Y , Z ⊗ id X ) ◦ ( id Y ⊗ c − 1 Z,X )( y (0) ⊗ y (1) · x ⊗ z ) = ( c Y , Z ⊗ id X )( y (0) ⊗ S (( y (1) · x ) (1) ) · z ⊗ ( y (1) · x ) (0) ) ( 3.6 ) = ( c Y , Z ⊗ id X )( y (0) ⊗ S ( x (1) ) · z ⊗ y (1) · x (0) ) = ( S ( x (1) ) · z ) (0) ⊗ ( S ( x (1) ) · z ) (1) · y (0) ⊗ y (1) · x (0) ( 3.6 ) = S ( x (1) ) · z (0) ⊗ z (1) · y (0) ⊗ y (1) · x (0) ( 3.6 ) = S ( x (1) ) · z (0) ⊗ ( z (1) · y ) (0) ⊗ ( z (1) · y ) (1) · x (0) = ( id Z ⊗ c X,Y )( S ( x (1) ) · z (0) ⊗ x (0) ⊗ z (1) · y ) = ( id Z ⊗ c X,Y ) ◦ ( c − 1 Z,X ⊗ id Y )( x ⊗ z (0) ⊗ z (1) · y ) = ( id Z ⊗ c X,Y ) ◦ ( c − 1 Z,X ⊗ id Y ) ◦ ( id X ⊗ c Y , Z )( x ⊗ y ⊗ z ) , pro ving that c is pseud osymmetric. Con versely , assu me that c is pseudosymmetric. W e consider the tw o usual Y etter-Drinfeld structures on th e v ector space H : the ﬁrst one, denoted by H 1 , is H w ith the u sual (regular) left mo du le structure and with como d u le structur e ρ 1 ( h ) = h 2 ⊗ h 3 S − 1 ( h 1 ), and the second, denoted by H 2 , is H with mo du le structure given by h · g = h 2 g S − 1 ( h 1 ) and como d ule structur e ρ 2 ( h ) = h 1 ⊗ h 2 . W e prov e ﬁrst that H is co commutativ e. Let h ∈ H ; we will apply the pseudosymm etry condition (3.4) for X = H 1 , Y = H 2 , Z = H 1 on the elemen t 1 ⊗ h ⊗ 1: ( c Y , Z ⊗ id X ) ◦ ( id Y ⊗ c − 1 Z,X ) ◦ ( c X,Y ⊗ id Z )(1 ⊗ h ⊗ 1) = ( c Y , Z ⊗ id X ) ◦ ( id Y ⊗ c − 1 Z,X )( h 1 ⊗ h 2 ⊗ 1) = ( c Y , Z ⊗ id X )( h 1 ⊗ h 2 S ( h 4 ) ⊗ h 3 ) = ( h 2 S ( h 4 )) 2 ⊗ [( h 2 S ( h 4 )) 3 S − 1 (( h 2 S ( h 4 )) 1 )] · h 1 ⊗ h 3 , 15 ( id Z ⊗ c X,Y ) ◦ ( c − 1 Z,X ⊗ id Y ) ◦ ( id X ⊗ c Y , Z )(1 ⊗ h ⊗ 1) = ( id Z ⊗ c X,Y ) ◦ ( c − 1 Z,X ⊗ id Y )(1 ⊗ 1 ⊗ h ) = ( id Z ⊗ c X,Y )(1 ⊗ 1 ⊗ h ) = 1 ⊗ h 1 ⊗ h 2 , so w e obtain ( h 2 S ( h 4 )) 2 ⊗ [( h 2 S ( h 4 )) 3 S − 1 (( h 2 S ( h 4 )) 1 )] · h 1 ⊗ h 3 = 1 ⊗ h 1 ⊗ h 2 . By app lying id ⊗ ε ⊗ id w e get h 1 S ( h 3 ) ⊗ h 2 = 1 ⊗ h, whic h, by making con vo lution with S ( h ) ⊗ 1, b ecomes S ( h 1 ) h 2 S ( h 4 ) ⊗ h 3 = S ( h 1 ) ⊗ h 2 , and so we obtain S ( h 2 ) ⊗ h 1 = S ( h 1 ) ⊗ h 2 , whic h implies ∆ cop ( h ) = ∆( h ), i.e. H is co comm utativ e. W e prov e now that H is comm utativ e. Note ﬁrst that cocomm utativit y implies c H 2 ,H 1 ( b ⊗ a ) = a ⊗ b , for all a, b ∈ H . Let no w g , h ∈ H ; we w ill apply the p seudosymmetry condition (3.4) for X = H 1 , Y = H 2 , Z = H 2 on the elemen t 1 ⊗ g ⊗ h : ( c Y , Z ⊗ id X ) ◦ ( id Y ⊗ c − 1 Z,X ) ◦ ( c X,Y ⊗ id Z )(1 ⊗ g ⊗ h ) = ( c Y , Z ⊗ id X ) ◦ ( id Y ⊗ c − 1 Z,X )( g 1 ⊗ g 2 ⊗ h ) = ( c Y , Z ⊗ id X )( g 1 ⊗ h ⊗ g 2 ) = h 1 ⊗ h 3 g 1 S − 1 ( h 2 ) ⊗ g 2 , ( id Z ⊗ c X,Y ) ◦ ( c − 1 Z,X ⊗ id Y ) ◦ ( id X ⊗ c Y , Z )(1 ⊗ g ⊗ h ) = ( id Z ⊗ c X,Y ) ◦ ( c − 1 Z,X ⊗ id Y )(1 ⊗ h 1 ⊗ h 3 g S − 1 ( h 2 )) = ( id Z ⊗ c X,Y )( h 1 ⊗ 1 ⊗ h 3 g S − 1 ( h 2 )) = h 1 ⊗ ( h 3 g S − 1 ( h 2 )) 1 ⊗ ( h 3 g S − 1 ( h 2 )) 2 , and so w e obtain h 1 ⊗ h 3 g 1 S − 1 ( h 2 ) ⊗ g 2 = h 1 ⊗ ( h 3 g S − 1 ( h 2 )) 1 ⊗ ( h 3 g S − 1 ( h 2 )) 2 . By applying id ⊗ ε ⊗ id w e get h 1 ⊗ h 3 g S − 1 ( h 2 ) = h ⊗ g , wh ic h implies h 3 g S − 1 ( h 2 ) h 1 = g h, that is hg = g h and hence H is comm utativ e.  Corollary 3.8 F or H a c ommutative and c o c ommutative Hopf algebr a, the double br aiding T X,Y ( x ⊗ y ) = ( c Y , X ◦ c X,Y )( x ⊗ y ) = y (1) · x (0) ⊗ x (1) · y (0) is a str ong twine on H Y D H . Deﬁnition 3.9 If H is a Hopf algebr a and R ∈ H ⊗ H is a quasitriangular structur e, we wil l say that R is pseudotriangular i f R 12 R − 1 31 R 23 = R 23 R − 1 31 R 12 . (3.7) If R is a pseudotriangular stru cture then it is easy to see that the br aidin g on H M giv en b y c M ,N : M ⊗ N → N ⊗ M , c M ,N ( m ⊗ n ) = R 2 · n ⊗ R 1 · m , is pseud osymmetric. Also, it is ob vious that if R is triangular (i.e. R 21 R = 1 ⊗ 1) then R is pseud otriangular, b ecause in this case (3.7) b ecomes the quan tum Y ang-Baxter equ ation R 12 R 13 R 23 = R 23 R 13 R 12 . W e ha ve also the Hopf-algebraic coun terpart of Theorem 3.6: 16 Prop osition 3.10 L et ( H, R ) b e a quasitriangular Hopf algebr a. Then R is pseudotriangular if and only