Twisted automorphisms of Hopf algebras

Twisted automorphism s of Hopf algebras A. Da vydo v No v e m ber 23, 2018 Department of Mathematics, Division of Info rmation and Communication Sciences, Macquarie Univ ersity , Sydney , NSW 2 109, Australia davydo v@math.mq.edu.au Abstract Twisted homomorphisms of bialgebras are bialgebra homomorphisms from t he first in to Drinfeld twistings of the second. They p ossess a com- p osition op eration extend ing composition of b ialgebra h omomorphisms. Gauge transf ormations of tw ists, compatible with adjacent homomor- phisms, give rise to gauge transformation of tw isted homomorphisms, whic h b ehav e nicely with resp ect to comp ositions. Here we study (gauge classes of ) tw isted automorphisms of cocomm utative Hopf algebras. After revising w ell-kn ow n relations b etw een twists, twisted forms of bialgebras and R -matrices (for commutativ e b ialgebras) w e describe twisted auto- morphisms of un iversa l env eloping algebras. 1 In tro duction The a im of the paper is to study “hidden” symmetrie s o f bia lg ebras whic h man- ifest themselves in representation theory . It is very w ell-known that ca teg ories of repr esentations (mo dules) of bialgebra s are examples of so-called monoidal c ate gories . It is less acknowledged that rela tions b etw e en bialgebras ( homomor- phisms of bialgebr as ) do not c apture all relatio ns ( monoidal funct ors ) b etw ee n their r epresentation categories. An algebraic notion whic h do es the job ( bi- Galois (c o)algebr a ) is known only to sp ecialists. They fully repre s ent monoidal relations betw een repre s entation categor ies but are sometimes not very ea sy to work with. F or example, c o mp osition of mono idal functors co rresp onds to tensor pro duct of Galois (co)algebr as and in some situations is quite tr icky to calculate explicitly . A t the same time monoida l functor s of interest could have some additio na l prop er ties which put restrictio ns o n the co rresp onding a lgebraic ob jects a nd a llow one to ha ve an alternative and per haps simpler description. Note that a ny representation c a tegory is equipp ed for free with a mo no idal functor to the category o f vector spaces, the functor forgetting the action of the bialgebra ( fo r getful functor ). Here w e deal with monoidal functors betw een rep- resentation categories which preserve (not nece ssarily mono idally) the forgetful 1 functors. The corr esp onding algebraic notion is of twiste d homomorphism . After defining them in the first s ection w e examine algebr aic co unt erparts of the com- po sition (comp o sition of twisted homomorphisms) and natural transformations of mo noidal functors ( gauge tra nsformations ). The ma in ob ject to study for us is the catego ry o f twisted automorphis ms of a coco mmutative Hopf algebra. W e sta rt b y re-examining the actions o f t wisted automo rphisms on twists (twisted homomorphisms from the ground field) a nd R -matrices. After tha t we trea t the case o f a universal env elo ping algebra of a Lie a lgebra (ov er formal pow er s e ries). It turns o ut that an y twisted automorphism is a bialg ebra a utomorphism to g ether with an inv ariant t wist ( sep ar ate d ca se). The ga uge classes of inv ariant twists for m an ab elian g roup isomorphic to the inv arian t elements of the exterior squar e of the Lie alge bra. The g roup of gauge cla sses o f twisted automorphisms is a c r ossed pro duct of the group of automorphisms o f the Lie alg ebra and the gro up o f inv ariant twists. Throughout the pap er k b e a ground field, whic h is supp o s ed to b e alge- braically closed of characteristic zero. Ac kno wledgmen t The paper was started during the author’s vis it to the Max-P lanck Institut f¨ ur Mathematik (Bonn). The author would like to thank this institution for hospitality and ex cellent w o rking co nditio ns. The work on the pa p er was sup- po rted by Austra lian Research Council gr a nt DP0066 3 514. The author thanks V. Ly akhovsky and A. Stolin for stimulating discussions. Sp ecial thanks are due to R. Stree t for inv aluable s uppo rt during the work o n the pape r . 2 Twisted homomorphisms of b ialgebras and monoidal functors b et w een categories of mo dules 2.1 Twisted homomorphisms of bialgebras Here we recall the notions of twisted ho momorphisms o f bialgebr as a nd their transformatio ns and show how they enrich the category of bialgebr as making it a 2-categor y . A twiste d homomorphi sm of bialgebra s ( H, ∆ , ε ) → ( H ′ , ∆ ′ , ε ′ ) is a pa ir ( f , F ) where f : H → H ′ is a homomorphism of a lgebras a nd F is an in vertible element of H ′ ⊗ H ′ ( f -twist or simply twist ) such that F ∆ ′ ( f ( x )) = ( f ⊗ f )∆( x ) F, ∀ x ∈ H , (1) ( F ⊗ 1)(∆ ′ ⊗ I )( F ) = (1 ⊗ F )( I ⊗ ∆ ′ )( F ) , 2-co cycle condition ( ε ⊗ I )( F ) = ( I ⊗ ε )( F ) = 1 , normalisatio n . F or example, a homomo rphism of bialg ebras f : H → H ′ is a twisted homo- morphism with the ident it y t wist ( f , 1). A twisted homomorphism is sep ar ate d 2 if the first comp onent f : H → H ′ is a homo mo rphism o f bialgebras. F o r a separated twisted homomo rphism the condition (1) amounts to the inv a riance of the twist with r esp ect to the sub-bialgebra f ( H ) ⊂ H ′ : ∆ ′ ( f ( x )) F = F ∆ ′ ( f ( x )) , ∀ x ∈ H . Note that the 2-co cycle and nor ma lisation conditio ns for a twist F imply the coasso cia tivity a nd c ounitality resp ectively o f the twiste d copro duct on H ′ : (∆ ′ ) F ( x ) = F ∆ ′ ( x ) F − 1 . The c o ndition (1) says that f : H → ( H ′ ) F is a homomorphism of bialg ebras, where H ′ F = ( H ′ , (∆ ′ ) F ) is the t wis ted bialgebr a. Note also that the condition on F ∈ H ′ ⊗ H ′ equiv alent to the coas so ciativity of the twisted copr o duct (∆ ′ ) F , is H ′ -inv ariance of (∆ ′ ⊗ I )( F ) − 1 ( F ⊗ 1) − 1 (1 ⊗ F )( I ⊗ ∆ ′ )( F ) . One could call such elemen ts qu asi-twists . Then a quasi-twiste d homomorphism of bia lgebras ( f , F ) : H → H ′ is a bialgebra ho momorphisms f : H → H ′ F for a quasi-twist F . Although, b eing a partial case o f the gener al picture of t wists for quasi-bialg ebras (see [4]), quasi-twisted homomorphism a re interesting in their own right. F or insta nce, Drinfeld’s rectification of K ohno’s mono dro my theo - rem establishes the ex istence of a quas i-twisted isomorphism betw een quantised and clas s ical universal env eloping a lgebras U q ( g )[[ h ]] → U ( g )[[ h ]] (see [4]). F o r another example see [2 ], where it was proved that finite gr oups have the same character tables if and only if there is a qua s i-twisted isomorphism b etw een their group algebras (o v er algebraically clo sed field of character istic zero ). Being m uc h more versatile, q uasi-twisted homomor phisms lack one imp orta nt pro p erty v alid for twisted homomor phisms. Ther e is no composition opera tion defined for gen- eral quas i-twisted homomo rphisms. It is related to the fact that (in c o ntrast to twists) the image of a quasi-twist under a bialgebra homomor phism is not necessarily a quasi-twist. F or succes ive twisted homomorphisms ( H, ∆ , ε ) ( f ,F ) / / ( H ′ , ∆ ′ , ε ′ ) ( f ′ ,F ′ ) / / ( H ′′ , ∆ ′′ , ε ′′ ) we define their c omp osition as ( f ′ , F ′ ) ◦ ( f , F ) = ( f ′ f , f ′ ( F ) F ′ ) . Here f ′ ( F ) = ( f ′ ⊗ f ′ )( F ). It is not hard to verify that the r esult is a twisted homomorphism ( H , ∆ , ε ) → ( H ′′ , ∆ ′′ , ε ′′ ) and that the comp osition is a sso cia - tive. See also section 3. Note that separated twisted homomo rphisms a r e closed under compo s ition. By a gauge tr ansformation ( f , F ) → ( f ′ , F ′ ) of t wisted homomorphisms ( f , F ) , ( f ′ , F ′ ) : ( H, ∆ , ε ) → ( H ′ , ∆ ′ , ε ′ ) we will mean an element a of H ′ such that af ( x ) = f ′ ( x ) a, ∀ x ∈ H , (2) 3 F ∆ ′ ( a ) = ( a ⊗ a ) F ′ . (3) W e will depict it gra phically as follows: ( H, ∆ , ε ) ( f ,F ) % % ( f ′ ,F ′ ) 9 9 ( H ′ , ∆ ′ , ε ′ ) a   Note that the c o ndition (3) together with no r malisation conditions for F and G implies ε ( a ) = 1. Note a lso that a twisted homo morphism gauge isomorphic to a sepa rated t wisted ho mo morphism is no t necessarily separated. F or succes sive gauge transforma tio ns ( H, ∆ , ε ) ( f ,F )   ( g,G ) / / ( j,J ) ? ? ( H ′ , ∆ ′ , ε ′ ) a   b   the comp osition b.a : ( f , F ) → ( j, J ) is simply the pro duct b a in H ′ . Again it is quite straightforward to check that this is a trans fo rmation. Note that if H ′ is a Hopf a lg ebra, an y gauge transfor mation b etw een twisted homomorphisms into H ′ is inv e r tible. This fact can b e chec ked b y pure algebraic computations. A sketc h of a different pro of will be given in sectio n 3. W e can also define comp ositions of tra ns formations and twisted homomor- phisms in the following t wo situations: ( H, ∆ , ε ) ( f ,F ) % % ( f ′ ,F ′ ) 9 9 ( H ′ , ∆ ′ , ε ′ ) ( g,G ) / / ( H ′′ , ∆ ′′ , ε ′′ ) a   ( H, ∆ , ε ) ( f ,F ) / / ( H ′ , ∆ ′ , ε ′ ) ( g,G ) & & ( g ′ ,G ′ ) 8 8 ( H ′′ , ∆ ′′ , ε ′′ ) b   we define it to b e ( g , G ) ◦ a = g ( a ) in the first c a se and b ◦ ( f , F ) = b in the second. The follo wing pr o p erties in tertwinin g compos itions of twisted homo- morphisms and g auge transforma tions are quite stra ightforw ard consequences of the definitions: ( a.b ) ◦ ( f , F ) = ( a ◦ ( f , F )) . ( b ◦ ( f , F )) , 4 ( g , G ) ◦ ( a.b ) = (( g , G ) ◦ a ) . (( g, G ) ◦ b ) , ( a ◦ ( f , F )) . (( g , G ) ◦ b ) = (( g , G ) ◦ b ) . ( a ◦ ( f , F )) . F or an alternative explana tion s ee section 3 . Note tha t the structures describ ed ab ov e ex tend the categor y of bialg ebras and twisted homomorphisms to a 2- categor y Tw with g auge tr ansformatio ns as 2-cells (2-morphisms). Twisted homomorphisms hav e certain