Universal deformation formulas and braided module algebras

UNIVERSAL DEF ORMA TION F ORMULAS AND BRAIDED MODULE ALGEBRAS JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI Abstract. W e study formal deformations of a crossed pro duct S ( V )# f G , of a p olynomial al gebra with a group, induced from a unive rsal deformation formula introduced by Witherspo on. These deformations ari se from braided actions of Hopf algebras generated by automorphisms and skew deriv ations. W e show that they are nont rivial in the cha racteristic f ree context , eve n if G is inﬁnite, by showing that their inﬁnitesimals are not cob oundaries. F or this we construct a new complex which computes the Ho c hschild cohomology of S ( V )# f G . In tro duction In [G-Z] Giaquinto and Z hang dev elop the notion of a universal de fo rmation formula based on a bia lgebra H , extending earlier for mulas based on universal env eloping algebras of Lie algebr as. Each one of these formulas is calle d universal beca use it provides a formal deformation for any H -mo dule algebra. In the sa me pap er the authors construct the ﬁrst family of such for mulas based on no nc o mm utative bialgebras , namely the env eloping algebra s of central extensio ns of a Heisenberg Lie algebra L . Another of these for m ulas, based on a Hopf a lg ebra H q ov er C , where q ∈ C × is a para meter, gener ated by group like e le men ts σ ± 1 and t wo skew primitive elements D 1 , D 2 , were o btained in the gener ic c a se b y the same authors, but were not published. In [W] the author generalizes this formula to include the case where q is a ro ot of unity , and she uses it to construct for mal de fo rmations of a crossed pro duct S ( V )# f G , where S ( V ) is the po lynomial algebra and the group G acts linear ly on V . More pr e cisely , she deals with deformations whose inﬁnitesimal sends V ⊗ V to S ( V ) w g , where g is a central element of G . In this pap er we prove that some r e sults established in [W] under the hypothesis that G is a ﬁnite group, remain v alid for arbitra ry groups, and with C replaced b y an a r bitrary ﬁeld. F or ins tance we show that the deter minan t of the action of g on V is a lw ays 1. Moreov er, w e do not only co nsider standard H q -mo dule alg ebra structures on S ( V )# f G , but als o the more general o nes introduced in [G-G1], and we work with actions which dep end on tw o c e n tral elements g 1 and g 2 of G and t wo p olynomials P 1 and P 2 . When the actions a re the standard ones, g 1 = 1 and P 1 = 1, we o btain the case consider ed in [W]. Fina lly , in Subsection 3.2 we show how to extend the e xplicit formulas obtained previously , to non central g 1 and g 2 . 2000 Mathematics Subject Classiﬁc ation. Primary 16S80; Secondary 16S35. Key wor ds and phr ases. Crossed pro duct; Deformation; Ho c hsch ild cohomology . Supported by UBACYT 095, PIP 112-200801-009 00 (CONICET) and PUCP-DAI-2009-004 2. Supported by UBACYT 095 and PIP 112-200801-00900 (CONICET). Supported b y PU CP-DAI -2009-0042, Lucet 90-DAI-L005, SFB 478 U. M ¨ unster, Konrad A de- nauer Stiftung. The second author thanks the app ointme nt as a visiting professor “C´ atedra Jos ´ e T ol a Pasquel” and the hospitality duri ng his stay at the PUCP . 1 2 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI As was noted by Withersp oo n, these formulas nec e ssarily inv olve all co mponents of S ( V )# f G corr espo nding to the elements of a union o f co njugacy classes of G . The pap er is or ganized as follows: in the ﬁr s t section we review the concept of braided mo dule algebr a introduced in [G-G1], we adapt the notio n o f Universal Deformation F ormula (UDF) to the braided c o n text, and we show that ea c h one of these for m ulas pro duces a deforma tion on any bra ided H -mo dule alg ebra whose transp osition (see Deﬁnition 1.6) s atisfy a suitable hypothesis. W e re mark that, when the bialg e bra H is sta ndard, the use of braided mo dule algebr a gives rise to more defor mations than the o nes obtained using only mo dule algebr a s, b ecause the transp osition can b e diﬀerent fro m the ﬂip. With this in mind, although we are going to work with the standar d Hopf alg ebra H q , we establish the ba sic pro per ties of UDF’s in the bra ided case, b ecause it is the most appr o priate se tting to deal with