if the lazy twist F = R 21 R satisﬁes the c ondition F 12 F 23 = F 23 F 12 (i.e. F is n eat , in the terminolo gy of [29]). Example 3.11 If H is a commutativ e Hopf alge br a, then an y quasitriangular structure on H is pseu dotriangular. F or instance, if k has c haracteristic zero and cont ains a primitive ro ot of unit y of degree n , then the group algebra of the cyclic group Z n admits a certain qu asitriangular structure (constructed in [25], [31]) w hic h is not triangular for n ≥ 3. Thus, for n ≥ 3, the catego ry of representa tions of Z n admits a pseudosymmetric braiding whic h is not symmetric. Remark 3.12 Let H b e a ﬁnite dimensional Hopf algebra. It is w ell-kno wn that the cate gory of Y etter-Drinfeld mo dules H Y D H is b raided equiv alen t to the category D ( H ) M of left mo dules o v er the Drinfeld doub le of H (realized on H ∗ cop ⊗ H and with quasitriangular structure giv en b y R = P ( ε ⊗ e i ) ⊗ ( e i ⊗ 1), where { e i } , { e i } are dual b ases in H and H ∗ ). Thus, via T heorem 3.7, we obtain that R is pseudotriangular if and only if H is comm utativ e an d co commutativ e. In p articular, if G is a ﬁnite, noncommutat ive group then ( D ( k [ G ]) , R ) is quasitriangular but not pseudotriangular. Deﬁnition 3.13 ([23]) L et ( H, R ) b e a quasitriangular Hopf algebr a. The element R i s c al le d almost-triangular i f R 21 R is c entr al in H ⊗ H . Remark 3.14 By Pr op osition 3.10 it f ol lows that an almost-triangular structur e is pseudotr i- angular. The c onverse is not true, a c ounter example is pr ovide d by Pr op osition 3.15 b elow. Assume now that char ( k ) 6 = 2 and consid er the 2 n +1 -dimensional Hopf algebra E ( n ) gener- ated by c , x 1 ,..., x n with r elations c 2 = 1, x 2 i = 0, x i c + cx i = 0 and x i x j + x j x i = 0, for all i, j ∈ { 1 , ..., n } , and coal gebra structur e ∆( c ) = c ⊗ c , ∆( x i ) = 1 ⊗ x i + x i ⊗ c , for all i ∈ { 1 , ..., n } . The quasitriangular structures of E ( n ) ha ve b een classiﬁed in [27], they are in bijection with n × n m atrices with en tries in k , an d m oreo v er the qu asitriangular structure R A corresp ondin g to the matrix A is giv en b y an explicit formula, generalizing the cases n = 1 from [32] and n = 2 from [17]. By [27] and [8] we kno w that R A is triangular if and on ly if the matrix A is symmetric. Prop osition 3.15 F or any n × n matrix A , the quasitriangular structur e R A is pseudotrian- gular, and it is almost-triangular if and only if A is symmetric (thus the only almost-triangular structur es of E ( n ) ar e the triangular ones). Pr o of. W e p resen t ﬁrst an alternativ e description for the quasitriangular structure R A . F or ev ery a ∈ k and i, j ∈ { 1 , ..., n } w e deﬁn e the element T i,j ( a ) := 1 ⊗ 1 + a ( x i ⊗ cx j ) ∈ E ( n ) ⊗ E ( n ) . (3.8) It is easy to see that T i,j ( a ) is a lazy t wist, T i,j ( a ) T i,j ( b ) = T i,j ( a + b ) and T i,j ( a ) T k ,l ( b ) = T k ,l ( b ) T i,j ( a ), f or all a, b ∈ k and i, j, k , l ∈ { 1 , ..., n } . If A = ( a ij ) i,j =1 ,...,n is an n × n matrix, w e deﬁne the elemen t T A := n Y i,j =1 T i,j ( a ij ) ∈ E ( n ) ⊗ E ( n ) (3.9) 17 (note that the ord er of the f actors do es not m atter since they all commute) . It is clear that if B is another n × n m atrix then T A T B = T A + B . One can also see that the elemen t T A is giv en b y the formula T A = 1 ⊗ 1 + X | P | = | F | ( − 1) | P | ( | P |− 1) 2 det ( P, F ) x P ⊗ c | P | x F , (3.10) where the s um is made o v er all nonempty sub sets P , F of { 1 , ..., n } such that | P | = | F | , and if P = { i 1 < i 2 < · · · < i s } and F = { j 1 < j 2 < · · · < j s } then det ( P, F ) is the determinan t of the s × s matrix obtained at the in tersection of the rows i 1 , ..., i s and columns j 1 , ..., j s of the matrix A , and x P = x i 1 · · · x i s , x F = x j 1 · · · x j s . In particular we obtain T 0 = 1 ⊗ 1 and T − 1 A = T − A . Deﬁne no w the elemen t R := 1 2 (1 ⊗ 1 + c ⊗ 1 + 1 ⊗ c − c ⊗ c ) ∈ E ( n ) ⊗ E ( n ) , whic h is a triangular structure for E ( n ). F rom the f orm ula for the quasitriangular structure R A in [27] and (3.10) w e immediately obtain R A = RT A . (3.11) If w e denote by A t the transp ose of a matrix A , then w e kno w from [8] th at R − 1 A = ( R A t ) 21 , (3.12) a consequence of whic h is the r elation ( R A ) 21 R B = T B − A t , for any n × n matrices A and B . W e record also the ob vious relation R A T B = R A + B , as w ell as ( T A ) 21 R B = R B − A t . Let now A b e an n × n matrix; we will pro v e that R A is p seudotriangular. In view of (3.12), what w e need to pro ve is the relatio n ( R A ) 12 ( R A t ) 13 ( R A ) 23 = ( R A ) 23 ( R A t ) 13 ( R A ) 12 . (3.13) W e will actually p ro ve something m ore general, namely ( R A ) 12 ( R B ) 13 ( R C ) 23 = ( R C ) 23 ( R B ) 13 ( R A ) 12 , (3.14) for an y n × n matrices A , B and C . W e in tro du ce the follo wing n otation, for a ∈ k and i, j ∈ { 1 , ..., n } : T i,j ( a ) 12 c := 1 ⊗ 1 ⊗ 1 + ax i ⊗ cx j ⊗ c, T i,j ( a ) 1 c 3 := 1 ⊗ 1 ⊗ 1 + ax i ⊗ c ⊗ cx j , T i,j ( a ) c 23 := 1 ⊗ 1 ⊗ 1 + c ⊗ ax i ⊗ cx j . By direct computation one can pro v e the f ollo win g relations: T i,j ( a ) 23 R 13 = R 13 T i,j ( a ) c 23 , T i,j ( a ) c 23 R 12 = R 12 T i,j ( a ) 23 , T i,j ( a ) 13 R 12 = R 12 T i,j ( a ) 1 c 3 , T i,j ( a ) 13 R 23 = R 23 T i,j ( a ) 1 c 3 , T i,j ( a ) 12 R 13 = R 13 T i,j ( a ) 12 c , T i,j ( a ) 12 c R 23 = R 23 T i,j ( a ) 12 . 18 One can also see that, f or all i, j, k , l , p, q ∈ { 1 , ..., n } and x, y , z ∈ k , all the elemen ts T i,j ( x ) 23 , T k ,l ( y ) 12 and T p,q ( z ) 1 c 3 comm ute with eac h other. Using all these facts together with the form ulae (3.11) and (3.9) w e obtain ( R A ) 12 ( R B ) 13 ( R C ) 23 = R 12 R 13 R 23 ( T A ) 12 ( T B ) 1 c 3 ( T C ) 23 , ( R C ) 23 ( R B ) 13 ( R A ) 12 = R 23 R 13 R 12 ( T C ) 23 ( T B ) 1 c 3 ( T A ) 12 , and the righ t hand sides are equal b ecause of the ab ov e-mentio ned comm utation relations to- gether with the fact that R satisﬁes the Y ang-Baxter equation. W e pro v e no w that R A is almost-triangular if and only if A is sym metric. Let B b e an n × n matrix; it is easy to see that T B is cen tral in E ( n ) ⊗ E ( n ) if and only if B = 0, b ecause if B 6 = 0 then T B do es not comm u te with 1 ⊗ c . W e ha v e seen ab ov e that ( R A ) 21 R A = T A − A t , and s o ( R A ) 21 R A is cen tral if and only if A = A t .  Remark 3.16 W e consider the group Z 2 ( E ( n )) as in Section 2, and insid e it the set G n := { T A , R A } , where A is an n × n matrix. If w e den ote b y ∗ the multiplica tion in Z 2 ( E ( n )), then w e hav e T A ∗ T B = T A T B = T A + B , R A ∗ T B = R A T B = R A + B , T A ∗ R B = ( T A ) 21 R B = R B − A t , R A ∗ R B = ( R A ) 21 R B = T B − A t , and so G n is a sub group of Z 2 ( E ( n )) (note th at the in ve rs e of R A in this group is R A t ). The ab o ve formulae imp ly G n ≃ Z 2 ⋉ ( M n ( k ) , +), a semidirect pro duct, where the action of Z 2 on ( M n ( k ) , +) is given by A · g = − A t ( g is the generator of Z 2 ), and the corresp ondence is giv en b y T A 7→ (1 , A ), R A 7→ ( g , A ). F or n = 1 ( E (1) is Sw eedler’s 4-dimensional Hopf algebra), one can pro ve by direct computation that G 1 = Z 2 ( E (1)). 4 La ycles, pseudot wistors and R-matrices W e recall the follo wing concept and result from [24 ]: Prop osition 4.1 ([24]) L et C b e a monoida l c ate gory, A an algebr a in C with multiplic ation µ and unit u , T : A ⊗ A → A ⊗ A a morphism in C such that T ◦ ( u ⊗ id A ) = u ⊗ id A and T ◦ ( id A ⊗ u ) = id A ⊗ u . Assume that ther e exist two morphisms ˜ T 1 , ˜ T 2 : A ⊗ A ⊗ A → A ⊗ A ⊗ A in C such that ( id A ⊗ µ ) ◦ ˜ T 1 ◦ ( T ⊗ id A ) = T ◦ ( id A ⊗ µ ) , (4.1) ( µ ⊗ id A ) ◦ ˜ T 2 ◦ ( id A ⊗ T ) = T ◦ ( µ ⊗ id A ) , (4.2) ˜ T 1 ◦ ( T ⊗ id A ) ◦ ( id A ⊗ T ) = ˜ T 2 ◦ ( id A ⊗ T ) ◦ ( T ⊗ id A ) . (4.3) Then ( A, µ ◦ T , u ) is also an algebr a in C , denote d by A T . The morphism T is c al le d a pseu- dot wistor and the two morphisms ˜ T 1 , ˜ T 2 ar e c al le d the companions of T . If C is the c ate gory of k - ve ctor sp ac es, ˜ T 1 = ˜ T 2 = T 13 and T 12 ◦ T 23 = T 23 ◦ T 12 , then T is c al le d a twistor for A . Prop osition 4.2 L et C b e a monoidal c ate gory and T a laycle on C . If ( A, µ, u ) is an algebr a in C , then T A,A is a pseudo twistor for A , with c omp anions ˜ T 1 := T b A,A,A and ˜ T 2 := T f A,A,A , wher e T b and T f ar e the families deﬁne d by (2.3 ) and (2.4). 