natural involutiv e symmetry . Lemma 2.1 .1. If ( f , F ) : H → H ′ is a twiste d homomorphism, then t ( f , F ) = ( f , t ( F )) : H co → ( H ′ ) co is a twiste d homomorphism b etwe en bialgebr as with opp osite c omultiplic ation. Pr o of. The condition (1) for ( f , t ( F )) is the transp os e (the result of applying t ) of the co ndition (1 ) for ( f , F ). The sa me with the no rmalisatio n condition for t ( F ). The 2-co cycle equation for t ( F ) is the 2-co cycle equation for F where the first and the last tenso r factor have bee n interchanged: ( t ( F ) ⊗ 1)( t ∆ ′ ⊗ I ) t ( F ) = ( t 2 t 1 t 2 )((1 ⊗ F )( I ⊗ ∆ ′ )( F )) = ( t 1 t 2 t 1 )(( F ⊗ 1)(∆ ′ ⊗ I )( F )) = (1 ⊗ t ( F ))( I ⊗ t ∆ ′ ) t ( F ) . Note that if we ha d a ga uge transfor mation a : ( f , F ) → ( g , G ), then a : t ( f , F ) → t ( g , G ) is als o a g a uge transformation. Moreov er, a ll compositio ns are compatible with t . In (2-)categor ical langua ge t : Tw → Tw is a 2-functor (2-isomor phism). W e call a twisted homomorphism ( f , F ) : H → H ′ of coco mm utative bialge- bras symmetric if t ( f , F ) = ( f , F ). Note that a twisted homomor phism ga ug e isomorphic to a s y mmetric t wis ted homomorphism is symmetric its e lf. 2.2 Twisted homomorphisms and Galois (co) algebras It was obse r ved by Drinfeld [3] that a twist F on a bialgebra H defines a new coalgebr a structure on H : ∆ F ( x ) = F ∆( x ) , x ∈ H . Coasso c ia tivity for ∆ F is equiv alent to the 2-co cycle co ndition on F . The prop- erty ∆ F ( xy ) = ∆ F ( x )∆( y ) g ua rantees that the r ight H -mo dule structure on H is a n H - mo dule coalgebra s tr ucture on ( H, ∆ F ). Recall that a coalgebr a ( C, δ ) is a n H - mo dule c o algebr a if C is an H -mo dule a : C ⊗ H → C and δ ( xy ) = δ ( x )∆( y ) , x ∈ C, y ∈ H. An H -mo dule coalg ebra C is Galois if the comp osition C ⊗ H δ ⊗ I / / C ⊗ C ⊗ H I ⊗ a / / C ⊗ C 5 is an isomorphism. It is not hard to see that if H is a Hopf algebra ( H, ∆ F ) is a Galois H -module coalgebra. Moreov er Ga lois H -module c o algebra s of the form ( H, ∆ F ) are characterised by the so-ca lled norma l basis prop erty . A Galois H - mo dule coalg ebra C has a normal b asis if C is isomorphic to H as a n H -mo dule. F or a twisted homomo rphism ( f , F ) : H → H ′ , the H ′ -Galois coa lgebra ( H ′ , ∆ F ) comes equipp ed with the left H -action H ⊗ H ′ → H ′ sending x ⊗ y to f ( x ) y . It follows from the definition of t wisted homomorphism that this action preserves the copro duct ∆ F : F ∆ ′ ( f ( x ) y ) = F ∆ ′ ( f ( x ))∆ ′ ( y ) = ( f ⊗ f )∆( x ) F ∆ ′ ( y ) . Comp osition o f twisted homomorphisms co r resp onds to the following op er- ation on Galois co algebra s. Let C b e a Galois H ′ -mo dule coa lgebra and C ′ be a Galo is H ′′ -mo dule coalgebra with compa tible left H ′ -action. Then the tensor pro duct of H ′ -mo dules C ⊗ H ′ C ′ is a Galois H ′′ -coalge br a with resp ect to the copro duct t 23 ( δ ⊗ δ ′ ). Moreov er a left H -action on C compatible with H ′ -mo dule coalgebra structure will pa s s on to C ⊗ H ′ C ′ . F or succes sive t wisted homomorphisms ( H, ∆ , ε ) ( f ,H ) / / ( H ′ , ∆ ′ , ε ′ ) ( f ′ ,F ′ ) / / ( H ′′ , ∆ ′′ , ε ′′ ) the tensor pro duct ( H ′ , ∆ F ) ⊗ H ′ ( H ′′ , ∆ F ′ ) is iso morphic to H ′′ as an H − H ′′ - bimo dule via x ⊗ y 7→ f ′ ( x ) y . Moreov er, the tensor pr o duct of comult iplications ∆ F ⊗ ∆ F ′ is car ried out into the comultiplication ∆ f ( F ) F ′ . Indeed, the tensor pro duct of com ultiplications follow ed b y the isomorphism sends x ⊗ y int o ( f ′ ⊗ f ′ )( F ∆ ′ ( x )) F ′ ∆ ′′ ( y ) which can be r ewritten as ( f ′ ⊗ f ′ )( F ∆ ′ ( x )) F ′ ∆ ′′ ( y ) = ( f ′ ⊗ f ′ )( F )( f ′ ⊗ f ′ )(∆ ′ ( x )) F ′ ∆ ′′ ( y ) = ( f ′ ⊗ f ′ )( F ) F ′ ∆ ′′ ( f ′ ( x ))∆ ′′ ( y ) = ( f ′ ⊗ f ′ )( F ) F ′ ∆ ′′ ( f ′ ( x ) y ) . Finally ( f ′ ⊗ f ′ )( F ) F ′ ∆ ′′ ( f ′ ( x ) y ) is the copro duct ∆ f ( F ) F ′ applied to f ′ ( x ) y . A ga uge transformatio n a ∈ H ′ of twisted homomorphisms ( f , F ) , ( f ′ , F ′ ) : ( H, ∆ , ε ) → ( H ′ , ∆ ′ , ε ′ ) defines a ho momorphism x 7→ ax of Galois H ′ -mo dule coalgebr as. P r eserv a tion of H ′ -mo dule structure is obvious. Compatible H - mo dule str ucture is preser ved b y the pro p erty (2) o f ga uge tra nsformations, compatibility with coalgebr a structures follows from the prop erty (3): F ∆ ′ ( ax ) = F ∆ ′ ( a )∆ ′ ( x ) = ( a ⊗ a ) F ′ ∆( x ) . Define a bicatego r y Gal w ith ob jects being Hopf alg ebras, arrows from H to H ′ being Galois right H ′ -mo dule coalgebra s with compatible left H -action and comp osition given by tensor pro duct, and homomor phisms of H − H ′ -bimo dule coalgebr as as 2-ce lls . The ab ov e constr uction defines a psedofunctor Tw → Gal . In the case of finite dimensiona l Hopf algebras we ca n repla ce Galois c oal- gebra with the more familiar notion of Galois alg ebra. Note that for a finite dimensional Hopf algebra H , a Galo is H -module co a lgebra must be finite di- mensional (of the same dimension). Note als o that the dua l algebr a to a Galois 6 H - c oalgebr a will be a Ga lois H -algebra. Recall that an H -algebr a R is an al- gebra and an H -mo dule in suc h wa y that the multiplication ma p µ is a map of H - mo dules: µ ∆( x )( a ⊗ b ) = xµ ( a ⊗ b ) = x ( ab ) , x ∈ H , a, b ∈ R. The cr osse d pr o duct R ∗ H of H with an H -algebra R is an algebra , which a s a vector space is iso morphic to the tenso r pro duct R ⊗ H . Denote elements of R ∗ H corresp onding to tensor s a ⊗ x b y a ∗ x . The pr o duct on R ∗ H is given by the rule: ( a ∗ x )( b ∗ y ) = X ( x ) ax 1 ∗ x 2 y . Here we use the so -called Swidler’s notation for the copro duct ∆( x ) = P ( x ) x 1 ⊗ x 2 . F ollowing [10] w e c all a n H - algebra R Galoi s if the homo morphism of algebras A ∗ H → E nd ( A ) , a ∗ x 7→ ( b 7→ ax ( a )) is a n isomorphism. F or a finite dimensional H the dual of the Galois coalgebr a ( H, ∆ H ) is the dual space H ∗ with the m ultiplica tion: ( l ∗ F l ′ )( x ) = ( l ⊗ l ′ )∆ F ( x ) = ( l ⊗ l ′ )( F ∆( x )) . This ca n be seen as an ordinary multiplication on H ∗ t wisted b y F with res pe c t to the (rig ht) H -action on H ∗ : l ∗ F l ′ = X l F 1 ( l ′ ) F 2 . Here F = P F 1 ⊗ F 2 and l y ( x ) = l ( xy ). 3 Categorical in terpretations Most of the material included in this section is pretty standar d (for exa mple, see [6]; for more categ o rically o r iented treatment see [13]). 3.1 Monoidal categories and functors A monoidal c ate gory is a categ ory G with a functor ⊗ : G × G − → G ( X, Y ) 7→ X ⊗ Y ( tensor pr o duct ), a natura l collection of isomorphisms ( asso ciativity c onstr aint ) ϕ X,Y ,Z : X ⊗ ( Y ⊗ Z ) → ( X ⊗ Y ) ⊗ Z for any X , Y , Z ∈ G which satisfies the following p entagon axiom : ( X ⊗ ϕ Y , Z,W ) ϕ X,Y ⊗ Z,W ( ϕ X,Y ,Z ⊗ W ) = ϕ X,Y ,Z ⊗ W ϕ X ⊗ Y ,Z,W 7 and an ob ject 1 ( unit ob ject) together with natural isomorphisms ρ X : X ⊗ 1 → X λ X : 1 ⊗ X → X such that λ 1 = ρ 1 λ X ⊗ Y = λ X ⊗ I : 1 ⊗ X ⊗ Y → X ⊗ Y , ρ X ⊗ I = I ⊗ λ Y : X ⊗ 1 ⊗ Y → X ⊗ Y , ρ X ⊗ Y = I ⊗ ρ Y : X ⊗ Y ⊗ 1 → X ⊗ Y for any X , Y ∈ G . Here I deno te the iden tit y mor phism. It is known that the first t wo conditions follow from the rest. W e for mulate them for the s ake of symmetry . The celebra ted Ma cLane coherenc e theorem [9] says that there is a unique isomorphism b etw een any t wo bra cket ar rangements on the tensor pro ducts of ob jects X 1 , ..., X n , which is a comp osition o f (tensor pro ducts o f ) the asso ciativ- it y constr aints. This fact a llows to omit br a ck ets in tensor pro ducts. It is also easy to see that the unit ob ject is unique up to an isomo r phism. Mor e precisely , any mono ida l c a tegory is monoidally equiv alent to a strict monoidal category (a monoidal ca tegory with identit y asso cia tivity and unit ob ject constr aints) (see, for example, [7]). A monoidal functor be tw een monoida l catego ries G and H is a functor F : G → H with a natural co lle c tion o f is o morphisms (the so-called m onoidal structur e ) F X,Y : F ( X ⊗ Y ) → F ( X ) ⊗ F ( Y ) for any X, Y ∈ G satisfying the c oher enc e axiom: ( I ⊗ F Y , Z ) F X,Y ⊗ Z = ( F X,Y ⊗ I ) F X ⊗ Y ,Z (4) for any ob jects X , Y , Z ∈ G . A natural trans fo rmation f : F → G o f monoida l functors F and G is monoidal if G X,Y f X ⊗ Y = ( f X ⊗ f Y ) F X,Y for any X , Y ∈ G . Here w e will b e mostly interested in tensor categor ies and functors (linear ov er the gr ound field k ). Recall that a monoidal ca tegory is ten sor if it is k - linear a b elian and the tensor pro duct is k -linear and bi-exact. F or a monoida l functor betw e en tenso r categories to b e tensor we will a sk it to b e k -linear and left exac t. Denote b y T ens the 2-ca tegory of tensor categ ories, with tensor functors and mono idal natural transformations . 3.2 Categories of mo dules o v er bialgebras Recall that the comultiplication in H can b e used to define a structure o f a n H - mo dule on the tensor pro duct M ⊗ k N of t w o H -modules: h ∗ ( m ⊗ n ) = ∆( h )( m ⊗ n ) h ∈ H , m ∈ M , n ∈ N . 