arbitrary transp ositions. In the s e c ond section we recall the deﬁnitions of the Hopf a lg ebra H q and of the UDF e x p q considered in [W , Section 3], which we are going to s tudy . W e also intro duce the co nc e pt of a go o d transp osition of H q on a n algebra A , and we study s ome of its prop erties. Perhaps the most impo rtant result in this s ection is Theorem 2.4, in which we obtain a description of all the H q -mo dule algebras ( A, s ), with s a go o d tra nspo sition. This is the ﬁrst of several results in which we give a systematic account of the necessar y and suﬃcient conditions that an alg ebra (in genera l a cro ssed pro duct S ( V )# f G ) must satisfy in order to supp ort a br aided H q -mo dule alg e bra structure satisfying suitable h yp othesis. In Section 4 of [W], using the UDF ex p q the author constructs a larg e family of deforma tions whose inﬁnitesima l sends V ⊗ V to S ( V ) w g , wher e g is a central element of G . Using cohomolog ical metho ds she proves that if G is ﬁnite, these defor mations are non trivia l, that the action of g on V has determinant 1 and that the co dimension o f g V is 0 o r 2. In the ﬁrs t part of Section 3 we study a la r ger family of defor mations and we prov e that the last tw o r esults hold for this family even if G is inﬁnite and the characteristic o f k is non zer o. Finally , in Sectio n 4 we show that, under very general hypo thesis, the deforma tio ns construc ted in the previo us section are non trivial. O nc e ag ain, we do not ass ume character istic zero, nor that the group G is ﬁnite. One o f the interesting p oints in this pap er is the metho d developed to deal with the cohomolo gy of S ( V )# f G when k [ G ] is non semisimple. As far as w e know it is the ﬁr st time tha t this type of co chain complexe s is us ed to prov e the non triviality of a Ho chsc hild co cycle. 1 Preliminaries After introducing some ba s ic notations we rec a ll br ieﬂy the concepts of bra ided bialgebra and braided Hopf a lgebra following the presentation given in [T1] (se e also [T2], [L1 ], [F-M-S], [A-S], [D], [So] and [B-K -L-T]). Then we review the no tio n of braided mo dule algebra intro duced in [G-G1], we rec a ll the concept of Universal Deformation F ormula based on a bialgebra H , due to Giaquinto a n Zha ng, and we show that s uc h a UDF pro duces a forma l defor mation when it is applied to an H -br aided mo dule algebra, satisfying suitable hypothes is, generalizing slightly a result in [G-Z]. In this pap er k is a ﬁeld, k × = k \ { 0 } , all the vector space s are ov er k , and ⊗ = ⊗ k . Moreov er we will use the usual no tation ( i ) q = 1 + q + · · · + q i − 1 and ( i )! q = (1) q · · · ( i ) q , for q ∈ k × and i ∈ N . Let V , W b e vector spa ces a nd let c : V ⊗ W → W ⊗ V be a k -linea r map. Recall that: UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 3 - If V is a n algebr a , then c is compatible with the algebra structure o f V if c   ( η ⊗ W ) = W ⊗ η and c   ( µ ⊗ W ) = ( W ⊗ µ )   ( c ⊗ V )   ( V ⊗ c ), where η : k → V and µ : V ⊗ V → V denotes the unit a nd the multiplication map of V , resp ectively . - If V is a co a lgebra, then c is compatible with the coalgebr a structure of V if ( W ⊗ ǫ )   c = ǫ ⊗ W and ( W ⊗ ∆)   c = ( c ⊗ V )   ( V ⊗ c )   (∆ ⊗ W ), where ǫ : V → k and ∆ : V → V ⊗ V denotes the counit and the co m ultiplication map of V , r espe c tiv ely . Of cours e, there are simila r co mpatibilities when W is an algebr a or a coalge br a. 1.1 Braided bialgebras a nd braided Hopf algebras Deﬁnition 1.1. A br aide d bialgebr a is a vector space H endow ed with an alge bra structure, a co algebra structure and a bra iding op erator c ∈ Aut k ( H ⊗ 2 ) (called the br ai d of H ), such tha t c is c ompatible with the a lgebra and coalg ebra structure s of H , ∆   µ = ( µ ⊗ µ )   ( H ⊗ c ⊗ H )   (∆ ⊗ ∆), η is a coalgebr a mor phism a nd ǫ is a n algebra morphism. F urthermore, if there exists a k -linea r map S : H → H , which is the inv erse of the identit y map for the conv olution pro duct, then we say tha t H is a br aide d Hopf algebr a and we call S the antip o de of H . Usually H denotes a braided bialgebr a, understanding the structure maps, and c denotes its braid. If necessa ry , we will use notations as c H , µ H , etcetera. R emark 1 .2 . Assume that H is a n a lgebra and a