19 Pr o of. W e p ro ve (4.1). The naturalit y of T implies T A,A ◦ ( id A ⊗ µ ) = ( id A ⊗ µ ) ◦ T A,A ⊗ A , and from (2.3) we obtain T A,A ◦ ( id A ⊗ µ ) = ( id A ⊗ µ ) ◦ T b A,A,A ◦ ( T A,A ⊗ id A ), q.e.d. Similarly one can pro ve (4.2), wh ile (4.3) follo ws immediately by usin g (2.3), (2.4) and (2.2).  Corollary 4.3 If T is a laycle on a monoidal c ate gory C and ( A, µ , u ) is an algebr a in C , then ( A, µ ◦ T A,A , u ) is also an algebr a in C . Remark 4.4 If C i s a monoidal c ate gory and c is a br aiding on C , then, by [5], the double br aiding c 2 X,Y := c Y , X ◦ c X,Y is a twine on C , in p articular a laycle. Thus, P r op osition 4.2 gener alizes the fact (pr ove d in [24], Cor ol lary 6.8) that a double br aiding induc es a pseudotwistor on ev ery algebr a in C . Deﬁnition 4.5 L et C b e a mono idal c ate gory, ( A, µ, u ) an algebr a in C and T : A ⊗ A → A ⊗ A a pseudotwistor with c omp anions ˜ T 1 and ˜ T 2 . We say that T is a strong pseudotwistor if T is invertible and the fol lowing c onditions ar e satisﬁe d: ˜ T 2 ◦ ( id A ⊗ T ) = ( id A ⊗ T ) ◦ ˜ T 1 , (4.4) ˜ T 1 ◦ ( T ⊗ id A ) = ( T ⊗ id A ) ◦ ˜ T 2 . (4.5) In this c ase, we denote T A ⊗ A,A := ˜ T 2 ◦ ( id A ⊗ T ) = ( id A ⊗ T ) ◦ ˜ T 1 , T A,A ⊗ A := ˜ T 1 ◦ ( T ⊗ id A ) = ( T ⊗ id A ) ◦ ˜ T 2 . Remark 4.6 If T X,Y is a laycle on a monoidal c ate gory C and ( A, µ, u ) is an algebr a in C , then, by (2.2), it fol lows that T A,A is a str ong pseudotwistor for A . Lemma 4.7 If T i s a str ong pseudotwistor, then the fol lowing r elations hold: ( T ⊗ id A ) ◦ T A ⊗ A,A = T A ⊗ A,A ◦ ( T ⊗ id A ) , (4.6) ( id A ⊗ T ) ◦ T A,A ⊗ A = T A,A ⊗ A ◦ ( id A ⊗ T ) , (4.7) T A ⊗ A,A ◦ ( T ⊗ id A ) = T A,A ⊗ A ◦ ( id A ⊗ T ) , (4.8) ( T ⊗ id A ) ◦ ˜ T 2 ◦ ( id A ⊗ T ) = ( id A ⊗ T ) ◦ ˜ T 1 ◦ ( T ⊗ id A ) . (4.9) Pr o of. Straigh tforward computation, using (4.4), (4.5 ) and (4.3).  Our next r esults are the analog ues for pseu d ot wistors of the facts that comp osition of layc les is a la ycle and the in verse of a la ycle is a la ycle. Prop osition 4.8 L et C b e a monoidal c ate gory, ( A, µ , u ) an algebr a in C and T , D : A ⊗ A → A ⊗ A two str ong pseudotwistors for A , suc h that D A,A ⊗ A ◦ ( id A ⊗ T ) = ( id A ⊗ T ) ◦ D A,A ⊗ A , (4.10) D A ⊗ A,A ◦ ( T ⊗ id A ) = ( T ⊗ id A ) ◦ D A ⊗ A,A . (4.11) Then U := T ◦ D is a pseudotwistor for A , with c omp anions ˜ U 1 := T A,A ⊗ A ◦ ˜ D 1 ◦ ( T − 1 ⊗ id A ) and ˜ U 2 := T A ⊗ A,A ◦ ˜ D 2 ◦ ( id A ⊗ T − 1 ) . If mor e over we have T A,A ⊗ A ◦ ( id A ⊗ D ) = ( id A ⊗ D ) ◦ T A,A ⊗ A , (4.12) T A ⊗ A,A ◦ ( D ⊗ id A ) = ( D ⊗ id A ) ◦ T A ⊗ A,A , (4.13) then U is also a str ong pseudotwistor. 20 Pr o of. W e chec k (4.1)–(4.3) for U : U ◦ ( id A ⊗ µ ) = T ◦ D ◦ ( id A ⊗ µ ) ( 4.1 ) = ( id A ⊗ µ ) ◦ ˜ T 1 ◦ ( T ⊗ id A ) ◦ ˜ D 1 ◦ ( D ⊗ id A ) = ( id A ⊗ µ ) ◦ T A,A ⊗ A ◦ ˜ D 1 ◦ ( D ⊗ id A ) = ( id A ⊗ µ ) ◦ ˜ U 1 ◦ ( U ⊗ id A ) , U ◦ ( µ ⊗ id A ) = T ◦ D ◦ ( µ ⊗ id A ) ( 4.2 ) = ( µ ⊗ id A ) ◦ ˜ T 2 ◦ ( id A ⊗ T ) ◦ ˜ D 2 ◦ ( id A ⊗ D ) = ( µ ⊗ id A ) ◦ T A ⊗ A,A ◦ ˜ D 2 ◦ ( id A ⊗ D ) = ( µ ⊗ id A ) ◦ ˜ U 2 ◦ ( id A ⊗ U ) , ˜ U 1 ◦ ( U ⊗ id A ) ◦ ( id A ⊗ U ) = T A,A ⊗ A ◦ ˜ D 1 ◦ ( D ⊗ id A ) ◦ ( id A ⊗ T ) ◦ ( id A ⊗ D ) = T A,A ⊗ A ◦ D A,A ⊗ A ◦ ( id A ⊗ T ) ◦ ( id A ⊗ D ) ( 4.10 ) = T A,A ⊗ A ◦ ( id A ⊗ T ) ◦ D A,A ⊗ A ◦ ( id A ⊗ D ) ( 4.8 ) = T A ⊗ A,A ◦ ( T ⊗ id A ) ◦ D A ⊗ A,A ◦ ( D ⊗ id A ) ( 4.11 ) = T A ⊗ A,A ◦ D A ⊗ A,A ◦ ( T ⊗ id A ) ◦ ( D ⊗ id A ) = T A ⊗ A,A ◦ ˜ D 2 ◦ ( id A ⊗ D ) ◦ ( T ⊗ id A ) ◦ ( D ⊗ id A ) = ˜ U 2 ◦ ( id A ⊗ U ) ◦ ( U ⊗ id A ) , pro ving that U is a p seudot wistor for A . W e assum e now that (4.12) and (4.13 ) hold and we pro ve (4.4 ) and (4.5) for U : ( id A ⊗ U ) ◦ ˜ U 1 = ( id A ⊗ T ) ◦ ( id A ⊗ D ) ◦ T A,A ⊗ A ◦ ˜ D 1 ◦ ( T − 1 ⊗ id A ) ( 4.12 ) = ( id A ⊗ T ) ◦ T A,A ⊗ A ◦ ( id A ⊗ D ) ◦ ˜ D 1 ◦ ( T − 1 ⊗ id A ) = ( id A ⊗ T ) ◦ T A,A ⊗ A ◦ D A ⊗ A,A ◦ ( T − 1 ⊗ id A ) ( 4.11 ) = ( id A ⊗ T ) ◦ T A,A ⊗ A ◦ ( T − 1 ⊗ id A ) ◦ D A ⊗ A,A ( 4.8 ) , ( 4.7 ) = T A ⊗ A,A ◦ D A ⊗ A,A = T A ⊗ A,A ◦ ˜ D 2 ◦ ( id A ⊗ D ) = ˜ U 2 ◦ ( id A ⊗ U ) , ( U ⊗ id A ) ◦ ˜ U 2 = ( T ⊗ id A ) ◦ ( D ⊗ id A ) ◦ T A ⊗ A,A ◦ ˜ D 2 ◦ ( id A ⊗ T − 1 ) ( 4.13 ) = ( T ⊗ id A ) ◦ T A ⊗ A,A ◦ ( D ⊗ id A ) ◦ ˜ D 2 ◦ ( id A ⊗ T − 1 ) = ( T ⊗ id A ) ◦ T A ⊗ A,A ◦ D A,A ⊗ A ◦ ( id A ⊗ T − 1 ) ( 4.10 ) = ( T ⊗ id A ) ◦ T A ⊗ A,A ◦ ( id A ⊗ T − 1 ) ◦ D A,A ⊗ A ( 4.8 ) , ( 4.6 ) = T A,A ⊗ A ◦ D A,A ⊗ A = T A,A ⊗ A ◦ ˜ D 1 ◦ ( D ⊗ id A ) = ˜ U 1 ◦ ( U ⊗ id A ) , 21 sho wing that U is a strong p seudot wistor.  Corollary 4.9 If T is a str ong pseudotwisto r for an algebr a ( A, µ, u ) in a monoidal c ate gory C , then T ◦ T is also a str ong pseudotwistor for A . Prop osition 4.10 L et C b e a monoidal c ate gory, ( A, µ, u ) an algebr a in C and T : A ⊗ A → A ⊗ A a str ong pseudotwistor for A such that the c omp anions ˜ T 1 and ˜ T 2 ar e i nv e rtible. Then the inverse V := T − 1 is also a str ong pseudotwistor for A , with c omp anions ˜ V 1 = ˜ T − 1 2 and ˜ V 2 = ˜ T − 1 1 . Pr o of. Straigh tforward computation, using (4.1)–(4.3) for T together with (4.4) and (4.5).  Remark 4.11 L e t C b e a monoidal c ate gory, ( A, µ , u ) an algebr a in C , T : A ⊗ A → A ⊗ A a str ong pseudotwistor for A and D a laycle on C . Then T and D := D A,A satisfy (4.10) and (4.11), henc e T ◦ D is a pseudotwisto r for A . Our next resu lt is the analog ue f or pseudot wistors of the fact that if σ , σ ′ are cohomolo gous lazy cocycles on a Hopf alge b r a H then the algebras H ( σ ) an d H ( σ ′ ) are isomorphic: Prop osition 4.12 L et C b e a monoid al c ate gory, ( A, µ, u ) an algebr a in C , T : A ⊗ A → A ⊗ A a str ong pseudotwisto r for A and R X : X → X a family of natur al isomorphisms in C suc h that R I = id I . Then we have an algebr a isomorp hism R A : A T ◦ D 1 ( R ) A,A ≃ A T . Pr o of. Note ﬁrst that T ◦ D 1 ( R ) A,A is a pseudot wistor by Remark 4.11. W e compu te: R A ◦ µ ◦ T ◦ D 1 ( R ) A,A = R A ◦ µ ◦ T ◦ R − 1 A ⊗ A ◦ ( R A ⊗ R A ) natural ity of R = µ ◦ R A ⊗ A ◦ T ◦ R − 1 A ⊗ A ◦ ( R A ⊗ R A ) natural ity of R = µ ◦ T ◦ ( R A ⊗ R A ) , ﬁnishing the pro of.  If T is a layc le on a m onoidal category C , then, by [5], T is a t wine if and only if the follo wing condition is satisﬁed: ( T f X,Y ,Z ⊗ id W ) ◦ ( id X ⊗ T b Y , Z,W ) = ( id X ⊗ T b Y , Z,W ) ◦ ( T f X,Y ,Z ⊗ id W ) , (4.14) for all X, Y , Z, W ∈ C . Note that the families T f , T b coincide resp ectiv ely to the families A , B from the Deﬁnition 1.5 of a pure-braided structure, and (4.14) coincides with (1.13). W e are th us led to the follo wing concept and terminology: Deﬁnition 4.13 L et C b e a monoidal c ate gory, A an algebr a in C and T : A ⊗ A → A ⊗ A a pseudotwisto r with c omp anions ˜ T 1 and ˜ T 2 . We c al l T a pure pseudot wistor if ( ˜ T 2 ⊗ id A ) ◦ ( id A ⊗ ˜ T 1 ) = ( id A ⊗ ˜ T 1 ) ◦ ( ˜ T 2 ⊗ id A ) . (4.15) Corollary 4.14 A twine on a monoidal c ate gory C induc es a pur e pseudotwistor on every al- gebr a A in C . In p articular, if c is a br aiding on C then c 2 A,A is a pur e pseudotw istor f or A . 22 Remark 4.15 Obvio us ly , a ps eudot wistor for whic h ˜ T 1 = ˜ T 2 = id A ⊗ A ⊗ A is p ure. Here are some concrete (but non u n ital) examples of suc h pseudot wistors: (i) take A an asso ciativ e alge b r a, R = R 1 ⊗ R 2 ∈ A ⊗ A and deﬁne T ( a ⊗ b ) = aR 1 ⊗ R 2 b , for all a, b ∈ A . (ii) tak e A an asso ciativ e algebra, f : A → A a linear map satisfying f ( ab ) = af ( b ) for all a, b ∈ A , and T ( a ⊗ b ) = f ( a ) ⊗ b . I f ins tead f satisﬁes f ( ab ) = f ( a ) b , then tak e T ( a ⊗ b ) = a ⊗ f ( b ). (iii) tak e A an associativ e algebra, δ : A → A ⊗ A a linear map suc h that δ ( ab ) = ( a ⊗ 1) δ ( b ) f or all a, b ∈ A and T : A ⊗ A → A ⊗ A , T ( a ⊗ b ) = δ ( a )(1 ⊗ b ). If in stead δ satisﬁes δ ( ab ) = δ ( a )(1 ⊗ b ), then tak e T ( a ⊗ b ) = ( a ⊗ 1) δ ( b ). Note that example (i) w as inspired b y a construction in [2], while (ii) and (iii) are related to some constructions in [22] in v olving so-ca lled (an ti-) d ip terous algebras. Example 4.16 If A is an asso ciativ e algebra and T : A ⊗ A → A ⊗ A is a twistor, then it is easy to see that T is pur e. Example 4.17 W e recall some facts from [24]. Let (Ω , d ) b e a DG alg ebra, that is Ω = L n ≥ 0 Ω n is a graded algebra and d : Ω → Ω is a lin ear map with d (Ω n ) ⊆ Ω n +1 for all n ≥ 0, d 2 = 0 and d ( ω ζ ) = d ( ω ) ζ + ( − 1) | ω | ω d ( ζ ) f or all h omogeneous ω and ζ , w h ere | ω | is the degree of ω . The F edoso v pro duct ([16], [11]), giv en b y ω ◦ ζ = ω ζ − ( − 1) | ω | d ( ω ) d ( ζ ) , for h omogeneous ω and ζ , giv es a new asso ciativ e algebra structure on Ω. W e consider C to b e the monoidal category of Z 2 -graded v ector sp aces, and regard Ω as a Z 2 -graded algebra (i.e. an alge br a in C ) by pu tting ev en comp onents in degree zero and odd comp onents in degree one. Deﬁne the linear map T : Ω ⊗ Ω → Ω ⊗ Ω , T ( ω ⊗ ζ ) = ω ⊗ ζ − ( − 1) | ω | d ( ω ) ⊗ d ( ζ ) , for homogeneous ω and ζ . Then T is a p seudot wistor f or Ω in C , aﬀording the F edoso v pro du ct. Its companions are giv en (for homogeneous ω , ζ , η ) b y ˜ T 1 ( ω ⊗ ζ ⊗ η ) = ˜ T 2 ( ω ⊗ ζ ⊗ η ) = ω ⊗ ζ ⊗ η − ( − 1) | ω | + | ζ | d ( ω ) ⊗ ζ ⊗ d ( η ) . W e claim that T is a pure pseudot wistor. Indeed, a straigh tforw ard computation sho ws that ( ˜ T 2 ⊗ id ) ◦ ( id ⊗ ˜ T 1 )( ω ⊗ ζ ⊗ η ⊗ ν ) = ( id ⊗ ˜ T 1 ) ◦ ( ˜ T 2 ⊗ id )( ω ⊗ ζ ⊗ η ⊗ ν ) = ω ⊗ ζ ⊗ η ⊗ ν − ( − 1) | ω | + | ζ | d ( ω ) ⊗ ζ ⊗ d ( η ) ⊗ ν − ( − 1) | ζ | + | η | ω ⊗ d ( ζ ) ⊗ η ⊗ d ( ν ) − ( − 1) | ω | + | η | d ( ω ) ⊗ d ( ζ ) ⊗ d ( η ) ⊗ d ( ν ) , for all homogeneous ω , ζ , η , ν . W e recall the follo wing result from [24]: Prop osition 4.18 ([24]) L et ( A, µ, u ) b e an algebr a in a monoida l c ate gory C , let R, P : A ⊗ A → A ⊗ A twisting maps b etwe en A and itself such that R is invertible, and assume that ( P ⊗ id A ) ◦ ( id A ⊗ P ) ◦ ( P ⊗ id A ) = ( id A ⊗ P ) ◦ ( P ⊗ id A ) ◦ ( id A ⊗ P ) , (4.16) ( R ⊗ id A ) ◦ ( id A ⊗ R ) ◦ ( R ⊗ id A ) = ( id A ⊗ R ) ◦ ( R ⊗ id A ) ◦ ( id A ⊗ R ) , (4.17) ( P ⊗ id A ) ◦ ( id A ⊗ P ) ◦ ( R ⊗ id A ) = ( id A ⊗ R ) ◦ ( P ⊗ id A ) ◦ ( id A ⊗ P ) , (4.18) ( R ⊗ id A ) ◦ ( id A ⊗ P ) ◦ ( P ⊗ id A ) = ( id A ⊗ P ) ◦ ( P ⊗ id A ) ◦ ( id A ⊗ R ) . (4.19) Deﬁne T : A ⊗ A → A ⊗ A , T := R − 1 ◦ P . Then T is a pseudotwisto r with c omp anions ˜ T 1 = ( R − 1 ⊗ id A ) ◦ ( id A ⊗ T ) ◦ ( R ⊗ id A ) , ˜ T 2 = ( id A ⊗ R − 1 ) ◦ ( T ⊗ id A ) ◦ ( id A ⊗ R ) . 23 Our next result is the analogue for pseudot wistors of the fact from [5] that a family of the t yp e T X,Y = c ′ Y , X ◦ c X,Y , with c , c ′ braidings, is a t wine: Prop osition 4.19 Assume that the hyp otheses of Pr op osition 4.18 hold. Then: (i) T is a pur e pseudotwistor; (ii) assume that mor e over P is also inve rtible and ( P ⊗ id A ) ◦ ( id A ⊗ R ) ◦ ( R ⊗ id A ) = ( id A ⊗ R ) ◦ ( R ⊗ id A ) ◦ ( id A ⊗ P ) , (4.20) ( R ⊗ id A ) ◦ ( id A ⊗ R ) ◦ ( P ⊗ id A ) = ( id A ⊗ P ) ◦ ( R ⊗ id A ) ◦ ( id A ⊗ R ) (4.21) (these c onditions app e ar in [24] to o and they imply that R is also a twisting map b etwe e n A T and itself ). Then T is a str ong pseudotwistor. Pr o of. W e chec k (4.15): ( ˜ T 2 ⊗ id A ) ◦ ( id A ⊗ ˜ T 1 ) = ( id A ⊗ R − 1 ⊗ id A ) ◦ ( T ⊗ id A ⊗ id A ) ◦ ( id A ⊗ R ⊗ id A ) ◦ ( id A ⊗ R − 1 ⊗ id A ) ◦ ( id A ⊗ id A ⊗ T ) ◦ ( id A ⊗ R ⊗ id A ) = ( id A ⊗ R − 1 ⊗ id A ) ◦ ( id A ⊗ id A ⊗ T ) ◦ ( T ⊗ id A ⊗ id A ) ◦ ( id A ⊗ R ⊗ id A ) = ( id A ⊗ R − 1 ⊗ id A ) ◦ ( id A ⊗ id A ⊗ T ) ◦ ( id A ⊗ R ⊗ id A ) ◦ ( id A ⊗ R − 1 ⊗ id A ) ◦ ( T ⊗ id A ⊗ id A ) ◦ ( id A ⊗ R ⊗ id A ) = ( id A ⊗ ˜ T 1 ) ◦ ( ˜ T 2 ⊗ id A ) . Assume now that P is in vertible and (4.2 0 ), (4.21) hold. Ob viously T is in vertible, and w e only ha ve to chec k (4.4) and (4.5): ˜ T 2 ◦ ( id A ⊗ T ) = ( id A ⊗ R − 1 ) ◦ ( T ⊗ id A ) ◦ ( id A ⊗ R ) ◦ ( id A ⊗ T ) = ( id A ⊗ R − 1 ) ◦ ( R − 1 ⊗ id A ) ◦ ( P ⊗ id A ) ◦ ( id A ⊗ P ) ( 4.20 ) = ( id A ⊗ R − 1 ) ◦ ( id A ⊗ P ) ◦ ( R − 1 ⊗ id A ) ◦ ( id A ⊗ R − 1 ) ( P − 1 ⊗ id A ) ◦ ( id A ⊗ R ) ◦ ( P ⊗ id A ) ◦ ( id A ⊗ P ) ( 4.18 ) = ( id A ⊗ R − 1 ) ◦ ( id A ⊗ P ) ◦ ( R − 1 ⊗ id A ) ◦ ( id A ⊗ R − 1 ) ◦ ( id A ⊗ P ) ◦ ( R ⊗ id A ) = ( id A ⊗ T ) ◦ ˜ T 1 , ˜ T 1 ◦ ( T ⊗ id A ) = ( R − 1 ⊗ id A ) ◦ ( id A ⊗ T ) ◦ ( R ⊗ id A ) ◦ ( T ⊗ id A ) = ( R − 1 ⊗ id A ) ◦ ( id A ⊗ R − 1 ) ◦ ( id A ⊗ P ) ◦ ( P ⊗ id A ) ( 4.21 ) = ( R − 1 ⊗ id A ) ◦ ( P ⊗ id A ) ◦ ( id A ⊗ R − 1 ) ◦ ( R − 1 ⊗ id A ) ( id A ⊗ P − 1 ) ◦ ( R ⊗ id A ) ◦ ( id A ⊗ P ) ◦ ( P ⊗ id A ) ( 4.19 ) = ( R − 1 ⊗ id A ) ◦ ( P ⊗ id A ) ◦ ( id A ⊗ R − 1 ) ◦ ( R − 1 ⊗ id A ) ◦ ( P ⊗ id A ) ◦ ( id A ⊗ R ) = ( T ⊗ id A ) ◦ ˜ T 2 , ﬁnishing the pro of.  24 Remark 4.20 If T is a b r aided twistor as intr o duc e d in [24], a c omputation identic al to the one i n the pr o of of Pr op osition 4.19 (i) shows that T is a pur e pseudot wistor. Example 4.21 Let A b e an algebra and F a braid system o v er A as in tro d uced by Durdevich in [14], that is a collectio n of b ijectiv e t wisting maps b et w een A an d itself, satisfying the condition ( α ⊗ id A ) ◦ ( id A ⊗ β ) ◦ ( γ ⊗ id A ) = ( id A ⊗ γ ) ◦ ( β ⊗ id A ) ◦ ( id A ⊗ α ) , ∀ α, β , γ ∈ F . F or α, β ∈ F deﬁn e the map T α,β : A ⊗ A → A ⊗ A , T α,β := α − 1 ◦ β . By [24] we kn ow that T is a pseu dot wistor for A , and by Prop osition 4.19 it follo ws that it is a p ure strong pseudot wistor. W e introdu ce no w the catego rical ve rs ion of Borc herds’ R -matrices: Prop osition 4.22 L et C b e a monoidal c ate gory, ( A, µ, u ) an algebr a in C and T : A ⊗ A → A ⊗ A a morphism in C such that T ◦ ( u ⊗ id A ) = u ⊗ id A and T ◦ ( id A ⊗ u ) = id A ⊗ u . Assume that ther e exist two morp hisms T 1 , T 2 : A ⊗ A ⊗ A → A ⊗ A ⊗ A in C such that ( id A ⊗ µ ) ◦ ( T ⊗ id A ) ◦ T 1 = T ◦ ( id A ⊗ µ ) , (4.22) ( µ ⊗ id A ) ◦ ( id A ⊗ T ) ◦ T 2 = T ◦ ( µ ⊗ id A ) , ( 4.23) ( T ⊗ id A ) ◦ T 1 ◦ ( id A ⊗ T ) = ( id A ⊗ T ) ◦ T 2 ◦ ( T ⊗ id A ) . (4.24) Then ( A, µ ◦ T , u ) is also an algebr a in C , denote d by A T . The morphism T is c al le d an R -matrix and the two morphisms T 1 , T 2 ar e c al le d the companions of T . Obviously, the original c onc ept of R -matrix is obtaine d for C b eing the c ate gory of k - ve ctor sp ac es and T 1 = T 2 = T 13 . Pr o of. Obvio us ly u is a unit for ( A, µ ◦ T ); w e c h eck the associativit y of µ ◦ T : ( µ ◦ T ) ◦ (( µ ◦ T ) ⊗ id A ) = ( µ ◦ T ) ◦ ( µ ⊗ id A ) ◦ ( T ⊗ id A ) ( 4.23 ) = µ ◦ ( µ ⊗ id A ) ◦ ( id A ⊗ T ) ◦ T 2 ◦ ( T ⊗ id A ) ( 4.24 ) = µ ◦ ( µ ⊗ id A ) ◦ ( T ⊗ id A ) ◦ T 1 ◦ ( id A ⊗ T ) = µ ◦ ( id A ⊗ µ ) ◦ ( T ⊗ id A ) ◦ T 1 ◦ ( id A ⊗ T ) ( 4.22 ) = µ ◦ T ◦ ( id A ⊗ µ ) ◦ ( id A ⊗ T ) = ( µ ◦ T ) ◦ ( id A ⊗ ( µ ◦ T )) , ﬁnishing the pro of.  Prop osition 4.23 L et C b e a monoidal c ate gory, ( A, µ, u ) an algebr a in C and T : A ⊗ A → A ⊗ A an inv ertible morphism in C . Then T is a pseudotwisto r if and only if it is an R -matrix. Mor e pr e cisely, i f T is a pseudotwistor with c omp anions ˜ T 1 , ˜ T 2 then T is an R -matrix with c omp anions T 1 = ( T − 1 ⊗ id A ) ◦ ˜ T 1 ◦ ( T ⊗ id A ) and T 2 = ( id A ⊗ T − 1 ) ◦ ˜ T 2 ◦ ( id A ⊗ T ) ; c onversely, if T is an R -matrix with c omp anions T 1 , T 2 then T is a pseudotwisto r with c omp anions ˜ T 1 = ( T ⊗ id A ) ◦ T 1 ◦ ( T − 1 ⊗ id A ) and ˜ T 2 = ( id A ⊗ T ) ◦ T 2 ◦ ( id A ⊗ T − 1 ) . Pr o of. Straigh tforward computation.  Corollary 4.24 L et C b e a monoidal c ate gory and T a laycle on C . If ( A, µ , u ) is an algebr a in C , then T A,A is an R -matrix for A , with c omp anions T 1 := T f A,A,A and T 2 := T b A,A,A , wher e T b and T f ar e the families deﬁne d by (2.3 ) and (2.4). 