8 The co asso cia tivity axiom for the co pro duct implies that the obvious a sso cia- tivit y c o nstraint for vector spaces ϕ : L ⊗ ( M ⊗ N ) → ( L ⊗ M ) ⊗ N ϕ ( l ⊗ ( m ⊗ n )) = ( l ⊗ m ) ⊗ n is H -linear. The counit defines an H -module structur e on the ground field k and the counit axiom guara ntees that this is a unit ob ject. Thus the catego ry H M od of (left) mo dules ov er a bialgebra b ec omes monoidal. F or a homomo r phism of alg e bras f : H → H ′ define by f ∗ : H ′ M od → H - M od the i nverse imag e functor, whic h turns an H ′ -mo dule M into an H -module f ∗ ( M ). Here as a vector space , f ∗ ( M ) is the same as M but with a new mo dule structure x.m = f ( x ) m for x ∈ H and m ∈ M . On the level o f categor ies of mo dules, t wis ted homo morphisms a nd gauge transforma tions take the following meaning. Prop ositi on 3.2 .1. F or a twiste d homomorp hism ( f , F ) : H → H ′ the inverse image functor f ∗ : H ′ − M od → H − M od b e c omes ten s or, with the monoidal structur e given by multiplic ation with the twist: F M ,N : f ∗ ( M ⊗ H ) → f ∗ ( M ) ⊗ f ∗ ( N ) , m ⊗ n 7→ F ( m ⊗ n ) . Comp ositions of twiste d homomorphisms and c orr esp onding functors ar e r elate d as fol lows: (( f , F ) ◦ ( g , G )) ∗ = ( g, G ) ∗ ◦ ( f , F ) ∗ . A gauge tr ansformation a : ( f , F ) → ( g , G ) defines a monoidal natur al tr ansfor- mation a : ( f , F ) ∗ → ( g , G ) ∗ : a M : f ∗ ( M ) → g ∗ ( M ) , m 7→ am. Comp ositions of gauge tr ansformations c orr esp ond to c omp ositions of natur al tr ansformations. Pr o of. It is straightforw ard to see that the condition (1) guarantees H -linearity of the monoidal constraint F M ,N while the 2-co cycle condition for F is equiv alent to the coherence a x iom (4). Similarly , the condition (2) for a gauge transfor - mation a s ays tha t a M is a mo rphism o f H - mo dules a nd the condition (3 ) is equiv alent to the monoidality of a M . 2-catego rically the propo sition 3.2 .1 says that the inv erse image functor co n- struction defines a 2-functor Tw → T ens co ntra v a riant on morphis ms and cov ariant on 2 -cells. W e can extend this to a bifunctor Gal → T ens a s follows. Prop ositi on 3.2.2. A Galois H ′ -mo dule c o algebr a C with c omp atible H - action defines a tensor functor C ⊗ H ′ : H ′ M od → H M od with the monoidal structu r e C ⊗ H ′ ( M ⊗ N ) δ ⊗ I   C ⊗ H ′ M ⊗ C ⊗ H ′ N ( C ⊗ C ) ⊗ H ′ ( M ⊗ N ) / / ( C ⊗ C ) ⊗ H ′ ⊗ H ′ ( M ⊗ N ) t 23 O O 9 Homomorphisms of H - H ′ -bimo dule c o algebr as define monoidal natur al tr ansfor- mations. Pr o of. It follows fro m the Galo is prop erty that the ab ov e monoida l s tr ucture is an isomorphism. Coasso ciativit y o f the copro duct implies the co herence axiom. Obviously , comp osition of functors cor resp onds to the tensor pr o duct o f coalgebr as. 4 Twisted automorphisms of a Hopf algebra In this section w e will describe Cat-groups of twisted automor phisms along with some of their natural Ca t-subgro ups . One o f the rea s ons (which will b e particularly imp or tant in the las t sec tio n) why Cat-gro ups are more natural ob jects to deal with ra ther than gr o ups of (classes of ) t wisted automor phisms is the following. Supp ose that we wan t to define an actio n of a group G on a bialgebra H b y twisted homo morphisms. A homomor phis m from the gro up G int o the g r oup O ut Tw ( H ) of clas ses of twisted homomorphism would just not hav e enough information. A homomorphism from G to the group Aut Tw ( H ) of t wisted a utomorphisms w ould do, but that would not capture all the cases. The right ans wer is a (monoidal) functor G → A ut Tw ( H ) of Cat-gro ups (a map of crossed modules of gr o ups). 4.1 Categorical groups, crossed mo dules and their maps Recall that a c ate goric al gr oup is a monoidal ca tegory in whic h every ar row is invertible (monoida l group oid) and for every ob ject X there is an ob ject X ∗ with an ar row e X : X ∗ ⊗ X → I (a dual ob ject). A categor ic a l gr oup is s trict (or a C at-gr oup ) if it is strict as a monoida l catego ry a nd e X can be c hosen to be an ident it y . In o ther words a Cat-g roup is a categorical group whose ob jects form a group (with the tensor product). A slight mo difica tion of Mac Lane’s c o herence theorem [9] s ays that a ny categor ic al group is mono idally equiv alent to a str ict one. Note that Cat-groups are group ob jects in the categor y of categories (th us the name). F or an y ob ject A o f a 2-category A the catego ry A ut A ( A ) of a utomorphisms of A (in vertible 1-morphisms A → A ), with the comp ositio n as the tensor pro d- uct and bijective natural transformations a s morphisms, is a Ca t-group. It is well known (see [1] for the history) that Cat-g roups are the same as crossed modules of Whitehead ([15]). Recall that a cr osse d mo dule of groups is a pair of gro ups P, C with a (left) action of P on C (b y group automor phisms): P × C → C , ( p, c ) 7→ p c and a homo morphism of g roups P ∂ ← C 10 such that ∂ ( p c ) = p∂ ( c ) p − 1 , ∂ ( c ) c ′ = cc ′ c − 1 . F or a Cat- group G the cor resp onding cross ed complex consists of the g roup o f ob jects P , the gr oup of morphisms X → I into the identit y ob ject with the pro duct: ( X a → I ) . ( Y b → I ) = X ⊗ Y a ⊗ b − → I ⊗ I = I , the action Y ( X a → I ) = Y ⊗ X ⊗ Y ∗ Y ⊗ a ⊗ Y ∗ − → Y ⊗ I ⊗ Y ∗ = I , and the ho momorphism ∂ : C → P sending X → I int o X . Relative simplicity and compactness of the notion of crossed module as a description for Cat-gr oups has its downside. A monoidal functor b etw ee n Cat- groups will corre s p o nd to something less obvious (and less well-known) than a homomorph ism of cros s ed modules (a pair of gr oup homomorphisms preser ving all the str uctures). That will describe o nly strict monoida l functors. T o deal with genera l mo noidal functors w e need the following weak er rela tion. A map of cr ossed mo dules ( P , C ) → ( E , N ) is a triple ( τ , ν, θ ) where τ and ν are ma ps making the diagram c ommutativ e P τ   C ∂ o o ν   E N ∂ o o and θ : P × P → N such that τ ( pq ) = ∂ ( θ ( p, q )) τ ( p ) τ ( q ) , p, q ∈ P, ν ( ab ) = θ ( ∂ ( a ) , ∂ ( b )) ν ( a ) ν ( b ) , a, b ∈ C, θ ( p , q r ) τ ( p ) θ ( q , r ) = θ ( pq , r ) θ ( p, q ) , p, q , r ∈ P , θ (1 , q ) = 1 = θ ( p, 1) , p, q ∈ P , ν ( p a ) = θ ( p, ∂ ( a )) τ ( p ) ν ( a ) . Complete inv a riants of a categ o rical-g roup G with resp ect to monoida l equiv- alences are π 0 ( G ) , π 1 ( G ) , φ ∈ H 3 ( π 0 ( G ) , π 1 ( G )) , where the first is the gr oup of isomorphis m classe s o f o b jects, the second is the ab elian gr oup ( π 0 ( G )-module) Aut G ( I ) of automorphis ms of the unit ob ject and the third is a coho mo logy clas s (the asso ciator ). In the crossed mo dule se tting π 0 = cok e r ( ∂ ) , π 1 = ker ( ∂ ) . 11 Note that the imag e o f ∂ is normal so the cokernal has sense. The class φ is defined as follows: ch o ose a sectio n σ : coke r ( ∂ ) → P and a map a : cok e r ( ∂ ) × cok er ( ∂ ) → C such that σ ( f g ) = ∂ ( a ( f , g )) σ ( f ) σ ( g ) , f , g ∈ cok er ( ∂ ) . Then for an y f , g , h ∈ cok er ( ∂ ) the expressio n a ( f , g h ) σ ( f ) a ( g , h ) a ( f , g ) − 1 a ( f g , h ) is a lwa ys in the k ernel o f ∂ and is a gro up 3-co c ycle of coke r ( ∂ ) with coefficients in k er ( ∂ ). The cohomolog y class φ do es not depend on the choices made. Analogously , co mplete inv ariants of a monoidal functor F : G → F betw een categoric al groups with resp ect to monoidal isomorphisms a re π 0 ( F ) : π 0 ( G ) → π 0 ( F ) , π 1 ( F ) : π 1 ( G ) → π 1 ( F ) , θ ( F ) : π 0 ( G ) × π 0 ( G ) → π 1 ( F ) , where the first is the homomor phism of gro ups, the second is the homomor phism of π 0 ( G )-modules a nd the third is really a class in C 2 ( π 0 ( G ) , π 1 ( F )) /B 2 ( π 0 ( G ) , π 1 ( F )) such that d ( θ ( F )) = π 1 ( F ) ∗ ( φ ( G )) − π 0 ( F ) ∗ ( φ ( F )) Here π 0 ( F ) ∗ : C ∗ ( π 0 ( F ) , π 1 ( F )) → C ∗ ( π 0 ( G ) , π 1 ( F )) , π 1 ( F ) ∗ : C ∗ ( π 0 ( G ) , π 1 ( G )) → C ∗ ( π 0 ( G ) , π 1 ( F )) are the maps of co c ha in complexes induced by the group homomo r phisms π 0 ( F ), π 1 ( F ). 4.2 Cat-groups of t wisted automorphisms As a pa rt of the 2 -categor y Tw the inv er tible twisted endomo r phisms of a Hopf algebra H (t wisted automorphisms) and gaug e transforma tions b etw een them form a Cat-group A ut Tw ( H ). The co rresp o nding c r ossed mo dule of groups has the form Aut Tw ( H ) ∂ ← H × ε . Here Aut Tw ( H ) is the gr oup of t wisted a utomorphisms of H with re s p e ct to the comp osition, H × ε is the g roup of inv er tible ele ments x of H such that ε ( x ) = 1, and ∂ sends x into the pa ir (an inner twisted automorphism) ( x ( ) , ( x ⊗ x )∆( x ) − 1 ) where the fir st compone nt is the conjuga tion automorphism: x ( ) : H → H, x ( y ) = xy x − 1 . The a ction of Aut Tw ( H ) on H × is given by the action of the fir s t co mp o nent ( f , F )( y ) = f ( y ). There a r e t wo imp or ta nt Cat-subgr oups in A ut Tw ( H ). The first is the full Cat-subgro up A ut 1 Tw ( H ) of t wisted automorphisms w ith the iden tity as the first compo nent. I ts cros sed module is Aut 1 Tw ( H ) ∂ ← ( Z ( H ) ε ) × . 12 Here Aut 1 Tw ( H ) is the group o f invariant twists on H (inv ertible elements of H ⊗ H comm uting with the image ∆( H ) a nd satisfying the 2-cocy cle condition), ( Z ( H ) ε ) × is the group o f in vertible elements of the cen tre of counit 1: ε ( x ) = 1 . Again ∂ assigns to x the in v ar iant t wist ( x ⊗ x )∆( x ) − 1 . The action of Aut 1 Tw ( H ) on ( Z ( H ) ε ) × is tr ivial. The s e cond is the full Cat-subg roup A ut bialg ( H ) of bialgebr a automorphisms of H . Here the crossed mo dule is Aut bialg ( H ) ∂ ← G ( H ) , where Aut bialg ( H ) is the g roup of a utomorphisms of H as a bialgebra, G ( H ) = { x ∈ H , ∆( x ) = x ⊗ x } is the group of gro up-like elemen ts of H a nd ∂ sends x in to the conjuga tion automorphism. The action of Aut bialg ( H ) on G ( H ) is obvious. Note that the Cat-s ubg roup A ut 1 Tw ( H ) is what migh t be called normal : the comp onents of its cros sed mo dule ar e normal s ubgroups in the comp onents o f the cr ossed c omplex for Aut Tw ( H ) a nd the action of Aut Tw ( H ) o n H × preserve the subgroup Z ( H ) × . The Cat-subgr o up A u t bialg ( H ) is no t in general norma l. In the nex t part we will characterise it as the stabiliser of a certain action. Recall that an action of a Cat-gr oup (monoidal categ o ry) G on a category A is a monoidal functor G → E nd ( A ) into the catego r y o f endofunctors o n A . F or an ob ject A ∈ A the stabiliser S t G ( A ) is the catego ry o f pairs ( G, g ), where G is a n o b ject and g : G ( A ) → A is an isomor phis m in A . A morphism of pair s ( G, g ) → ( F, f ) is a morphism x : G → F in G such that the diagram G ( A ) x A / / g ! ! D D D D D D D D F ( A ) f } } z z z z z z z z A commute. 