coalg ebra and c ∈ Aut k ( H ⊗ 2 ) is a s olution of the bra iding eq uation, which is co mpa tible with the algebr a and coalgebr a s tr uctures of H . Let H ⊗ c H b e the algebra with underlying vector spa ce H ⊗ 2 and multiplication map given by µ H ⊗ c H := ( µ ⊗ µ )   ( H ⊗ c ⊗ H ). It is ea sy to see that H is a bra ided bia lgebra with braid c if and o nly if ∆ : H → H ⊗ c H and ǫ : H → k a re morphisms of algebr as. Deﬁnition 1.3. Let H and L b e braide d bialgebra s . A map g : H → L is a morphism of br aid e d bialgebr a s if it is an algebra homomorphis m, a coalg ebra ho- momorphism and c   ( g ⊗ g ) = ( g ⊗ g )   c . Let H and L b e braided Hopf algebr as. It is well k no wn that if g : H → L is a morphism of bra ided bia lgebras, then g   S = S   g . 1.2 Braided mo dule algebras Deﬁnition 1. 4. Let H b e a braided bialgebra . A left H -br aide d sp ac e ( V , s ) is a vector spa ce V , endow ed with a bijective k - line a r map s : H ⊗ V → V ⊗ H , which is compatible with the bialgebra structur e o f H and satisﬁes ( s ⊗ H )   ( H ⊗ s )   ( c ⊗ V ) = ( V ⊗ c )   ( s ⊗ H )   ( H ⊗ s ) (compatibility of s with the br aid). Let ( V ′ , s ′ ) be another left H -braide d space. A k -linear map f : V → V ′ is said to be a morphism of left H -br aide d sp ac es , from ( V , s ) to ( V ′ , s ′ ), if ( f ⊗ H )   s = s ′   ( H ⊗ f ). W e let LB H denote the categor y of all left H -braided spa ces. I t is easy to chec k that this is a mo noidal category with: - unit ( k , τ ), where τ : H ⊗ k → k ⊗ H is the ﬂip, - tensor pr o duct ( V , s V ) ⊗ ( U, s U ) := ( V ⊗ U, s V ⊗ U ), where s V ⊗ U is the map s V ⊗ U := ( V ⊗ s U )   ( s V ⊗ U ), - the usual as so ciativit y and unit constr ain ts. 4 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI Deﬁnition 1 .5. W e will say that ( A, s ) is a left H -br aide d algebr a or simply a left H -algebr a if it is an algebra in LB H . W e let ALB H denote the catego ry o f left H -braided algebra s . Deﬁnition 1.6. Let A b e an alg ebra. A left tr ansp o sition of H o n A is a bijective map s : H ⊗ A → A ⊗ H , satisfying: (1) ( A, s ) is a left H -br aided spac e , (2) s is compa tible with the algebr a structur e o f A . R emark 1.7 . A left H -bra ided alge bra is a pa ir ( A, s ) co nsisting o f an a lgebra A and a left tra ns positio n s of H on A . Let ( A ′ , s ′ ) b e ano ther left H -br aided a lgebra. A map f : A → A ′ is a mor phism of left H -braided alg ebras, from ( A, s ) to ( A ′ , s ′ ), if and only if it is a morphis m of standard alge br as and ( f ⊗ H )   s = s ′   ( H ⊗ f ). Note that ( H , c ) is an algebr a in LB H . Hence, one ca n consider left and right ( H, c )-mo dules in this mo noidal ca teg ory . Deﬁnition 1 .8. W e will s ay that ( V , s ) is a left H -br aide d m o dule or simply a left H -mo dule to mean that it is a left ( H, c )-mo dule in LB H . W e let H ( LB H ) denote the categ ory of left H -braided mo dules. R emark 1.9 . A left H -br aided spac e ( V , s ) is a left H -mo dule if and only if V is a standard left H -mo dule and s   ( H ⊗ ρ ) = ( ρ ⊗ H )   ( H ⊗ s )   ( c ⊗ V ) , where ρ denotes the a ction of H on V . F urthermor e, a map f : V → V ′ is a morphism of left H -mo dules , from ( V , s ) to ( V ′ , s ′ ), if and only if it is H -linear and ( f ⊗ H )   s = s ′   ( H ⊗ f ). Given left H -mo dules ( V , s V ) and ( U, s U ), with actions ρ V and ρ U resp ectively , we let ρ V ⊗ U denote the diago nal action ρ V ⊗ U := ( ρ V ⊗ ρ U )   ( H ⊗ s V ⊗ U )   (∆ H ⊗ V ⊗ U ) . The following prop osition says in particular that ( k , τ ) is a left H - mo dule via the trivial action and that ( V , s V ) ⊗ ( U, s U ) is a left H -mo dule via ρ V ⊗ U . Prop osition 1.10 (see [G-G1 ]) . The c ate gory H ( LB H ) , of left H -br aide d mo dules, endowe d with the usu al asso ciativity and unit c onstr aints, is monoidal. Deﬁnition 1.11. W e say that ( A, s ) is a left H -br aide d mo dule algebr a or simply a left H - mo d ule algebr a if it is an algebra in H ( LB H ). W e let H ( ALB H ) denote the categor y of left H -braided mo dule a lgebras. R emark 1.12 . ( A, s ) is a left H -mo dule algebra if and only if the following facts hold: (1) A is an alge br a, (2) s is a left tr anspo sition of H on A , (3) A is a standar d le ft H -mo dule, (4) s   ( H ⊗ ρ ) = ( ρ ⊗ H )   ( H ⊗ s )   ( c ⊗ A ), (5) µ A   ( ρ ⊗ ρ )   ( H ⊗ s ⊗ A )   (∆ H ⊗ A ⊗ A ) = ρ   ( H ⊗ µ A ), (6) h · 1 = ǫ ( h )1 fo r all h ∈ H , where ρ de no tes the a ction of H on A . So, ( A, s ) is a le ft H -mo dule algebra if a nd only if it is a left H -algebr a, a left H -mo dule and s atisﬁes conditions (5) and (6). UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 5 In the sequel, given a map ρ : H ⊗ A → A , s ometimes we will write h · a to denotes ρ ( h ⊗ a ). R emark 1.1 3 . If X genera tes H as a k -algebr a, then co nditions (4), (5) and (6) o f the ab ov e rema rk ar e satisﬁed if and o nly if s ( h ⊗ l · a ) = ( ρ ⊗ H )   ( H ⊗ s )   ( c ⊗ A )( h ⊗ l ⊗ a ) , h · ( ab ) = µ A   ( ρ ⊗ ρ )   ( H ⊗ s ⊗ A )(∆( h ) ⊗ a ⊗ b ) , h · 1 = ǫ ( h ) , for all a, b ∈ A and h, l ∈ X . Let ( A ′ , s ′ ) b e another left H -mo dule algebra . A map f : A → A ′ is a morphism of left H -m o dule algebr as , from ( A, s ) to ( A ′ , s ′ ), if and only if it is an H -linea r morphism of sta nda rd alg ebras that satisﬁes ( f ⊗ H )   s = s ′   ( H ⊗ f ). 1.3 Bialgebra actions and universal deformation form ulas Most o f the re sults of [G-Z, Sectio n 1] rema in v alid in our more g eneral context, with the same a rguments a nd minimal changes. In particular Theorem 1.1 5 b elow holds. Let H be a braided bialg ebra. Giv en a left H -mo dule algebra ( A, s ) a nd an element F ∈ H ⊗ H , we let F l : A ⊗ A → A ⊗ A denote the ma p deﬁned by F l ( a ⊗ b ) := ( ρ ⊗ ρ )   ( H ⊗ s ⊗ A )( F ⊗ a ⊗ b ) , where ρ : H ⊗ A → A is the action o f H on A . W e let A F denote A endow ed with the multiplication map µ A   F l . Deﬁnition 1.14. W e say that F ∈ H ⊗ H is a twisting element (b ase d on H ) if (1) ( ǫ ⊗ id)( F ) = (id ⊗ ǫ )( F ) = 1, (2) [(∆ ⊗ id)( F )]( F ⊗ 1) = [(id ⊗ ∆)( F )](1 ⊗ F ) in H ⊗ c H ⊗ c H , (3) ( c ⊗ H )   ( H ⊗ c )( F ⊗ h ) = h ⊗ F , for all h ∈ H . Theorem 1.15. L et ( A, s ) b e a left H -mo dule algebr a. If F ∈ H ⊗ H is a twisting element such that ( s ⊗ H )   ( H ⊗ s )( F ⊗ a ) = a ⊗ F , for al l a ∈ A , then A F is an asso ci ative algebr a with unit 1 A . The notions of braided bialg ebra, left H -braided mo dule alg ebra and twisting element make sense in arbitrary monoidal categorie s. Here we consider the monoi- dal categor y M [[ t ]] deﬁned as follows: - the ob jects ar e the k [[ t ]]- mo dules of the form M [[ t ]] wher e M is a k -vector space, - the ar rows ar e the k [[ t ]]-linea r maps, - the tensor pro duct is the completation M [[ t ]] b ⊗ k [[ t ]] N [[ t ]] of the alge br aic tensor pr o duct M [[ t ]] ⊗ k [[ t ]] N [[ t ]] with resp ect to the t - adic top ology , - the unit and the asso ciativity constra ins are the evident o nes. W e identify M [[ t ]] b ⊗ k [[ t ]] N [[ t ]] with ( M ⊗ N )[[ t ]] by the map Θ : M [[ t ]] b ⊗ k [[ t ]] N [[ t ]] → ( M ⊗ N )[[ t ]] given by Θ ( mt i ⊗ nt j ) := ( m ⊗ n ) t i + j . 6 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI If A is a k -alg ebra, then A [[ t ]] is an a lgebra in M [[ t ]] v ia the multiplication map ( A ⊗ A )[[ t ]] µ / / A [[ t ]] P ( a i ⊗ b i ) t i  / / P a i b i t i , where a i b i = µ A ( a i ⊗ b i ). The unit ma p is the cano nical inclusion k [[ t ]] ֒ → A [[ t ]]. If H is a braided bialg ebra ov er k , then H [[ t ]] is a braided bialg e bra in M [[ t ]]. The multiplication and unit maps are as ab ov e. The co multiplication and c ounits are the maps H [[ t ]] ∆ / / ( H ⊗ H )[[ t ]] P h i t i  / / P ∆ H ( h i ) t i and H [[ t ]] ǫ / / k [[ t ]] P h i t i  / / P ǫ H ( h i ) t i , and the braid op era to r is the map ( H ⊗ H )[[ t ]] c [[ t ]] / / ( H ⊗ H )[[ t ]] P ( h i ⊗ l i ) t i  / / P c H ( h i ⊗ l i ) t i . If ( A, s ) is a H -mo dule alge br a, then ( A [[ t ]] , s [[ t ]]), where s [[ t ]] is the map ( H ⊗ A )[[ t ]] s [[ t ]] / / ( A ⊗ H )[[ t ]] P ( h i ⊗ a i ) t i  / / P s ( h i ⊗ a i ) t i , is an H [[ t ]]-mo dule algebra , via ( H ⊗ A )[[ t ]] ρ / / A [[ t ]] P ( h i ⊗ a i ) t i  / / P ρ A ( h i ⊗ a i ) t i . A twisting element based on H [[ t ]] in M [[ t ]] is an element F ∈ H [[ t ]] b ⊗ k [[ t ]] H [[ t ]] satisfying conditions (1)–(3) of Deﬁnition 1.1 4 . It is easy to chec k that a p ow er series F = P F i t i ∈ ( H ⊗ H )[[ t ]] corres ponds via Θ − 1 to a twisting element if and only if (1) ( ǫ ⊗ id)( F 0 ) = (id ⊗ ǫ )( F 0 ) = 1 and ( ǫ ⊗ id)( F i ) = (id ⊗ ǫ )( F i ) = 0 for i ≥ 1, (2) F or all n ≥ 0 , X i + j = n (∆ ⊗ id)( F i )( F j ⊗ 1) = X i + j = n (id ⊗ ∆)( F i )(1 ⊗ F j ) in H ⊗ c H ⊗ c H , (3) ( c ⊗ H )   ( H ⊗ c )( F n ⊗ h ) = h ⊗ F n , for all h ∈ H and n ≥ 0. W e will say that F is a n u niversal deformation formula (UDF) b ase d on H if, moreov er, F 0 = 1 ⊗ 1. Theorem 1. 