25 5 A c haracterization of generalized double braidings Let C b e a monoidal category and A an algebra in C . If T is a pseudotwisto r for A and R : A ⊗ A → A ⊗ A is an in v ertible t wisting map suc h that the companions of T are given by the form u lae ˜ T 1 = ( R − 1 ⊗ id A ) ◦ ( id A ⊗ T ) ◦ ( R ⊗ id A ) , (5.1) ˜ T 2 = ( id A ⊗ R − 1 ) ◦ ( T ⊗ id A ) ◦ ( id A ⊗ R ) , (5.2) then, by [24], Theorem 6.6, it follo w s that R ◦ T is a t wisting map b etw een A and itself. This result has th e follo wing categorical analogue, with la ycles replacing pseu d ot wistors and braidin gs replacing t wisting m ap s : Theorem 5.1 L et C b e a monoida l c ate gory, T a laycle and d a br aiding on C , such tha t for al l X, Y , Z ∈ C the fol lowing r elations hold: T X ⊗ Y , Z = ( id X ⊗ T Y , Z ) ◦ ( d − 1 X,Y ⊗ id Z ) ◦ ( id Y ⊗ T X,Z ) ◦ ( d X,Y ⊗ id Z ) , (5.3) T X,Y ⊗ Z = ( T X,Y ⊗ id Z ) ◦ ( id X ⊗ d − 1 Y , Z ) ◦ ( T X,Z ⊗ id Y ) ◦ ( id X ⊗ d Y , Z ) . (5.4) Then the families d ′ X,Y := d X,Y ◦ T X,Y and d ′′ X,Y := T Y , X ◦ d X,Y ar e also br aidings on C . M or e over, T is a twine and d ′′ X,Y = T Y , X ◦ d ′ X,Y ◦ T − 1 X,Y (thus ( C , d ′ ) and ( C , d ′′ ) ar e br aide d isomorph ic). Pr o of. Note ﬁr st that (5.3) and (5.4) are the analogues of (5.1 ) and (5.2), b ecause they are resp ectiv ely equiv alen t to T b X,Y ,Z = ( d − 1 X,Y ⊗ id Z ) ◦ ( id Y ⊗ T X,Z ) ◦ ( d X,Y ⊗ id Z ) , T f X,Y ,Z = ( id X ⊗ d − 1 Y , Z ) ◦ ( T X,Z ⊗ id Y ) ◦ ( id X ⊗ d Y , Z ) . Also, as consequences of (2.2), (5.3) and (5.4) w e obtain the follo wing relations: T X,Y ⊗ Z = ( d − 1 X,Y ⊗ id Z ) ◦ ( id Y ⊗ T X,Z ) ◦ ( d X,Y ⊗ id Z ) ◦ ( T X,Y ⊗ id Z ) , (5.5) T X ⊗ Y , Z = ( id X ⊗ d − 1 Y , Z ) ◦ ( T X,Z ⊗ id Y ) ◦ ( id X ⊗ d Y , Z ) ◦ ( id X ⊗ T Y , Z ) . ( 5.6) No w we chec k (1.6) and (1.7) for d ′ : d ′ X,Y ⊗ Z = d X,Y ⊗ Z ◦ T X,Y ⊗ Z ( 1.6 ) , ( 5.4 ) = ( id Y ⊗ d X,Z ) ◦ ( d X,Y ⊗ id Z ) ◦ ( T X,Y ⊗ id Z ) ◦ ( id X ⊗ d − 1 Y , Z ) ◦ ( T X,Z ⊗ id Y ) ◦ ( id X ⊗ d Y , Z ) ( 5.5 ) = ( id Y ⊗ d X,Z ) ◦ ( id Y ⊗ T X,Z ) ◦ ( d X,Y ⊗ id Z ) ◦ ( T X,Y ⊗ id Z ) = ( id Y ⊗ d ′ X,Z ) ◦ ( d ′ X,Y ⊗ id Z ) , d ′ X ⊗ Y , Z = d X ⊗ Y , Z ◦ T X ⊗ Y , Z ( 1.7 ) , ( 5.3 ) = ( d X,Z ⊗ id Y ) ◦ ( id X ⊗ d Y , Z ) ◦ ( id X ⊗ T Y , Z ) ◦ ( d − 1 X,Y ⊗ id Z ) ◦ ( id Y ⊗ T X,Z ) ◦ ( d X,Y ⊗ id Z ) ( 5.6 ) = ( d X,Z ⊗ id Y ) ◦ ( T X,Z ⊗ id Y ) ◦ ( id X ⊗ d Y , Z ) ◦ ( id X ⊗ T Y , Z ) = ( d ′ X,Z ⊗ id Y ) ◦ ( id X ⊗ d ′ Y , Z ) . 26 Th u s, d ′ is a braiding. It is obvious that d ′′ X,Y = T Y , X ◦ d ′ X,Y ◦ T − 1 X,Y , and it follo ws that d ′′ is also a braiding, b y using P r op osition 2.8. Th e fact that T satisﬁes (1.17) follo ws immediately b y using (5.5) and (5.6).  Corollary 5.2 L et H b e a Hopf algebr a, R ∈ H ⊗ H a qu asitriangular structur e and F ∈ H ⊗ H a lazy twist, such that (∆ ⊗ id )( F ) = F 23 R − 1 12 F 13 R 12 , (5.7) ( id ⊗ ∆)( F ) = F 12 R − 1 23 F 13 R 23 . (5.8) Then the elements R ′ = RF and R ′′ = F 21 R ar e also q u asitriangular structur es on H . Prop osition 5.3 L et C b e a monoidal c ate gory and c , c ′ br aidings on C . Then the inverse br aiding d X,Y := c − 1 Y , X and the laycle T X,Y = c ′ Y , X ◦ c X,Y satisfy the hyp otheses of The or em 5.1. Conse quently, the family d ′ X,Y = d X,Y ◦ T X,Y = c − 1 Y , X ◦ c ′ Y , X ◦ c X,Y is a br aiding on C , and the br aiding d ′′ c oincides with the original br aiding c ′ . Pr o of. W e chec k (5.3): ( id X ⊗ T Y , Z ) ◦ ( d − 1 X,Y ⊗ id Z ) ◦ ( id Y ⊗ T X,Z ) ◦ ( d X,Y ⊗ id Z ) = ( id X ⊗ c ′ Z,Y ) ◦ ( id X ⊗ c Y , Z ) ◦ ( c Y , X ⊗ id Z ) ◦ ( id Y ⊗ c ′ Z,X ) ◦ ( id Y ⊗ c X,Z ) ◦ ( c − 1 Y , X ⊗ id Z ) ( 1.6 ) , natur ality of c = ( id X ⊗ c ′ Z,Y ) ◦ ( c ′ Z,X ⊗ id Y ) ◦ ( id Z ⊗ c Y , X ) ◦ ( c Y , Z ⊗ id X ) ◦ ( id Y ⊗ c X,Z ) ◦ ( c − 1 Y , X ⊗ id Z ) ( 1.8 ) = ( id X ⊗ c ′ Z,Y ) ◦ ( c ′ Z,X ⊗ id Y ) ◦ ( c X,Z ⊗ id Y ) ◦ ( id X ⊗ c Y , Z ) ◦ ( c Y , X ⊗ id Z ) ◦ ( c − 1 Y , X ⊗ id Z ) ( 1.6 ) , ( 1.7 ) = c ′ Z,X ⊗ Y ◦ c X ⊗ Y , Z = T X ⊗ Y , Z . The pro of of (5.4) is similar and left to th e reader.  Remark 5.4 The or em 5.1 to gether with Pr op osition 5.3 pr ovide an alternative pr o of of the fact fr om [5] that the laycle T X,Y = c ′ Y , X ◦ c X,Y is a twine. Remark 5.5 If ( C , c ) is a br aide d monoidal c ate gory and we take the inverse br aiding d X,Y = c − 1 Y , X , then in gener al ( C , c ) and ( C , d ) ar e not br aide d isomorphic. 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