4.3 Action on t wists Here w e examine the (categorica l) a ction of the Cat-gro up A ut Tw ( H ) on the category of t wis ted ho mo morphisms Tw ( k , H ) given by the comp osition. Let us start with the ca tegory Tw ( k , H ). An y twisted ho mo morphism k → H mu st hav e the form ( ι, F ) where ι : k → H is the unit inclusio n and F ∈ H ⊗ 2 is a n in vertible element sa tisfying the 2- co cycle co ndition (a twist ). A gauge tra nsformation ( ι, F ) → ( ι, F ′ ) is an inv er tible element a ∈ H such tha t F ′ ( a ⊗ a ) = ∆( a ) F . The categ ory Tw ( k , H ) has a marked ob ject ( ι, 1). On the level of ob jects the action Aut Tw ( H ) × Tw ( k , H ) → Tw ( k , H ) has the form ( f , F ) ◦ ( ι, F ′ ) 7→ ( ι, F ′ f F ) . An ob ject of the stabilis er S t Aut Tw ( H ) ( ι, 1) is a triple ( f , F , a ), where ( f , F ) is an o b ject of Aut Tw ( H ) and a ∈ H is an inv er tible ele ment such that F = 13 ( a ⊗ a )∆( a ) − 1 (a transformation ( f , F ) ◦ ( ι, 1) → ( ι, 1)). The elemen t a can be in terpr eted as a gauge transforma tio n ( f , F ) → ( a ( ) ◦ f , 1). In particular , the comp ositio n a ( ) ◦ f o f f with conjuga tion with a is an automor phism of bialgebras . The naturality of this cons truction implies the following result. Prop ositi on 4.3.1. The inclusion A ut bialg ( H ) → S t Aut Tw ( H ) ( ι, 1) is an e quiv- alenc e of c ate gories. Now we de s crib e the stabilizer S t Aut 1 Tw ( H ) ( ι, 1) in the Cat-subg roup o f in- v ariant twists. Its ob jects are pa irs ( F , a ) where F is an inv ariant t wist on H and a ∈ H is a n inv ertible element such that F = ( a ⊗ a )∆( a ) − 1 (a trans- formation ( I , F ) ◦ ( ι, 1) → ( ι, 1)). Similarly , the ele ment a can b e seen as a gauge transformation ( I , F ) → ( a ( ) , 1). In particular, the co njugation a ( ) is an automorphism of bialg ebras. Thus we hav e the following. Prop ositi on 4.3. 2. The stabiliser S t Aut 1 Tw ( H ) ( ι, 1) is e quivalent to t he Cat- gr oup A ut inn bialg ( H ) with the cr osse d mo dule: Aut inn bialg ( H ) ∂ ← G ( H ) , (5) wher e Aut inn bialg ( H ) is the gr oup of bialgebr a automorphisms of H which ar e inner as algebr a automorphisms, i.e. the kernel of the homomorphism Aut bialg ( H ) → Ou t alg ( H ) . In particular, π 0 ( S t Aut 1 Tw ( H ) ( ι, 1)) is isomorphic to the kernel of O ut bialg ( H ) → Ou t alg ( H ). W e finish this se ction with a descr iption of the or bits of the Cat- g roup a ction of A ut Tw ( H ) on Tw ( k , H ) in terms of twisted for ms of the bialg ebra H . Recall that a t wis t F ∈ H ⊗ 2 allows us to define a new copr o duct on H : ∆ F ( x ) = F − 1 ∆( x ) F. W e call the bia lgebra T f ( F ) = ( H, ∆ F ) an F -twiste d form of H (or just a twiste d form ). Note that a gauge tr ansformatio n of twists F ′ ( a ⊗ a ) = ∆( a ) F defines a homomorphism o f bialgebras ( ) a : ( H , ∆ F ′ ) → ( H, ∆ F ). Indeed, ∆ F ( a − 1 xa ) = F − 1 ∆( a ) − 1 ∆( x )∆( a ) F = ( a ⊗ a ) − 1 ( F ′ ) − 1 ∆( x ) F ′ ( a ⊗ a ) = ( a ⊗ a ) − 1 ∆ F ′ ( x )( a ⊗ a ) . Thu s w e hav e a functor T f : Tw ( k , H ) → B ial g into the categ o ry B ial g o f bialgebras a nd their automorphisms. Prop ositi on 4. 3.3. The functor T f : Tw ( k , H ) → B ial g is st rictly c onstant on orbits of the actio n of A ut Tw ( H ) on Tw ( k , H ) , i.e. isomorphisms of twiste d forms f : ( H , ∆ F ) → ( H, ∆ F ′ ) ar e in 1-1 c orr esp ondenc e with t wiste d automor- phisms ( f , F ′′ ) : ( H, ∆) → ( H, ∆) such that ( f , F ′′ ) ◦ ( ι, F ) = ( ι, F ′ ) . 14 Pr o of. If ( f , F ′′ ) : ( H, ∆) → ( H , ∆) is a twisted automorphism such that F ′′ ( f ⊗ f )( F ) = F ′ then ( f ⊗ f )∆ F ( x ) = ( f ⊗ f )( F ) − 1 ( f ⊗ f )∆( x )( f ⊗ f )( F ) = ( F ′ ) − 1 F ′′ ( f ⊗ f )∆( x )( F ′′ ) − 1 F ′ = ( F ′ ) − 1 ∆( f ( x )) F ′ = ∆ F ′ ( f ( x )) so that f is an isomorphism o f bia lgebras ( H , ∆ F ) → ( H, ∆ F ′ ). Conv er sely , for an isomo rphism of bialg e bras f : ( H , ∆ F ) → ( H, ∆ F ′ ), the e le ment F ′′ = F ′ ( f ⊗ f )( F ) − 1 defines a structure of twisted isomorphism ( f , F ′′ ) : ( H, ∆) → ( H , ∆): F ′′ ( f ⊗ f )∆( x ) = F ′ ( f ⊗ f )( F ) − 1 ( f ⊗ f )∆( x ) = F ′ ( f ⊗ f )( F − 1 ∆( x )) = ( F ′ ) − 1 F ′ ∆( f ( x )) F ′ ( f ⊗ f )( F ) − 1 . In particular , the o rbits of the group ac tion of Aut Tw ( H ) on T w ( k , H ) are in 1-1 co rresp o ndence with isomor phism class es of twisted forms o f H . 4.4 Action on triangular structures Here w e re fine the action of t wis ted automor phisms on triangula r structures to the level of ca tegories . Recall that a tr iangular structu r e on a bialgebra H is an inv e r tible element R ∈ H ⊗ H (a universal R -matrix ) satisfying Rt ∆( x ) = ∆( x ) R ∀ x ∈ H , (6) along with t riangle e quations : ( I ⊗ ∆)( R ) = R 13 R 12 , (∆ ⊗ I )( R ) = R 13 R 23 , normalisation : ( ε ⊗ I )( R ) = ( I ⊗ ε )( R ) = 1 , and unitarity condition: R 21 = R − 1 . Here R 21 = (12) R is the transp osition of tensor factors o f R ∈ H ⊗ H , R 12 = R ⊗ 1, R 13 = ( I ⊗ (12))( R 12 ) etc. Note tha t any univ e r sal R - matrix on a co commutativ e bialgebra H sa tisfies the 2-co cycle condition: ( R ⊗ 1)(∆ ⊗ I )( R ) = R 12 R 13 R 23 = ( I ⊗ ∆)( R )(1 ⊗ R ) . Denote b y T r ( H ) the set of triangular structures on the bialgebra H . It w as obs e r ved b y Drinfeld (see als o [11]) that the map H ∗ → H , l 7→ ( l ⊗ I )( R ) (7) 15 is a homomorphism of algebra s and anti-homomorphism o f coalg ebras for a ny triangular structure R . In particular, its image is a (finite-dimensional) sub- bialgebra H R in H , R b elongs to H ⊗ 2 R (the so-ca lled minimal triangular s ub- bialgebra), and the map (7) factor s as follows H ∗ → H ∗ R ≃ H R → H , where the first surjection H ∗ → H ∗ R is dua l to the la st inclus ion H R → H and the isomor phism H ∗ R ≃ H R is self-dual. It is well known that the minimal triangular s ubalgebra for a co co mmut ative Hopf algebra p os sesses a quite simple description. Prop ositi on 4.4.1. F or a c o c ommu tative Hopf algebr a H t he set T r ( H ) of triangular stru ctur es is isomorphic to the set of p airs ( A, b ) , wher e A is a normal c ommutative c o c ommutative finite dimensional s u b-bialgebr a in H and b : A ∗ → A is an H -invari ant isomorphi sm of bi-algebr as. Pr o of. As a sub-biagebr a of H the minimal tr ia ngular sub-bialgebr a H R m ust b e co commutativ e. It is commutativ e since H ∗ R is a sub-bialg ebra of H ∗ . Norma lity of H R and H -inv ariance o f b : H ∗ R ≃ H R are equiv alent to the condition (6) for R . F or a t wisted automorphism ( f , F ) and a triangular structure R on H define a twiste d triangular structure: R ( f ,F ) = F 21 ( f ⊗ f )( R ) F − 1 . It is str a ightforw ard to v erify that the prop erties of the R -matrix are preser ved. Moreov er, ga uge isomo rphic twisted automor phisms act eq ually . Indeed, for g ( x ) = af ( x ) a − 1 and G = ∆( a ) F ( a ⊗ a ) − 1 , R ( g,G ) = ∆( a ) F 21 ( a ⊗ a ) − 1 ( a ⊗ a )( f ⊗ f )( R )( a ⊗ a ) − 1 ( a ⊗ a ) F − 1 t ∆( a ) − 1 = ∆( a ) R ( f ,F ) t ∆( a ) − 1 = R ( f ,F ) . Thu s an actio n o f the group O ut Tw ( H ) on the set T r ( H ) is defined. Recall that the Drinfeld element of a triangula r structure R o n a Hopf algebra H is u = µ ( I ⊗ S )( R ), where S : H → H is the antipo de and µ : H ⊗ H → H is the multiplication map. It was prov en in [5] that u (also see [8]) is a group- like element: ∆( u ) = u ⊗ u . Moreov er , if H is co co mmutative then u is ce ntral and of order 2. Thus we hav e a ma p u : T r ( H ) → ( G ( H ) ∩ Z ( H )) 2 from the s et of triangular structures to the 2-tors io n subgroup of group-like central elements. This ma p a dmits a section, whic h sends a gr oup-like in volution u to an R -ma trix R u = 1 2 (1 ⊗ 1 + 1 ⊗ u + u ⊗ 1 − u ⊗ u ) . Prop ositi on 4.4.2. F or a c o c ommutative Hopf algebr a the Drinfeld element map is c onstant on orbits of t he action of Ou t 1 Tw ( H ) on T r ( H ) . 16 Pr o of. It is q uite straightforw ard to chec k that the Drinfeld element is constant on o rbits. Indeed, it was pro v ed in [5] that the map µ ( I ⊗ S ) : ( H ⊗ 2 ) H → Z ( H ) is a homo mo rphism of alg ebras. Thus µ ( I ⊗ S )( F 21 RF − 1 ) = µ ( I ⊗ S )( F 21 ) µ ( I ⊗ S )( R ) µ ( I ⊗ S )( F ) − 1 equals µ ( I ⊗ S )( R ) since µ ( I ⊗ S )( F ) = 1. Categoric a lly triangula r structure s corr esp ond to symmetric structures o n the ca tegory of mo dules. Recall that a mo noidal ca tegory G is s ymmetric if it is equipp e d with a collectio n o f isomor phisms c X,Y : X ⊗ Y → Y ⊗ X natural in X, Y ∈ G and sa tisfying the follo wing axio ms: c X,Y c Y , X = 1 , symmetry , hexagon a xioms: c X,Y ⊗ Z = φ Y , Z,X ( Y ⊗ c X,Z ) φ − 1 Y , X,Z ( c X,Y ⊗ Z ) φ X,Y ,Z , c X ⊗ Y ,Z = φ − 1 Z,X,Y ( c X,Z ⊗ Y ) φ X,Z,Y ( X ⊗ c Y , Z ) φ − 1 X,Y ,Z . Note that the last condition is r edundant and included her e for the sake of symmetry . Prop ositi on 4.4.3. A triangular st ructur e R on a bialgebr a H defines a sym- metric st ructur e: c M ,N : M ⊗ N → N ⊗ N , m ⊗ n 7→ R ( n ⊗ m ) on t he c ate gory H − M od Pr o of. The condition (6 ) implies that c M ,N is a mor phism o f H -mo dules: c M ,N (∆( x )( m ⊗ n )) = Rt ∆( x )( n ⊗ m ) = ∆( x ) R ( n ⊗ m ) = ∆( x ) c M ,N ( m ⊗ n ) . T r iangle eq uations are equiv a lent to hexag on axio ms. Normalisation for R giv es the conditions c 1 ,N = I , c M , 1 = I . Unitarity for R implies symmetry for c . Monoidal auto equiv alences of a monoidal catego ry act natura lly on the set of symmetric structures of the categor y . F or a monoidal auto equiv alence F and a symmetry c define the new symmetry c F by F ( X ) ⊗ F ( Y ) c F F ( X ) ,F ( Y ) / / F ( Y ) ⊗ F ( X ) F ( X ⊗ Y ) F X,Y O O F ( c X,Y ) / / F ( Y ) ⊗ F ( X ) F Y ,X O O It is straig ht forward to see that this actio n corresp o nds to the action of twisted homomorphisms on R -matrice s . 