16. L et ( A, s ) b e a left H -mo dule algebr a . If F = P F i t i is an UDF b ase d on H , such t hat ( s ⊗ H )   ( H ⊗ s )( F i ⊗ a ) = a ⊗ F i for al l i ≥ 0 and a ∈ A , then, the c onstruction c onsider e d in The or em 1.15, applie d to the left H [[ t ]] -mo dule algebr a ( A [[ t ]] , s [[ t ]]) intro duc e d ab ove , pr o duc es a formal deformation of A . Pr o of. It is immediate.  UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 7 2 H q -mo dule algebra structures and deformations In this section, w e brieﬂy review the construction of the Hopf alg ebra H q and the UDF exp q based on H q considered in [W], we introduce the notion of a g o o d transp osition of H q on an alg ebra A , and we describ e all the braided H q -mo dule algebras whose transp osition is g oo d. Let q ∈ k × and let H be the algebr a genera ted by D 1 , D 2 , σ ± 1 , sub ject to the relations D 1 D 2 = D 2 D 1 , σ σ − 1 = σ − 1 σ = 1 and q σ D i = D i σ for i = 1 , 2, It is easy to check that H is a Hopf alg ebra with ∆( D 1 ) := D 1 ⊗ σ + 1 ⊗ D 1 , ǫ ( D 1 ) := 0 , S ( D 1 ) := − D 1 σ − 1 , ∆( D 2 ) := D 2 ⊗ 1 + σ ⊗ D 2 , ǫ ( D 2 ) := 0 , S ( D 2 ) := − σ − 1 D 2 , ∆( σ ) := σ ⊗ σ, ǫ ( σ ) := 1 , S ( σ ) := σ − 1 . If q is a primitive l -ro ot of unity with l ≥ 2, then the ideal I of H generated by D l 1 and D l 2 is a Hopf idea l. So , the quotient H /I is also a Hopf a lgebra. Let H q := ( H/ I if q is a primitive l -r oo t of unity with l ≥ 2, H if q = 1 or it is not a ro ot o f unity . The Hopf alg ebra H q was considered in the pap er [W], where it was proved tha t exp q ( tD 1 ⊗ D 2 ) :=            l − 1 X i =0 1 ( i )! q ( tD 1 ⊗ D 2 ) i if q is a primitive l -r oo t of unity ( l ≥ 2), ∞ X i =0 1 ( i ) q ! ( tD 1 ⊗ D 2 ) i if q = 1 or it is not a ro ot of unit y , is an UDF base d o n H q . 2.1 Go o d tra nsp ositions of H q on an a lgebra One o f our main purp oses in this pa per is to construc t formal deformation o f a lge- bras by using the UDF exp q ( tD 1 ⊗ D 2 ). By Theorem 1.16, it will b e suﬃcient to obtain examples of H q -mo dule algebras ( A, s ), whose underlying transp ositions s satisfy ( s ⊗ H q )   ( H q ⊗ s )( D 1 ⊗ D 2 ⊗ a ) = a ⊗ D 1 ⊗ D 2 for all a ∈ A . ( 2.1) Deﬁnition 2. 1. A k -linear map s : H q ⊗ A → A ⊗ H q is go o d if condition (2.1) is fulﬁlled. It is evident that s : H q ⊗ A → A ⊗ H q is go o d if and only if ther e exists a bijectiv e k -linea r map α : A → A such that s ( D 1 ⊗ a ) = α ( a ) ⊗ D 1 and s ( D 2 ⊗ a ) = α − 1 ( a ) ⊗ D 2 for all a ∈ A . Lemma 2.2. L et k [ σ ± 1 ] denote the subHopfalgebr a of H q gener ate d by σ . Each tr ansp osition s : H q ⊗ A → A ⊗ H q takes k [ σ ± 1 ] ⊗ A onto A ⊗ k [ σ ± 1 ] . Pr o of. Let τ b e the ﬂip. Since τ   s − 1   τ is a transp osition, it suﬃces to pr o ve that s ( σ ± 1 ⊗ a ) ∈ A ⊗ k [ σ ± 1 ] for all a ∈ A . W rite s ( σ ⊗ a ) = X ij k γ ij k ( a ) ⊗ σ i D j 1 D k 2 . 8 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI Since S 2 ( D 1 ) = q − 1 D 1 , S 2 ( D 2 ) = q D 2 and S 2 ( σ ± 1 ) = σ ± 1 , we have X ij k γ ij k ( a ) ⊗ σ i D j 1 D k 2 = s ( σ ⊗ a ) = s   ( S 2 ⊗ A )( σ ⊗ a ) = ( A ⊗ S 2 )   s ( σ ⊗ a ) = X ij k q k − j γ ij k ( a ) ⊗ σ i D j 1 D k 2 , and so γ ij k = 0 for j 6 = k . Using now that X ij γ ij j ( a ) ⊗ ∆( σ ) i ∆( D 1 ) j ∆( D 2 ) j = ( A ⊗ ∆)   s ( σ ⊗ a ) = ( s ⊗ H q )   ( H q ⊗ s )   (∆ ⊗ A )( σ ⊗ a ) = X ij i ′ j ′ γ i ′ j ′ j ′ ( γ ij j ( a )) ⊗ σ i ′ D j ′ 1 D j ′ 2 ⊗ σ i D j 1 D j 2 , it is easy to chec k that γ ij j = 0 if j > 0 (use that in each term of the r igh t side the exp onen t o f D 1 equals the exp onent of D 2 ). F or σ − 1 the s ame argument carr ies ov er. This ﬁnishes the pro of.  In the following re sult we o btain a characterization o f the go o d transp ositions of H q on an alg ebra A . Theorem 2.3. The fol lowing facts hold: (1) If s : H q ⊗ A → A ⊗ H q is a go o d tr ansp osi tion, then s ( σ ± 1 ⊗ a ) = a ⊗ σ ± 1 for al l a ∈ A and the map α : A → A , deﬁne d by s ( D 1 ⊗ a ) = α ( a ) ⊗ D 1 , is an algebr a homomorphism. (2) Given an algebr a automorphism α : A → A , ther e exists only one go o d tr ansp osition s : H q ⊗ A → A ⊗ H q such that s ( D 1 ⊗ a ) = α ( a ) ⊗ D 1 for al l a ∈ A . Pr o of. (1) By L e mma 2.2, we know tha t s induces b y res tr iction a tra nspo sition of k [ σ ± 1 ] on A . Hence, by [G-G1, Theorem 4.14], there is a sup eralgebra str uc tur e A = A + ⊕ A − such that s ( σ i ⊗ a ) = ( a ⊗ σ i if a ∈ A + , a ⊗ σ − i if a ∈ A − . Let α : A → A b e as in the statement. Since σ is a transp ositio n, if a ∈ A − , then α ( a ) ⊗ D 1 ⊗ σ + α ( a ) ⊗ 1 ⊗ D 1 = ( A ⊗ ∆)   s ( D 1 ⊗ a ) = ( s ⊗ H q )   ( H q ⊗ s )   (∆ ⊗ A )( D 1 ⊗ a ) = α ( a ) ⊗ D 1 ⊗ σ − 1 + α ( a ) ⊗ 1 ⊗ D 1 . So, A − = 0. Finally , α is an alg ebra ho momorphism, b ecause s ( h ⊗ 1) = 1 ⊗ h for ea c h h ∈ H q and s   ( H q ⊗ µ A ) = ( µ A ⊗ H q )   ( A ⊗ s )   ( s ⊗ A ) . (2) By item (1) a nd the comment pr e ceding Lemma 2.2, it must b e s ( σ ± 1 ⊗ a ) = a ⊗ σ ± 1 , s ( D 1 ⊗ a ) = α ( a ) ⊗ D 1 and s ( D 2 ⊗ a ) = α − 1 ( a ) ⊗ D 2 . So, necessar ily s ( σ i D j 1 D k 2 ⊗ a ) = α j − k ( a ) ⊗ σ i D j 1 D k 2 . W e leav e to the rea der the task to prov e that s is a g o o d transp osition.  UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 9 2.2 Some H q -mo dule algebra structures Let A b e an algebr a. Let us consider k -linear ma ps ς , δ 1 , δ 2 : A → A . It is evident that there is a (necessarily unique) a ction ρ : H q ⊗ A → A s uc h that ρ ( σ ⊗ a ) = ς ( a ) , ρ ( D 1 ⊗ a ) = δ 1 ( a ) and ρ ( D 2 ⊗ a ) = δ 2 ( a ) (2.2) for all a ∈ A , if a nd o nly if the maps ς , δ 1 and δ 2 satisfy the following conditions: (1) ς is a bijective map, (2) δ 1   δ 2 = δ 2   δ 1 , (3) q ς   δ i = δ i   ς for i = 1 , 2. (4) If q 6 = 1 and q l = 1, then δ l 1 = δ l 2 = 0. Let s : H q ⊗ A → A ⊗ H q be a go o d tr anspo sition a nd let α b e the as so c iated automorphism. Let ς , δ 1 and δ 2 be k -linear endo morphisms of A s a tisfying (1)– (4). Next, we determine the conditions that ς , δ 1 and δ 2 m ust satisfy in order that ( A, s ) bec omes an H q -mo dule algebr a via the action ρ deﬁned by (2.2). Theorem 2.4. ( A, s ) is an H q -mo dule algebr a via ρ if and only if: (5) ς is an algebr a aut omorph ism, (6) α   δ i = δ i   α for i = 1 , 2 , (7) α   ς = ς   α , (8) δ i (1) = 0 for i = 1 , 2 , (9) δ 1 ( ab ) = δ 1 ( a ) ς ( b ) + α ( a ) δ 1 ( b ) for al l a, b ∈ A , (10) δ 2 ( ab ) = δ 2 ( a ) b + ς  α − 1 ( a )  δ 2 ( b ) for al l a, b ∈ A . Pr o of. Assume that ( A, s ) is an H q -mo dule alg ebra and let τ : H q ⊗ H q → H q ⊗ H q be the ﬂip. E v alua ting the equality s   ( H q ⊗ ρ ) = ( ρ ⊗ H q )   ( H q ⊗ s )   ( τ ⊗ A ) successively o n D 1 ⊗ D i ⊗ a and D 1 ⊗ σ ⊗ a with i ∈ { 1 , 2 } a nd a ∈ A a r bitrary , we verify that items (6) and (7) are s atisﬁed. Item (8) follows fro m the fact that D 1 · 1 = D 2 · 1 = 0. Finally , using that σ · 1 = 1 and ev aluating the equality ρ   ( H q ⊗ µ A ) = µ A   ( ρ ⊗ ρ )   ( H q ⊗ s ⊗ A )   (∆ ⊗ A ⊗ A ) on σ ⊗ a ⊗ b and D i ⊗ a ⊗ b , with i = 1 , 2 and a, b ∈ A ar bitr ary , we see that items (5), (9) and (10) hold. So, conditions (5)–(1 0) are necessary . By Remar k 1.1 3, in o r der to verify that they are a ls o suﬃcient, it is enough to chec k that they imply that h · 1 = ǫ ( h ) , s ( h ⊗ l · a ) = ( ρ ⊗ H q )   ( H q ⊗ s )( l ⊗ h ⊗ a ) , h · ( ab ) = µ A   ( ρ ⊗ ρ )   ( H q ⊗ s ⊗ A )(∆( h ) ⊗ a ⊗ b ) , for all a, b ∈ A and h, l ∈ { D 1 , D 2 , σ ± 1 } . W e leav e this task to the reader.  Note that condition (8 ) in Theorem 2.4 is redundant since it can b e obtained by applying condition (9) and (10) with a = b = 1. 10 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI 3 H q -mo dule algebra structures on crossed pro ducts Let G be a gr oup endow ed with a repr esent ation on a k -vector spa ce V of dimen- sion n . Cons ider the symmetric k -a lgebra S ( V ) eq uipped with the unique action of G by auto morphisms that extends the actio n of G on V and ta k e A = S ( V )# f G , where f : G × G → k × is a normal co cycle. B y deﬁnition the k -alge br a A is a free left S ( V )-mo dule with basis { w g : g ∈ G } . Its pro duct is g iv en by ( P w g )( Qw h ) := P g Qf ( g , h ) w gh , where g Q denotes the action of g on Q . This sectio n is devoted to the study of the H q -mo dule algebras ( A, s ), with s go o d, that satisfy: s ( H q ⊗ V ) ⊆ V ⊗ H q , s ( H q ⊗ k w g ) ⊆ k w g ⊗ H q , σ · v ∈ V