17 5 Twisted automorphisms of univ ersal en v elop- ing algebras In this part we lo ok a t twisted a utomorphisms of universal env elo ping alg ebras ov er formal p ower series k [[ h ]]. 5.1 In v arian t t wists of U ( g )[[ h ]] Let F ∈ U ( g )[[ h ]] g be a t wist. Expand it as a for mal p ower ser ies in h with co efficients in U ( g ): F = ∞ X i =0 F i h i . Since inv ertible elements of a universal env eloping algebra ov er a field k of characteristic zero are trivia l (scala rs), the cons tant ter m F 0 of F must b e the ident it y . Let X = F l be the first no n- zero co efficie nt . The degre e l part of the 2-co cycle equation is the additive 2-co cycle condition for X : 1 ⊗ X + ( I ⊗ ∆)( X ) = X ⊗ 1 + (∆ ⊗ I )( X ) . F ollowing Drinfeld [4] consider the complex ( H ⊗∗ , ∂ ) with the differential ∂ : H ⊗ n → H ⊗ n +1 defined b y ∂ ( X ) = 1 ⊗ X + n X i =1 ( − 1) i ( I ⊗ i − 1 ⊗ ∆ ⊗ I ⊗ n − i − 1 )( X ) + ( − 1) n +1 ( X ⊗ 1) . (8) The co homolog y of this co mplex admits a s imple description. Prop ositi on 5.1.1. F or a universal enveloping algebr a H = U ( g ) t he alterna- tion map Al t n : H ⊗ n → Λ n H induc es an isomorphism of the n-t h c ohomolo gy of the c omplex (8) and Λ n g . Pr o of. Sk etch of the pro of (for details see [4]): By the Poincare-Bir khoff-Witt theo rem, the universal env eloping algebr a U ( g ) is isomo rphic as a coa lgebra to the symmetric alg ebra S ∗ ( g ). The co mplex (8) for H = S ∗ ( g ) breaks into graded pieces: S n ( g ) → ⊕ i 1 + i 2 = n S i 1 ( g ) ⊗ S i 2 ( g ) → ... → ⊕ i 1 + ... + i s = n ⊗ s j =1 S i j ( g ) → ... → ( g ) ⊗ n (9) The degree n piece is isomor phic to the cochain complex of the simplicial n -cub e tensored (ov er symmetric group) with ( g ) ⊗ n . In particular , for an additive 2-co cycle X ∈ U ( g ) ⊗ 2 there is a ∈ U ( g ) such that X = X + a ⊗ 1 + 1 ⊗ a − ∆( a ) , X = Alt 2 ( X ) = 1 2 ( X − X 21 ) . (10) Note that b oth X and X are g -inv ariant whic h makes ∂ ( a ) = a ⊗ 1 + 1 ⊗ a − ∆( a ) g -inv ariant . The last implies that a can be chosen to b e g - inv arian t (central). T o see it we prov e a slightly more general statement. 18 Lemma 5 .1.2. F or a un iversal envelo ping algebr a H = U ( g ) the alternation map Alt n : ( H ⊗ n ) H → (Λ n H ) H induc es an isomorphi sm of the n- th c ohomolo gy of the sub c omplex of H -invariant elements of (8) and the sp ac e of g -invariant skewsymmetric tensors (Λ n g ) g . Pr o of. The coa lgebra isomor phism be t ween U ( g ) a nd S ∗ ( g ) is g -inv ariant. The isomorphism b etw een the degr ee n piece (8) a nd the cochain complex of the sim- plicial n -cube tensore d with ( g ) ⊗ n is na tural in g a nd in particular g -inv ariant . F or the central a satisfying (10) the exponent e xp ( ah l ) defines a ga uge tra ns- formation o f inv ariant t wists F → F ′ where F ′ = 1 + X h l + .... . T hus we can assume (up to a gauge trans formation) tha t X ∈ (Λ 2 g ) g . Note that g -in v ariance of X implies [1 ⊗ X, ( I ⊗ ∆)( X )] = [ X ⊗ 1 , (∆ ⊗ I )( X )] = 0 . Hence the exp o nent exp ( X h l ) is a n inv ariant twist on U ( g )[[ h ]]: ( exp ( X h l ) ⊗ 1)(∆ ⊗ I )( exp ( X h l )) = exp (( X ⊗ 1 + (∆ ⊗ I )( X )) h l ) = exp ((1 ⊗ X + ( I ⊗ ∆)( X )) h l ) = (1 ⊗ e xp ( X h l ))( I ⊗ ∆)( exp ( X h l )) . W riting F as exp ( X h l ) ◦ F ′ we will have at leas t the firs t l comp onents o f F ′ being zer o . Iterating the argument we prov e the following. Prop ositi on 5.1.3. Any invaria nt twist on U ( g )[[ h ]] is gauge isomorphic to a pr o duct Q ∞ i =1 exp ( X i h i ) wher e X i ∈ (Λ 2 g ) g . Note that the comp onents X i are defined uniquely by the t wis t F . Thus the set π 0 ( A ut 1 Tw ( U ( g )[[ h ]])) of classes of inv ariant twists is iso morphic to (Λ 2 g ) g [[ h ]]. Another wa y to s e e it is to use the logarithmic map. Since, for an inv ariant twist F , the factor s 1 ⊗ F , ( I ⊗ ∆)( F ) and F ⊗ 1 , (∆ ⊗ 1)( F ) o f the 2- co cycle equation pair wise commute, the logarithm l og ( F ) is an additive 2-co cycle. Hence F is gaug e isomorphic to exp ( X ), where X = Al t 2 ( log ( F )) ∈ (Λ 2 g ) g [[ h ]]. T o examine the gro up structure on π 0 ( A ut 1 Tw ( U ( g )[[ h ]])) we will use the Baker-Campb ell-Hausdorff formula: exp ( X ) e xp ( Y ) = e xp ( X + Y ) exp ( A ( X, Y )) , where A ( X , Y ) is an element of the completion (with respect to the natura l grading) of the free Lie algebra on X , Y . Note that A ( X, Y ) = 1 2 [ X , Y ] + higher terms . Now for X, Y ∈ (Λ 2 g ) g [[ h ]] the comm utator [ X , Y ] is an additive g -inv ariant 2-co cycle and is symmetric. Thus there is a cen tra l a ( X , Y ) ∈ Z ( U ( g ))[[ h ]] suc h that [ X , Y ] = a ( X , Y ) ⊗ 1 + 1 ⊗ a ( X, Y ) − ∆( a ( X , Y )) . (11) 19 Note also that any Z ∈ (Λ 2 g ) g [[ h ]] m ust commut e with [ X , Y ]. Indeed, the first commutator in [[ X , Y ] , Z ] = [ a ( X , Y ) ⊗ 1 + 1 ⊗ a ( X , Y ) , Z ] − [∆( a ( X , Y )) , Z ] is z e ro by centrality of a ( X , Y ) while the sec o nd v a nishes b ecause of g -inv ariance of Z . In par ticular, the higher terms in A ( X, Y ) are all zero and A ( X , Y ) = ∂ ( 1 2 a ( X , Y )). Thus the exp o nent e xp ( 1 2 a ( X , Y )) is a gauge tra nsformation b e- t ween the in v ariant twists exp ( X ) exp ( Y ) and exp ( X + Y ), which pr ov es the following statement. Theorem 5.1.4 . The gr oup π 0 ( A ut 1 Tw ( U ( g )[[ h ]])) of classes of invaria nt twists is isomorphic to the addititve gr oup (Λ 2 g ) g [[ h ]] . Moreov er, we can calculate the asso cia tor class φ ∈ H 3 ( π 0 ( A ut 1 Tw ( U ( g )[[ h ]])) , π 1 ( A ut 1 Tw ( U ( g )[[ h ]]))) of the Cat-gr oup A ut 1 Tw ( U ( g )[[ h ]]) of inv ariant t wists. F or our choice o f gauge transformatio n b etw een the inv ariant t wis ts exp ( X ) exp ( Y ) and exp ( X + Y ) the logar ithm of the asso ciator on exp ( X ) , exp ( Y ) , exp ( Z ) (as an element of Z ( g )[[ h ]] ⊂ Z ( U ( g ) )[[ h ]]) equals 1 2 ( a ( X, Y ) + a ( X + Y , Z ) − a ( Y , Z ) − a ( X, Y + Z )) . (12) Since both π i ( A ut 1 Tw ( U ( g )[[ h ]])) , i = 0 , 1 are divisible ab elia n groups (vector spaces ov er k ) with trivial action of the first on the second, the cohomo logy group H 3 ( π 0 ( A ut 1 Tw ( U ( g )[[ h ]])) , π 1 ( A ut 1 Tw ( U ( g )[[ h ]]))) ≃ H 3 ((Λ 2 g ) g [[ h ]] , Z ( g )[[ h ]]) is isomorphic to the gro up H om ( Λ 3 ((Λ 2 g ) g [[ h ]]) , Z ( g )[[ h ]]) of skew-symmetric maps via the map which ta kes a group 3- co cycle into its a lternation. Clear ly , the alternation of the (logarithm of the) asso cia tor (1 2) is zero. Thus the cla ss φ is trivia l. Remark 5 .1.5. T angent Cat-Lie algebr a of A ut 1 Tw ( U ( g )) . W e can forma lis e the gro und ring dep endence of A ut 1 Tw ( U ( g )) in the for m of a Cat-g roup v a lued pseudo-functor k 7→ A ut 1 Tw ( U k ( g )) on the category of Artinian lo cal co mmutative algebras. This po int of view allows us to define in a sta ndard wa y the tangent Cat-Lie algebr a (crossed mo dule of L ie a lgebras ) for A ut 1 Tw ( U ( g )): Z 2 ∂ ← C 1 . 20 Here Z 2 is the Lie a lgebra (with re sp ect to the comm utator in U ( g ) ⊗ 2 ) of 2- co cycles of the s ubco mplex of g -inv ariants of (8) a nd C 1 is the ab elian Lie algebra of 1-co chains of the sa me sub co mplex. The action of Z 2 on C 1 is trivial a nd the commutator [ X, ∂ ( a )] is zero fo r any X ∈ Z 2 and a ∈ C 1 (th us fulfilling the a xioms of a cro ssed mo dule of L ie algebras). W riting [ X , Y ] = [ Alt 2 ( X ) , Al t 2 ( Y )] a s ∂ ( a ( X, Y )) for a ( X , Y ) = a ( Al t 2 ( X ) , Al t 2 ( Y )) a s b efore we can see that a ( X , [ Y , Z ]) = 0 so the Jacobia tor of the cr ossed mo dule o f Lie algebras is trivial. 5.2 Separation for t wisted automorphisms Here w e exa mine twisted a utomorphisms ( f , F ) : H → H where H = U ( g )[[ h ]]. The co nstant term (with resp ect to h ) o f F m ust be the identit y . T hus by condition (1) the constant term of f must be an auto morphism of the bialgebra H , hence must b e induced by an automorphism of the Lie algebra g . These allow us to assume without loss of genera lity (up to an automor phism o f g ) that f = I + ∞ X i =1 f i h i , F = 1 + ∞ X i =1 F i h i . Let X = F l be the firs t no n-zero co efficient. As b efore, the degree l part of the 2-co cycle equation is the additive 2-co cycle condition for X : 1 ⊗ X + ( I ⊗ ∆)( X ) = X ⊗ 1 + (∆ ⊗ I )( X ) . As b efor e we write X = X + a ⊗ 1 + 1 ⊗ a − ∆( a ) for X = Al t 2 ( X ) and some a ∈ U ( g ). The exp o nent exp ( ah l ) defines a gauge transformatio n of twisted automor phisms ( f , F ) → ( f ′ , F ′ ) where F ′ = 1 + X h l + .... . Hence w e can a ssume (up to a ga uge transfor mation) that X ∈ Λ 2 g . Now the left hand side of the degr e e l part of the condition (1), namely , X i + j = l ( f i ⊗ f j )∆( x ) − ∆( f l ( x )) = [ X , ∆( x )] is symmetric while the right hand side is anti-symmetric. That means bo th sides are z ero. In particular, X ∈ Λ 2 g is g - inv a riant and exp ( X h l ) is a twist on U ( g )[[ h ]]. W r iting ( f , F ) as (1 , e xp ( X h l )) ◦ ( f , F ′ ) we will have at least the first l c omp onents of F ′ being zero. Proce eding like tha t (using induction by the n um ber of first successive zero comp onents in F ) we prov e the follo wing. Prop ositi on 5.2.1 . Any twiste d automorphism of a universal enveloping alge- br a U ( g )[[ h ]] is gauge isomorphi c to a sep ar ate d twiste d automorphi sm, i.e. a twiste d automorphism of the form (1 , F ) ◦ ( f , 1) wher e F is an invariant twist on U ( g )[[ h ]] and f is a bialgebr a automorphism of U ( g )[[ h ]] . 