and σ · w g ∈ k w g , for all v ∈ V a nd g ∈ G . In Theo r em 3.5 we give a general characterizatio n of these mo dule a lgebras, a nd in Subsection 3.1 we consider a s p eciﬁc ca se which is more suitable for ﬁnding concr ete examples , and we study it in detail. Finally in Sub- section 3.2 we consider the ca se where the co cycle involv es several no n necessar ily central elements o f G . In the following prop osition we ch ara cterize the g o o d transp ositions s o f H q on A satisfying the hypothesis mentioned ab ov e. By theo r em 2 .3 this is eq uiv alent to require that the k -linear map α : A → A ass o cia ted with α , takes V to V and k w g to k w g for all g ∈ G . Prop osition 3.1. L et ˆ α : V → V b e a k -line ar m ap and χ α : G → k × a map. Ther e is a go o d tra nsp osition s : H q ⊗ A → A ⊗ H q , such that s ( D 1 ⊗ v ) = ˆ α ( v ) ⊗ D 1 and s ( D 1 ⊗ w g ) = χ α ( g ) w g ⊗ D 1 for al l v ∈ V and g ∈ G , if and only if ˆ α is a bije ctiv e k [ G ] -line ar map and χ α is a gr oup homomorphism. Pr o of. By Theorem 2 .3 we know that s exists if a n only if the k -linear map α : A → A deﬁned by α ( v 1 · · · v m w g ) := ˆ α ( v 1 ) · · · ˆ α ( v m ) χ α ( g ) w g , is an automo rphism. B ut, if this happ ens, then a) χ α is a mor phism since χ α ( g ) χ α ( h ) f ( g , h ) w gh = α ( w g ) α ( w h ) = α ( w g w h ) = χ α ( g h ) f ( g , h ) w gh for all g , h ∈ G , b) ˆ α is a bijective k [ G ]-linear map, since it is the r estriction and c orrestriction of α to V , and ˆ α ( g v ) = α ( w g ) ˆ α ( v ) α ( w − 1 g ) = χ α ( g ) w g ˆ α ( v )( χ α ( g ) w g ) − 1 = w g ˆ α ( v ) w − 1 g = g ˆ α ( v ) . Conv ersely , if ˆ α is a bijective map then α is also, and if ˆ α is a k [ G ]-linear map and χ α is a mor phism, then α ( w g ) ˆ α ( v ) = χ α ( g ) w g ˆ α ( v ) = g ˆ α ( v ) χ α ( g ) w g = ˆ α ( g v ) α ( w g ) and α ( w g ) α ( w h ) = χ α ( g ) w g χ α ( h ) w h = f ( g , h ) χ α ( g h ) w gh = α ( f ( g , h ) w gh ) , for all v ∈ V a nd g , h ∈ G , fro m which it follows eas ily tha t α is a morphism.  Let A = S ( V )# f G b e a s ab ov e. Throughout this section w e ﬁx a mor phism χ α : G → k × and a bijective k [ G ]-linear map ˆ α : V → V , and we let α : A → A denote the automor phism deter mined by ˆ α and χ α . Moreover we will call s : H q ⊗ A → A ⊗ H q UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 11 the go o d transp osition a sso ciated with α . O ur purp ose is to obtain all the H q -mo- dule algebr a structur es on ( A, s ) such that σ · v ∈ V and σ · w g ∈ k w g for all v ∈ V a nd g ∈ G . (3.3) Under these res trictions we obtain conditions which allow us to constr uct a ll H q - module structures in concrete examples . Thanks to Theorem 1.16 and the fact that exp q ( tD 1 ⊗ D 2 ) is an UDF based on H q , each one of thes e examples pro duces automatically a fo r mal deformation of A . First note that given an H q -mo dule algebra structure o n ( A, s ) s a tisfying (3.3), we can deﬁne k -linea r maps ˆ δ 1 : V → A, ˆ δ 2 : V → A and ˆ ς : V → V and maps δ 1 : G → A, δ 2 : G → A and χ ς : G → k × , by ˆ δ i ( v ) := D i · v , ˆ ς ( v ) := σ · v , δ i ( g ) := D i · w g and σ · w g := χ ς ( g ) w g . Lemma 3. 2. L et ˆ ς : V → V b e a k -line ar map and χ ς : G → k × b e a map. Then, the map ς : A → A deﬁne d by ς ( v 1 m w g ) := ˆ ς ( v 1 ) · · · ˆ ς ( v m ) χ ς ( g ) w g , is a k -algebr a automorphism if and only if ˆ ς is a bije ctive k [ G ] -line ar map and χ ς is a gr oup homomorphism. Pr o of. This was chec k ed in the pro of of P r opo sition 3.1.  Lemma 3.3. L et ˆ δ 1 : V → A and ˆ δ 2 : V → A b e k -line ar maps and let δ 1 : G → A and δ 2 : G → A b e maps. (1) The k - line ar map δ 1 : A → A given by δ 1 ( v 1 m w g ) := m X j =1 α ( v 1 ,j − 1 ) ˆ δ 1 ( v j ) ς ( v j +1 ,m w g ) + α ( v 1 m ) δ 1 ( g ) , wher e v hl = v h · · · v l , is wel l deﬁne d if and only if ˆ δ 1 ( v ) ˆ ς ( w ) + ˆ α ( v ) ˆ δ 1 ( w ) = ˆ δ 1 ( w ) ˆ ς ( v ) + ˆ α ( w ) ˆ δ 1 ( v ) for al l v , w ∈ V . (3.4) (2) The m ap δ 2 : A → A given by δ 2 ( v 1 m w g ) := m X j =1 ς  α − 1 ( v 1 ,j − 1 )  ˆ δ 2 ( v j ) v j +1 ,m w g + ς  α − 1  ( v 1 m ) δ 2 ( g ) is wel l deﬁne d if and only if ˆ δ 2 ( v ) w + ς  ˆ α − 1 ( v )  ˆ δ 2 ( w ) = ˆ δ 2 ( w ) v + ς  ˆ α − 1 ( w )  ˆ δ 2 ( v ) for al l v , w ∈ V . (3.5 ) Pr o of. W e prove the ﬁrst assertion and leav e the s econd o ne , which is similar , to the reader . The only if part follows immedia tely by noting that ˆ δ 1 ( v ) ˆ ς ( w ) + ˆ α ( v ) ˆ δ 1 ( w ) = δ 1 ( v w ) = δ 1 ( wv ) = ˆ δ 1 ( w ) ˆ ς ( v ) + ˆ α ( w ) ˆ δ 1 ( v ) . In order to prove the if part it suﬃces to chec k that δ 1 ( v 1 · · · v i − 1 v i +1 v i v i +2 · · · v m w g ) = δ 1 ( v 1 m w g ) for all i < m , which follows ea sily from the hypothesis.  