21 5.3 The Cat-group A ut Tw ( U ( g )[[ h ]]) Since any symmetric inv a riant t wist is iso morphic to a triv ia l one, the group Aut Tw ( U ( g )[[ h ]]) = π 0 ( A ut Tw ( U ( g )[[ h ]])) ≃ O ut bialg ( U ( g )[[ h ]]) ⋉ (Λ 2 g ) g [[ h ]] is the cross ed pro duct of the g roup of outer bialgebra automor phisms of U ( g )[[ h ]] and the gro up of ga uge cla s ses o f in v ariant twists. An y bialgebra automorphism of a univ ersal env eloping alg ebra is induced by a Lie alg ebra automorphism. Thu s the g roup Ou t bialg ( U ( g )[[ h ]]) in its turn is the cr ossed pro duct Aut ( g ) ⋉ (1 + hO utD er ( g )[[ h ]]) of the gro up of automorphisms of the Lie algebra g and the exponent of the Lie alg ebra hO utD er ( g )[[ h ]] of outer deriv ations of g [[ h ]] of degr ee ≥ 1. The action of the subgroup Aut ( g ) on the gr oup of ga uge classes of in v a riant t wists (Λ 2 g ) g [[ h ]] is f ( F ) = ( f ⊗ f )( F ) . The action of the degree ≥ 1 part (1 + hOutD e r ( g )[[ h ]]) is induced b y the action of the Lie a lgebra of deriv ations Der ( g ) on the space (Λ 2 ( g )) g : dX = ( d ⊗ I + I ⊗ d )( X ) . It is str a ightforw ard to see that inner deriv ations a ct trivially (t his is equiv alent to g -inv ariance). T o see that g -in v ariance is preserved b y this action we need to verify that d x d ( X ) = 0 for any x ∈ g . Here d x ( y ) = [ x, y ] is the inner deriv ation corres p o nding to x . Since d x d = dd x + d d ( x ) we ha v e d x dX = dd x ( X ) + d d ( x ) X = 0. Note that π 1 ( A ut Tw ( U ( g )[[ h ]])) = π 1 ( A ut 1 Tw ( U ( g )[[ h ]])) = Z ( g )[[ h ]]. Prop ositi on 5.3.1. The asso ciator φ ∈ H 3 ( Aut Tw ( U ( g )[[ h ]]) , Z ( U ( g )[[ h ]])) of the Cat-gr oup A ut Tw ( U ( g )[[ h ]]) is t he image of the asso ciator ψ ∈ H 3 ( Out bialg ( U ( g )[[ h ]]) , Z ( U ( g )[[ h ]])) of t he Cat-gr ou p A ut bialg ( U ( g )[[ h ]]) under the homomorphism o f gr oups O ut bialg ( U ( g )[[ h ]]) → Aut Tw ( U ( g )[[ h ]]) . Pr o of. W e need to chec k that the as so ciator is trivial if at least one of the arguments b elong s to the subgroup (Λ 2 g ) g [[ h ]]. W e hav e s een in sectio n 5.1 that it is tr ivial if all three a rguments a re from (Λ 2 g ) g [[ h ]]. In s ection 5.5 we will construct the solution a ( X , Y ) of (11) such that a ( g ( X ) , g ( Y )) = a ( X, Y ) for any automorphism g ∈ Aut ( g ) (a nd a ( dX, Y ) + a ( X, d Y ) = 0 for an y deriv ation d ∈ Der ( g )) thus co vering the case when t wo of the argument s of the as so ciator belo ng to (Λ 2 g ) g [[ h ]]. Fina lly , the fact O u t ( g ) a nd O utD e r ( g ) act on (Λ 2 g ) g guarantees that the as so ciator is trivial when o ne o f the ar guments belong s to the subgroup (Λ 2 g ) g [[ h ]]. 22 5.4 Twists on U ( g )[[ h ]] Recall that Tw ( k [[ h ]] , U ( g )[[ h ]]) is a complicated no ta tion for the gr oup oid of t wists on U ( g )[[ h ]]. W riting the twist a s a fo r mal power s eries F = P ∞ i =0 f i h i , we g et, for the degree n part o f the 2- c o cycle equation: X i + j = n, i,j >n ((1 ⊗ f i )( I ⊗ ∆)( f j ) − ( f i ⊗ 1)(∆ ⊗ I )( f j )) = 0 . The degr ee 1 par t is simply d ( f 1 ) = 0. Thus f 1 is an a dditive 2-co cycle and r = Al t 2 ( f 1 ) b elong s to Λ 2 ( g ). Up to a gaug e isomorphism we can a ssume that f 1 = r . The deg r ee 2 part re a ds as (1 ⊗ f 1 )( I ⊗ ∆)( f 1 ) − ( f 1 ⊗ 1)(∆ ⊗ I )( f 1 ) = d ( f 2 ) . (13) The left hand side is in general a 2-co cycle (this can b e chec ked directly). Thus the alternatio n o f the left hand s ide is a n element of Λ 3 ( g ). This element can be written ex plicitly if we assume a s b efore that f 1 = r is bi-primitive and skew-symmetric. Indeed, for s uch a c ho ice, the left hand side of (13) ha s the form r 23 ( r 13 + r 12 ) − r 12 ( r 13 + r 23 ) . Note tha t Alt 3 ( r 23 r 13 ) = 1 6 ( r 23 r 13 + r 31 r 21 + r 12 r 32 − r 13 r 23 − r 32 r 12 − r 21 r 31 ) , which for a skew-symmetric r eq ua ls 1 6 ([ r 23 , r 13 ] + [ r 23 , r 12 ] + [ r 13 , r 12 ]) . Note a lso that Alt 3 ( r 23 r 13 ) = Al t 3 ( r 23 r 12 ) = − Al t 3 ( r 12 r 13 ) = − Al t 3 ( r 12 r 23 ) . Thu s Alt 3 ( r 23 ( r 13 + r 12 ) − r 12 ( r 13 + r 23 )) = 4 6 ([ r 23 , r 13 ] + [ r 23 , r 12 ] + [ r 13 , r 12 ]) . T o gether with (13 ) it means tha t [ r 23 , r 13 ] + [ r 23 , r 12 ] + [ r 13 , r 12 ] = 0 . (14) This e quation is known as the classic al Y ang-Baxter equation ( CYBE ). Denote by C Y B ( g ) the se t of so lutions (in Λ 2 ( g )) of the classical Y ang -Baxter equa tion. Thu s w e hav e the following. 23 Prop ositi on 5.4.1. The ab ove defines a map π 0 ( Tw ( k [[ h ]] , U ( g )[[ h ]])) → C Y B ( g ) (15) fr om t he set of gauge isomorphism classes t o t he set of solutions to CYBE. A section to the map (15) w a s constructed in [3] (see a lso [6]). In par ticular, the map (15 ) is s urjective. The map (15) can b e extended to a map of g roup oids in the following wa y . F or a ga uge auto morphism a ∈ U ( g )[[ h ]] o f a t wist F ∈ Tw ( k [[ h ]] , U ( g )[[ h ]]) the condition ∆( a ) F = F ( a ⊗ a ) expanded in h gives X i + j = n ∆( a i ) f j = X i + j = n f j ( a i ⊗ 1 + 1 ⊗ a i ) . Since a 0 = f 0 = 1, the deg ree 1 par t is ∆( x ) + r = r + x ⊗ 1 + 1 ⊗ x , where x = a 1 and r = f 1 . Th us x b elo ngs to g . After cancellatio n the sk ew - symmetric degree 2 par t reads ∆( x ) r = r ( x ⊗ 1 + 1 ⊗ x ) or [ r , x ⊗ 1 + 1 ⊗ x ] = 0 . (16) Define the c entr aliser C g ( r ) of r as the Lie s ubalgebra in g of those x which satisfy (16). Denote b y C Y B ( g ) a disconnected gr oup oid with the set of (classes of ) ob jects C Y B ( g ) and with the ab elia n automor phism gro ups Aut C Y B ( g ) ( r ) = C g ( r ) wher e r ∈ C Y B ( g ). Then the map (15) lifts to a functor o f g roup oids Tw ( k [[ h ]] , U ( g )[[ h ]]) → C Y B ( g ) . The a ction of the cat-group A ut 1 Tw ( U ( g )[[ h ]]) on Tw ( k [[ h ]] , U ( g )[[ h ]]) co rre- sp onds to the a ction of (Λ 2 ( g )) g on C Y B ( g ) given by addition: for X ∈ (Λ 2 ( g )) g and r ∈ C Y B ( g ) the sum X + r b elong s to C Y B ( g ). The action of Aut ( g [[ h ]]) on C Y B ( g ) b oils down to the group action of Aut ( g ). Remark 5 .4.2. It seems that the constr uction of [3] (as w ell as [6]) can b e extended to a bijection C Y B h ( g ) → π 0 ( Tw ( k [[ h ]] , U ( g )[[ h ]])). Here we understand C Y B h ( g ) as the set of solutions to CYBE in Λ 2 ( g )[[ h ]]. g -equiv ariance of the construction from [3] w ould imply that the map C Y B h ( g ) → Tw ( k [[ h ]] , U ( g )[[ h ]]) is a functor if we define automorphism g roups in C Y B h ( g ) to b e exp onents o f Lie a lg ebras C g ( r )[[ h ]]. 5.5 Geometric description F or X ∈ Λ 2 ( g ) define its supp ort subsp ac e to be a ( X ) = { ( l ⊗ I )( X ) , l ∈ g ∗ } ⊂ g . Note tha t X belong s to Λ 2 a ( X ) and defines a linear isomorphism a ( X ) ∗ → a ( X ) , l 7→ ( l ⊗ )( X ) . 24 Thu s X is the Casimir element for the symplectic form b = b ( X ) on a ( X ): b (( l ⊗ I )( X ) , ( l ′ ⊗ I )( X )) = ( l ⊗ l ′ )( X ) ( I ⊗ b )( X ⊗ I ) = ( b ⊗ I )( I ⊗ X ) = I : a ( X ) → a ( X ) . The following ge o metric characterisation of solutions to CYBE w as o btained by Drinfeld [3]. Prop ositi on 5.5.1. The supp ort s = { ( I ⊗ l )( r ) , l ∈ g ∗ } of a solution r ∈ Λ 2 ( g ) to CYBE is a Lie sub algebr a of g . The symple ctic form b on s is a Lie 2-c o cycle. Pr o of. Applying ( I ⊗ l ⊗ m ) to CYBE w e get [( I ⊗ l )( r ) , ( I ⊗ m )( r )] + ( I ⊗ l ′ )( r ) + ( I ⊗ m ′ )( r ) = 0 , where l ′ ( x ) = l ([ x, ( I ⊗ m )( r )]) and m ′ ( x ) = m ([ x, ( l ⊗ I )( r )]. Thus s ⊂ g is a Lie s ubalgebra . T o see that the form b is a 2 -co cycle note that b ([( I ⊗ l )( r ) , ( I ⊗ m )( r )] , ( I ⊗ n )( r )) = ( n ⊗ m ⊗ l )([ r 12 , r 23 ] − [ r 13 , r 23 ) . Thu s the a lternation (in l , m, n ) of the left hand side is ( n ⊗ m ⊗ l ) applied to a m ultiple of CYBE . In terms of s and b the centraliser C g ( r ) is the sta biliser S t C g ( s ) ( b ) of the form b in the centraliser C g ( s ) of the Lie subalg ebra s in g . F or g - inv a riant X ∈ Λ 2 ( g ), its supp o rt a ( X ) is an ideal in g and the for m b ( X ) is g - inv arian t. Lemma 5.5 .2. F or any g -invariant X ∈ Λ 2 ( g ) g , [ X 13 , X 23 ] = 0 . (17) Pr o of. Since X 23 is ad ( g )-in v a riant [ X 13 , X 23 ] = − [ X 12 , X 23 ] whic h equals [ X 23 , X 12 ]. Again, b y ad ( g )-in v ariance of X 12 , w e hav e [ X 23 , X 12 ] = − [ X 13 , X 12 ] which co- incides with [ X 13 , X 21 ] since X 12 = − X 21 . Now, b y ad ( g )-in v a riance o f X 13 , [ X 13 , X 21 ] = − [ X 13 , X 23 ] which finally implies [ X 13 , X 23 ] = − [ X 13 , X 23 ] . It follows fro m the relation (17 ) that a ( X ) is a n a b elia n ideal for a ny X ∈ Λ 2 ( g ) g : [( l ⊗ I )( X ) , ( l ′ ⊗ I )( X )] = [( l ⊗ l ′ ⊗ I )([ X 13 , X 23 ]) = 0 . Thu s we assig n an ab elian ideal with an g -inv arian t symplectic form to an y element of Λ 2 ( g ) g . Conversely , for such a pair ( a , b ) the Casimir element X b of b ob viously b elo ngs to Λ 2 ( g ) g . Thus we have the following. Prop ositi on 5.5.3. The supp ort c onstruction establishes a bije ction b etwe en Λ 2 ( g ) g and the set of p airs ( a , b ) , wher e a ⊂ g is an ab elian ide al and b is a g -invariant symple ctic form of a 25 F or a Lie algebr a g denote by nil ( g ) its nilr adic al , i.e. the maximal nilp otent ideal ( = sum of all nilp otent ideals). Corollary 5.5.4 . F or any Lie algebr a g , Λ 2 ( g ) g = Λ 2 ( nil ( g )) g . In p articular Λ 2 ( g ) g = 0 if ni l ( g ) = 0 . Pr o of. An y ab elian idea l is obviously nilp otent. Thus it m ust b e contained in nil ( g ). Now w e desc r ib e vector s pa ce structur e of Λ 2 ( g ) in terms of pa ir s ( a , b ). Obviously , multiplication by scalar s is given b y the following rule: c ( a , b ) = ( a , cb ) , for c ∈ k \ { 0 } . The g eometric presentation for the addition is mo re in volv ed. F or X 1 , X 2 ∈ Λ 2 ( g ), the direct sum of their supp o r ts a 1 ⊕ a 2 is equipped with the symplectic for m b 1 ⊕ b 2 . Denote by ( a 1 ∩ a 2 ) ⊥ = { ( u 1 , u 2 ) ∈ a 1 ⊕ a 2 : b 1 ( u 1 , x ) = b 2 ( u 2 , x ) ∀ x ∈ a 1 ∩ a 2 } the orthog o nal co mplement of the an ti-diagonal image of a 1 ∩ a 2 in a 1 ⊕ a 2 . Denote by K the intersection ( a 1 ∩ a 2 ) ∩ ( a 1 ∩ a 2 ) ⊥ which coincides with the kernel k er ( b 1 | a 1 ∩ a 2 − b 2 | a 1 ∩ a 2 ) of the difference of the symplectic forms b i restricted to a 1 ∩ a 2 . The kernel of the surjection a 1 ⊕ a 2 → a 1 + a 2 ⊂ g coincides with the anti-diagonal image of a 1 ∩ a 2 and the sho rt exact sequence a 1 ∩ a 2 → a 1 ⊕ a 2 → a 1 + a 2 extends to a co mm utative diagra m with short exact r ows and co lumns: K / /   a 1 ∩ a 2 / /   (( a 1 ∩ a 2 ) /K ) ∗   ( a 1 ∩ a 2 ) ⊥ / /   a 1 ⊕ a 2 / /   ( a 1 ∩ a 2 ) ∗   a / / a 1 + a 2 / / K ∗ (18) W e can use the b ottom row to define a subspa ce a ⊂ g as the kernel of the map a 1 + a 2 → K ∗ induced b y the map a 1 ⊕ a 2 → ( a 1 ∩ a 2 ) ∗ : ( u 1 , u 2 ) 7→ ( x 7→ b 1 ( u 1 , x ) − b 2 ( u 2 , x )) x ∈ a 1 ∩ a 2 } . Then the left co lumn allows us to define a symplectic for m o n a . I ndee d, the kernel of the r estriction of b 1 ⊕ b 2 to ( a 1 ∩ a 2 ) ⊥ is K . Thus b 1 ⊕ b 2 induces a non-degenera te skew-symmetric bilinea r form b on ( a 1 ∩ a 2 ) ⊥ /K = a . 26 Prop ositi on 5.5.5. The subsp ac e a ⊂ g is the supp ort of the sum X 1 + X 2 ∈ Λ 2 ( g ) . The Casimir element of t he symple ctic form b c oincides with X 1 + X 2 . Pr o of. First we nee d to verify that the supp or t of X 1 + X 2 lies in a . F or that purp ose w e need a more explicit description of the map a 1 + a 2 → K ∗ . F or ( l 1 ⊗ I )( X 1 ) + ( l 2 ⊗ I )( X 2 ) ∈ a 1 + a 2 it gives a linear function K → k : if we write an element x ∈ K as x = ( m 1 ⊗ I )( X 1 ) = ( m 2 ⊗ I )( X 2 ) the v alue o f this function on x is ( l 1 ⊗ m 1 )( X 1 ) − ( l 2 ⊗ m 2 )( X 2 ) . Note that this expr ession is zero for any l i ∈ g ∗ such that ( l 1 ⊗ I )( X 1 ) = ( l 2 ⊗ I )( X 2 ) so it is w ell defined on a 1 + a 2 (do es not dep e nd on the presentation ( l 1 ⊗ I )( X 1 ) + ( l 2 ⊗ I )( X 2 )). The function assoc iated to a n ele ment ( l ⊗ I )( X 1 ) + ( l ⊗ I )( X 2 ) of the supp ort of X 1 + X 2 is clear ly zero . Thus the supp ort of X 1 + X 2 belo ngs to a . Now it suffices to chec k that ( b ⊗ I )( I ⊗ ( X 1 + X 2 )) is the iden tity on a , or that ( b ⊗ l )( I ⊗ ( X 1 + X 2 )) = l for a ny linear function l o n a . The bilinear form b on a assig ns to ( l 1 ⊗ I )( X 1 ) + ( l 2 ⊗ I )( X 2 ) , ( m 1 ⊗ I )( X 1 ) + ( m 2 ⊗ I )( X 2 ) ∈ a the n um ber ( l 1 ⊗ m 1 )( X 1 ) − ( l 2 ⊗ m 2 )( X 2 ) . In particular, for x = ( l 1 ⊗ I )( X 1 ) + ( l 2 ⊗ I )( X 2 ) ∈ a , ( b ⊗ l )( x ⊗ ( X 1 + X 2 )) = b ( x, ( I ⊗ l )( X 1 ) + ( I ⊗ l )( X 2 )) = ( l 1 ⊗ l )( X 1 ) + ( l 2 ⊗ l )( X 2 ) = l ( x ) . In pa rticular, if b oth X 1 , X 2 are g -inv ariant we get a geo metr ic description of the a dditio n o n Λ 2 ( g ) g . Remark 5 .5.6. According to prop osition 5.5 .3, the supp ort subspa ce a of X 1 + X 2 m ust b e an ab elian ideal. The fact that a is an ideal follows from the g -equiv ariance o f the construction for a , while the abe lia n prope rty can b e chec ked by direct compu- tation. First note that the commutan t [ a 1 , a 2 ] lies in K . As a consequence, the map a 1 + a 2 → K ∗ is a homo morphism of Lie algebr as (with ab elian s tructure on K ∗ ). Th us a is a L ie subalgebra . T o see that the commutan t [ a , a ] is zer o, it is e nough to chec k that b 1 ([ u 1 + u 2 , v 1 + v 2 ] , y ) = 0 for any u 1 + u 2 , v 1 + v 2 ∈ a and an y y ∈ a 1 . W r iting [ u 1 + u 2 , v 1 + v 2 ] = [ u 1 , v 2 ] + [ u 2 , v 1 ] = [ u 1 , v 2 ] − [ v 1 , u 2 ] , we need to verify that b 1 ([ u 1 , v 2 ] , y ) = b 1 ([ v 1 , u 2 ] , y ). Indeed, by g -inv ariance of b i and the defining relations for u 1 + u 2 , v 1 + v 2 (together with [ a 1 , a 2 ] ⊂ K ), we hav e the chain of equalities: b 1 ([ u 1 , v 2 ] , y ) = − b 1 ( u 1 , [ y , v 2 ]) = − b 2 ( u 2 , [ y , v 2 ]) = b 2 ([ y , u 2 ] , v 2 ) = b 1 ([ y , u 2 ] , v 1 ) = − b 1 ( y , [ v 1 , u 2 ]) = b 1 ([ v 1 , u 2 ] , y ) . 27 Note also that the arrows of the dia gram (18) are homomor phisms of v ector spaces and not of Lie algebras . T o turn it into a dia gram of Lie algebras it suffices to introduce an appropriate Lie algebra structure on the direct sum a 1 ⊕ a 2 . Denote b y a 1 ⊲ ⊳ a 2 the Lie alg ebra of pair s ( x 1 , x 2 ), x i ∈ a i with the brack et: [( x 1 , x 2 ) , ( y 1 , y 2 )] = 1 2 ([ x 2 , y 1 ] + [ x 1 , y 2 ] , [ x 2 , y 1 ] + [ x 1 , y 2 ]) . Then the maps o f the diag ram K / /   a 1 ∩ a 2 / /   (( a 1 ∩ a 2 ) /K ) ∗   ( a 1 ∩ a 2 ) ⊥ / /   a 1 ⊲ ⊳ a 2 / /   ( a 1 ∩ a 2 ) ∗   a / / a 1 + a 2 / / K ∗ (19) bec ome homomo rphisms o f Lie alg ebras (one should think of ob jects in the rig ht column as a be lian Lie algebras). When X 1 is g -in v ariant and X 2 is a solution to CYBE the sum X 1 + X 2 is a lso a solution to the CYBE. In this case the supp or t spa ce a is still a Lie algebra. Again the diagram (18) can b e made into a dia gram of Lie alg ebras. One needs to think of a 1 ⊲ ⊳ a 2 as the Lie algebra with the brack et: [( x 1 , x 2 ) , ( y 1 , y 2 )] = 1 2 ([ x 2 , y 1 ] + [ x 1 , y 2 ] , [ x 2 , y 1 ] + [ x 1 , y 2 ] + 2 [ x 2 , y 2 ]) . In other w ords, the diagram (19) defines an action of the set of a b e lian ideals with inv ariant symplectic for ms (which coincides with the vector space (Λ 2 g ) g ) on the s e t of suba lgebras with non-degenerate 2-co cycles (the set C Y B ( g )). Now we give a geometric description of the ma p (Λ 2 g ) g ⊗ (Λ 2 g ) g → ( S 3 g ) g , (20) which for X 1 , X 2 gives a cob oundary a fo r the comm utator [ X 1 , X 2 ] = a ⊗ 1 + 1 ⊗ a − ∆( a ) . Here we identify ( S ∗ g ) g with the c entre Z ( U ( g )) = U ( g ) g of the universal en- veloping algebra b y mea ns of the K irillov-Duflo isomor phism. F or t wo ab elia n ideals a 1 , a 2 ⊂ g denote by b = [ a 1 , a 2 ] their comm utan t. Note that orthogonal co mplements b ⊥ b i with resp ect to in v a riant symplectic forms b i on a i hav e the fo llowing commutation pro p erty: [ b ⊥ b 1 , a 2 ] = [ a 1 , b ⊥ b 2 ] = 0 . (21) 28 Indeed, for x ∈ b ⊥ b 1 , y ∈ a 2 , b 1 ([ x, y ] , z ) = − b 1 ( x, [ z , y ]) = 0 , ∀ z ∈ a 1 . Hence [ x, y ] = 0. Similar ly for [ a 1 , b ⊥ b 2 ]. Now define a map b ∗ ⊗ b ∗ → b by sending l 1 ⊗ l 2 int o [ x 1 , x 2 ], where x i ∈ a i are defined b y b i ( x i , u ) = l i ( u ) , ∀ u ∈ b . The elements x i are defined up to b ⊥ b i so in view o f the rela tions (21) the co m- m utator [ x 1 , x 2 ] is well defined. Ob vious ly , the map b ∗ ⊗ b ∗ → b cor resp onds to a 3- vector c ∈ b ⊗ 3 , which is g -inv ariant by the constructio n. T o see that this 3-vector is symmetric, think of it a s a ma p ( b ∗ ) ⊗ 3 → k , l 1 ⊗ l 2 ⊗ l 3 7→ l 3 ([ x 1 , x 2 ]) . Find y i ∈ a i such that b i ( y i , u ) = l 3 ( u ) , ∀ u ∈ b . Then l 3 ([ x 1 , x 2 ]) = b 1 ( y 1 , [ x 1 , x 2 ]) = − b 1 ([ y 1 , x 2 ] , x 1 ) = l 1 ([ y 1 , x 2 ]) , which in terms o f the 3-vector c means tha t c 13 = c . Similarly l 3 ([ x 1 , x 2 ]) = b 2 ( y 2 , [ x 1 , x 2 ]) = − b 2 ([ x 1 , y 2 ] , x 2 ) = l 2 ([ x 1 , y 2 ]) , which means that c 23 = c . Lemma 5 .5.7. The element a c orr esp onding to the invariant c ∈ S 3 ( g ) g via the isomorphism S ∗ ( g ) g → Z ( U ( g )) satisfies the e quation [ X 1 , X 2 ] = a ⊗ 1 + 1 ⊗ a − ∆( a ) . Pr o of. Since b = [ a 1 , a 2 ] is is otropic with r esp ect to the forms b i we can c hoo se subspaces L i ⊂ a i , which a re Lagrangia n with res p ec t to b i resp ectively and contain b . W rite a i = L i ⊕ L ∗ i . A choice of ba ses L 1 = < e i >, L 2 = < f j > allows us to write X 1 = X i e i ∧ e i , X 2 = X j f j ∧ f j . Here e i ( f j ) is the dual ba sis in L ∗ 1 (resp ectively L ∗ 2 ). B y the choice, [ L 1 , L 2 ] = 0 so [ X 1 , X 2 ] = X i,j [ e i ∧ e i , f j ∧ f j ] = X i,j e i e j ⊙ [ e i , f j ] . Here ⊙ is the symmetric pro duct x ⊙ y = x ⊗ y + y ⊗ x . Since the comm utator pairing [ , ] : L ∗ 1 ⊗ L ∗ 2 → b factors throug h c : b ∗ ⊗ b ∗ → b we can write a = X i,j e i f j [ e i , f j ] = X s,t x s x t c ( x s , x t ) , 29 where x s is a basis of b . Finally no te tha t the element c belong s to S ∗ ( b ) g = U ( b ) g ⊂ Z ( U ( g )) so w e do not need to w orry a b out the particular c hoice of isomorphism S ∗ ( g ) g → Z ( U ( g )). Remark 5 .5.8. It follows from the in v a riance of the construction of c that c ( g ( X 1 ) , g ( X 2 )) = c ( X 1 , X 2 ) for an automo rphism g o f the Lie alg ebra g . W e can use a s y mmetric 3-vector c ∈ b ⊗ 3 on a v ecto r space b to define a structure of (meta- ab elian) Lie algebra g ( b , c ) on the vector space b ∗ ⊕ b ∗ ⊕ b : [( l 1 , m 1 , x 1 ) , ( l 2 , m 2 , x 2 )] = (0 , 0 , ( l 1 ⊗ m 2 ⊗ I − l 2 ⊗ m 1 ⊗ I )( c )) . The s ubspaces a 1 = b ∗ ⊕ 0 ⊕ b , a 2 = 0 ⊕ b ∗ ⊕ b are a b elian ideals in g ( b , c ). The symplectic forms b i on a i b 1 (( l 1 , 0 , x 1 ) , ( l 2 , 0 , x 2 )) = l 1 ( x 2 ) − l 2 ( x 1 ) , b 2 ((0 , m 1 , x 1 ) , (0 , m 2 , x 2 )) = m 1 ( x 2 ) − m 2 ( x 1 ) are g ( b , c )-inv ariant. Indeed, by c 13 = c b 1 ([(0 , m, x ) , ( l 1 , 0 , x 1 )] , ( l 2 , 0 , x 2 )) = b 1 ( − (0 , 0 , ( l 1 ⊗ m ⊗ I )( c )) , ( l 2 , 0 , x 2 )) = = ( l 1 ⊗ m ⊗ l 2 )( c ) , whic h coincides with b 1 (( l 1 , 0 , x 1 ) , [( l 2 , 0 , x 2 ) , (0 , m, x )]) = b 1 (( l 1 , 0 , x 1 ) , (0 , 0 , ( l 2 ⊗ m ⊗ I )( c )) ) = ( l 2 ⊗ m ⊗ l 1 )( c ) . Similarly , by c 23 = c b 2 ([(0 , m 1 , x 1 ) , ( l , 0 , x )] , (0 , m 2 , x 2 )) = b 1 ( − (0 , 0 , ( l ⊗ m 1 ⊗ I )( c )) , (0 , m 2 , x 2 )) = = ( l ⊗ m 1 ⊗ m 2 )( c ), which coincides with b 2 ((0 , m 1 , x 1 ) , [( l , 0 , x ) , (0 , m 2 , x 2 )]) = b 2 ((0 , m 1 , x 1 ) , (0 , 0 , ( l ⊗ m 2 ⊗ I )( c ))) = = ( l ⊗ m 2 ⊗ m 1 )( c ) . Clear ly , the 3-vector corres po nding to the pair ( a i , b i ) via (20) is c . The ab ov e construction is univ er sal in the follo wing sense. F or a pair ( a i , b i ) of ab elian ideals with symplectic inv ariant for ms, the sum a 1 + g 2 (whic h is a meta-ab elian ideal) maps on to g ( b , c ), wher e b = [ a 1 , a 2 ] a nd c is the 3-vector defined b y (20). Note that b is isotro pic in bo th a i so the maps a i → b ∗ defined by the forms b i factor through a i / b . Thus we hav e a diagram b   / / a 1 + a 2   / / a 1 / b ⊕ a 2 / b   b / / g ( b , c ) / / b ∗ ⊕ b ∗ The middle v ertical map is a homomorphis m of Lie algebras since the right vertical map is co mpatible with commut ator pairing s a 1 / b ⊗ a 2 → b (induced from a 1 + g 2 ) and b ∗ ⊗ b ∗ → b (in g ( b , c )). 30 Example 5. 5.9. Heisenb er g algebr a Let ( V , b ) be a symplectic vector s pa ce and g = H ( V , b ) = V ⊕ h c i its Heisen- ber g Lie algebra with the central gener ator c . Then the map v 7→ v ∧ c = v ⊗ c − c ⊗ v is an is omorphism V → (Λ 2 g ) g . Clearly v ∧ c is g - inv a riant: [ u ⊗ 1 + 1 ⊗ u, v ∧ c ] = [ u, v ] ∧ c = b ( u, v ) c ∧ c = 0 . T o se e that there are no other g -inv ariant elements in Λ 2 g note that Λ 2 g = Λ 2 V ⊕ V with the g -a ction on the fir st compone nt V ⊗ Λ 2 V → V being induced by the bilinear form b a nd he nc e having no inv ariants. The commutator (in U ( g ) ⊗ 2 ) of tw o ele ments from (Λ 2 g ) g [ v ∧ c, u ∧ c ] = b ( v, u )( c ⊗ c 2 + c 2 ⊗ c ) is a cobo undary a ⊗ 1+ 1 ⊗ a − ∆( a ). F or ex ample, we can c ho o se a : Λ 2 ((Λ 2 g ) g ) → U ( g ) to b e a ( v ∧ c, u ∧ c ) = b ( u, v ) 3 c 3 . The subalgebras in H ( V , b ) break in to tw o classes depending on whether or not they co ntain the cen tre. A s ubalgebra o f the first type has the form U ⊕ < c > for an ar bitrary subspace U ⊂ V , while the second t yp e is simply a n isotr opic subs pace U of V (isotropic means that the restriction of the symplectic form b to U is zero). A non-dege ne r ate 2-co c ycle on a subalgebra o f the second t yp e is a symplectic fo r m. F or a subalgebr a of the fir st type U ⊕ < c > with a non- degenerate 2-co cycle β denote by U ′ the o rthogo na l c omplement (with resp ect to β ) of c in U . Clearly the restr iction of β o n U ′ is non-degener ate. Moreov er, the 2-co cycle conditio n implies that U ′ is isotropic with r esp ect to b . This corr esp onds to the fact that any X ∈ C Y B ( g ) for g = H ( V , b ) can be uniquely written as Y + v ∧ c for Y ∈ C Y B ( g ) supp o r ted on a subalgebra of the second t y p e . In par ticular, (Λ 2 g ) g acts transitively on C Y B ( g ) with o rbits corres p o nding to subalgebras of the seco nd type with s ymplectic forms on them. 6 Crossed pro du cts with resp ect to actions b y t wisted automorphisms In this sec tio n we co nstruct crossed pro duct of a bialgebra with a gro up acting by t wisted automorphisms. W e say that a gr oup G acts by t wiste d automorphisms on a bialgebr a H if a map o f Cat-gro ups (a s in section 4.1): ( τ , θ ) : G → A ut Tw ( H ) is given: τ : G → Aut Tw ( H ) , τ ( g ) = ( g , F g ) , θ : G × G → Z ( H ) × , satisfying f ( θ ( g , h )) θ ( f , g h ) = θ ( f , g ) θ ( f g , h ) , (22) 31 f ( F g ) F f ∆( θ ( f , g )) = ( θ ( f , g ) ⊗ θ ( f , g )) F f g . (23) A cr osse d pr o duct o f a bia lgebra H with a g roup G with respect to the action by t wisted automo rphisms ( τ , θ ) is a bia lg ebra H ∗ ( τ ,θ ) G , which as a vector space is s panned b y sy mbols x ∗ f for x ∈ H , f ∈ G sub ject to the linearity in the first argument ( x + y ) ∗ f = x ∗ f + y ∗ f ; with the pro duct a nd the copro duct given by the formulas: ( x ∗ f )( y ∗ g ) = ( xf ( y ) θ ( f , g )) ∗ f g , ∆( x ∗ f ) = (∆( x ) F − 1 f ) ∗ ( f ⊗ f ) . It is quite stra ightforw ard to chec k that all bia lgebra axioms ar e satisfied. In- deed, asso ciativ ity of m ultiplication follows from the condition (22): ( x ∗ f )(( y ∗ g )( z ∗ h )) = ( x ∗ f )( y g ( z ) θ ( g , h ) ∗ g h ) = xf ( y ) f g ( z ) f ( θ ( g , h )) θ ( f , g h ) ∗ f g h coincides with (( x ∗ f )( y ∗ g ))( z ∗ h ) = ( xf ( y ) θ ( f , g ) ∗ f g )( z ∗ h ) = xf ( y ) θ ( f , g ) f g ( z ) θ ( f g , h ) ∗ f g h. Coasso c ia tivity of comultiplication follows from the 2-c o cycle prop er ty of twists: (∆ ⊗ I )∆( x ∗ f ) = (∆ ⊗ I )((∆( x ) F − 1 f ) ∗ ( f ⊗ f )) = (∆ ⊗ I )∆( x )(∆ ⊗ I )( F f ) − 1 ( F f ⊗ 1) − 1 ∗ ( f ⊗ f ⊗ f ) is e qual to ( I ⊗ ∆)∆( x ∗ f ) = ( I ⊗ ∆)((∆( x ) F − 1 f ) ∗ ( f ⊗ f )) = ( I ⊗ ∆)∆ ( x )( I ⊗ ∆)( F f ) − 1 (1 ⊗ F f ) − 1 ∗ ( f ⊗ f ⊗ f ) . Finally , the compa tibility of m ultiplication and comultiplication follo ws from the condition (23): ∆(( x ∗ f )( y ∗ g )) = ∆( x ∗ f )∆( y ∗ g ) = (∆( x ) F − 1 f ∗ ( f ⊗ f ))(∆( y ) F − 1 g ∗ ( g ⊗ g )) = ∆( x ) F − 1 f ( f ⊗ f )∆( y )( f ⊗ f )( F g ) − 1 ( θ ( f , g ) ⊗ θ ( f , g )) ∗ ( f g ⊗ f g ) , which, b y the definition of twisted homomorphism, co incides with ∆( x )∆( f ( y )) F − 1 f ( f ⊗ f )( f ⊗ f )( F g ) − 1 ( θ ( f , g ) ⊗ θ ( f , g )) ∗ ( f g ⊗ f g ) . A t the same time ∆( xf ( y ) θ ( f , g ) ∗ f g ) = ∆( x )∆( f ( y ))∆( θ ( f , g )) F − 1 f g ∗ ( f g ⊗ f g ) . As an ex ample we consider the c ase of a universal env eloping alg ebra. Let A b e a subspa ce of (Λ 2 g ) g . Define a map of Cat-groups A → A ut Tw ( U ( g )[[ h ]]) 32 by ass igning to X ∈ A the twisted automorphism ( I , e xp ( X h )) and defining θ ( X , Y ) a s exp ( 1 2 a ( X , Y ) h 2 ), where a ( X, Y ) ∈ Z ( U ( g )) is a so lution o f [ X , Y ] = a ( X , Y ) ⊗ 1 + 1 ⊗ a ( X, Y ) − ∆( a ( X , Y )) , satisfying the 2-co cyle condition a ( X , Y ) + a ( X + Y , Z ) = a ( Y , Z ) + a ( X, Y + Z ) . The cr ossed product U ( g )[[ h ]] ∗ A will ha ve the follo wing rules for m ultiplication and com ultiplication: ( x ∗ X )( y ∗ Y ) = xy exp ( 1 2 a ( X , Y ) h 2 ) ∗ ( X + Y ) , ∆( x ∗ X ) = ∆( x ) exp ( X h ) ∗ ( X ⊗ X ) . W riting formally X as exp ( l X h ) the rules turn in to the following exp ( l X h ) exp ( l Y h ) = e xp ( a ( X , Y ) h 2 + ( l X + l Y ) h ) , ∆( exp ( l X h )) = e xp ( X h )( exp ( l X h ) ⊗ exp ( l Y h )) . These can b e re solved b y setting [ l X , l Y ] = − a ( X , Y ) , (24) ∆( l X ) = X + l X ⊗ 1 + 1 ⊗ l X . (25) W e can for malise these equations b y formally adding new g e ne r ators l X , X ∈ A to U ( g ) s ub ject to the r elation (24), with the co multip lication extending the one on U ( g ) and satifying (2 5). The resulting o b ject U ( g )[ A, a ] is a bialge bra. Indeed, the only thing to chec k in this abstr a ct setting is that the relation (2 4) is pr eserved by the comultiplication: ∆([ l X , l Y ] + a ( X, Y )) = [ X , Y ] + [ l X , l Y ] ⊗ 1 + 1 ⊗ [ l X , l Y ] + ∆( a ( X , Y )) = ([ l X , l Y ] + a ( X, Y )) ⊗ 1 + 1 ⊗ ([ l X , l Y ] + a ( X, Y )) . Example 6. 0.10. Let g = H ( V , b ) = V ⊕ h c i b e the Heisenber g Lie a lg ebra of the symplectic vector s pace ( V , b ). Let A b e the space V mapp ed in to (Λ 2 g ) g by v 7→ v ∧ c . As in the example 5 .5.9, let a ( v , u ) = b ( u,v ) 3 c 3 . Then the extra generator s l v , v ∈ V of the a lg ebra U ( H ( V , b ))[ V , a ] satisfy [ l v , l u ] = b ( u, v ) 3 c 3 , with the com ultiplication defined by ∆( l v ) = v ⊗ c − c ⊗ v + l v ⊗ 1 + 1 ⊗ l v . 33 References [1] R. Brown, C.B. Spe ncer, G -gro upo ids, crossed mo dules and the fundamen- tal group oid of a topo logical group. Nederl. Ak ad. W etensch. Pr o c., Ser. A, 38 (1976), no. 4, 2 96–30 2. [2] A. Davydov, Finite gro ups with the same character tables, Drinfel’d al- gebras a nd Ga lois algebr as. Algebra (Mo scow, 1998), 99–1 11, de Gr uyter, Berlin, 200 0 . [3] V. G. Drinfel’d, Constant quasicla ssical solutions o f the Y ang-Baxter q uan- tum equatio n. (Russian) Do k l. Ak ad. Na uk SSSR 273 (1 983), no . 3, 531– 535. [4] V. G. Drinfel’d, Quasi- Ho pf algebras. (Russia n) Algebra i Analiz 1 (1989), no. 6, 114– 148; translation in Leningr ad Math. J. 1 (1990), no. 6 , 1419– 1457. [5] V. G. Drinfel’d, Almost co commutativ e Hopf algebra s. (Russian) Algebra i Analiz 1 (1989), no. 2 , 3 0–46 ; translation in Leningr a d Math. J. 1 (1990), no. 2, 321–34 2 [6] P . Etingof, O. Schiffmann, Le ctures on qua ntum groups. Lectures in Math- ematical Physics. International Pr ess, Bo ston, MA, 1998. 2 39 pp. [7] A. Joyal, R. Street, Br aided tensor categories. Adv. Math. 10 2 (19 93), no . 1, 20–7 8. [8] V. Lyubashe nko, Sup er- analisis and solutions of tria ngle equation. Ph. D. thesis, 1986 . [9] S. MacLane, Categor ies for the W orking Ma thematician, Spring er-V erlag, 1988. [10] S. Montgomery , Hopf a lgebras and their actions on r ing s. CBMS Regio nal Conference Series in Mathematics, 82 . American Mathematical So ciety , Providence, RI, 1993 . 23 8 pp [11] Radfor d D., Minimal quasitriangular Hopf alg ebras, J Algebr a, 157 (19 93), n 2, 2 85-3 15. [12] P . Sc hauen burg, Hopf bi-Galois extensions. Comm. Algebra 24 (19 96), no . 12, 3797– 3825 . [13] R. Street, Q uantum gro ups. A path to curr ent alge br a. Cam br idge Univ er- sity Pr ess, 20 07, 141 pp. [14] K.-H.Ulbrich, On Galois H-dimodule algebras, Math. J. Ok ay a ma Univ., 31 (1989), 79-84. [15] J. H. C. Whitehead, Com binato rial ho motopy . I,II. Bull. Amer. Math. So c. 55, (1949), 213– 2 45, 453– 4 96. 34