Lemma 3.4. As s u me that ς is an algebr a automorphism and δ 1 , δ 2 ar e wel l deﬁne d. The fol lowing facts hold: 12 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI (1) The m ap δ 1 satisﬁes δ 1 ( x 1 · · · x m ) = m X j =1 α ( x 1 . . . x j − 1 ) δ 1 ( x j ) ς ( x j +1 · · · x m ) for al l x 1 , . . . , x m ∈ k # f G ∪ V , if and only if (a) ˆ δ 1 ( g v ) χ ς ( g ) w g + ˆ α ( g v ) δ 1 ( g ) = δ 1 ( g ) ˆ ς ( v ) + χ α ( g ) w g ˆ δ 1 ( v ) , (b) f ( g , h ) δ 1 ( g h ) = δ 1 ( g ) χ ς ( h ) w h + χ α ( g ) w g δ 1 ( h ) , for al l v ∈ V and g , h ∈ G . (2) The m ap δ 2 satisﬁes δ 2 ( x 1 · · · x m ) = m X j =1 ς   α − 1 ( x 1 . . . x j − 1 ) δ 1 ( x j ) x j +1 · · · x m for al l x 1 , . . . , x m ∈ k # f G ∪ V , if and only if (a) ˆ δ 2 ( g v ) w g + ˆ ς  ˆ α − 1 ( g v )  δ 2 ( g ) = δ 2 ( g ) v + χ ς ( g ) χ − 1 α ( g ) w g ˆ δ 2 ( v ) , (b) f ( g , h ) δ 2 ( g h ) = δ 2 ( g ) w h + χ ς ( g ) χ − 1 α ( g ) w g δ 2 ( h ) , for al l v ∈ V and g , h ∈ G . Pr o of. W e prov e the ﬁrst a ssertion and leav e the second one to the reader. F or the only if part it s uﬃces to note that ˆ δ 1 ( g v ) ς ( w g ) + α ( g v ) δ 1 ( g ) = δ 1 ( g v w g ) = δ 1 ( w g v ) = δ 1 ( g ) ς ( v ) + α ( w g ) ˆ δ 1 ( v ) , f ( g , h ) δ 1 ( g h ) = δ 1 ( w g w h ) = δ 1 ( g ) ς ( w h ) + α ( w g ) δ 1 ( h ) , and to use the deﬁnitions o f ς ( w g ) and α ( w g ). W e pr o ve the suﬃcient part by induc- tion on r = m + 1 − i , wher e i is the ﬁrst index with x i ∈ k # f G (if x 1 , . . . , x m ∈ V we set r := 0 ). F o r r ∈ { 0 , 1 } the r esult follows immedia tely fro m the deﬁnition of δ 1 . Assume that it is true when r < r 0 and that m + 1 − i = r 0 . If x i = w g and x i +1 = v ∈ V , then δ 1 ( x 1 · · · x m ) = δ 1 ( y 1 · · · y m ) where y j =      x j if j / ∈ { i, i + 1 } , g v if j = i , w g if j = i + 1, and hence, by the inductive hypo thes is a nd item (a), δ 1 ( x 1 · · · x m ) = m X j =1 α ( y 1 . . . y j − 1 ) δ 1 ( y j ) ς ( y j +1 · · · y m ) = m X j =1 α ( x 1 . . . x j − 1 ) δ 1 ( x j ) ς ( x j +1 · · · x m ) . If x i = w g and x i +1 = w h , then δ 1 ( x 1 · · · x m ) = f ( g , h ) δ 1 ( y 1 · · · y m − 1 ) where y j =      x j if j < i , w gh if j = i , x j +1 if j > i , and hence, by the inductive hypo thes is a nd item (b), δ 1 ( x 1 · · · x m ) = m − 1 X j =1 f ( g , h ) α ( y 1 . . . y j − 1 ) δ 1 ( y j ) ς ( y j +1 · · · y m − 1 ) = m X j =1 α ( x 1 . . . x j − 1 ) δ 1 ( x j ) ς ( x j +1 · · · x m ) , UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 13 as we wan t.  Theorem 3 .5. L et ˆ δ 1 : V → A , ˆ δ 2 : V → A and ˆ ς : V → V b e k -line ar m aps and let δ 1 : G → A , δ 2 : G → A and χ ς : G → k × b e maps. Ther e is an H q -mo dule algebr a structur e on ( A, s ) , such t hat σ · v = ˆ ς ( v ) , σ · w g = χ ς ( g ) w g , D i · v = ˆ δ i ( v ) and D i · w g = δ i ( g ) for al l v ∈ V , g ∈ G and i ∈ { 1 , 2 } , if and only if (1) ˆ ς : V → V is a bije ctive k [ G ] -line ar map and χ ς is a gr oup homomorphism, (2) Conditions (3.4) and (3.5 ) in L emma 3.3 and items (1)(a), (1)(b), (2)(a) and (2)(b) in L emma 3.4 ar e satisﬁe d, (3) ˆ δ i   ˆ α = α   ˆ δ i , (4) χ α ( g ) δ i ( g ) = α  δ i ( g )  for al l g ∈ G , (5) ˆ ς   ˆ α = ˆ α   ˆ ς (6) The maps ς : A → A , δ 1 : A → A and δ 2 : A → A , intr o duc e d in L emmas 3.2 and 3.3, satisfy the fol lowing pr op erties: δ 2   ˆ δ 1 = δ 1   ˆ δ 2 , ˆ δ i   ˆ ς = q ς   ˆ δ i , δ 2   δ 1 = δ 1   δ 2 , χ ς ( g ) δ i ( g ) = q ς  δ i ( g )  , δ l 1 = δ l 2 = 0 if q 6 = 1 and q l = 1 . Pr o of. By Theorem 2.4 and the dis cussion above it, we know that to have a n H q -mo dule alg ebra structure on ( A, s ) sa tisfying the require ments in the statement is equiv alen t to have maps ς , δ 1 , δ 2 : A → A satisfying conditions (1)–(10) in Sub- section 2.2 and such that ς ( v ) = ˆ ς ( v ) , ς ( w g ) = χ ς ( g ) w g , δ i ( v ) = ˆ δ i ( v ) and δ i ( w g ) = δ i ( g ) for all v ∈ V , g ∈ G and i ∈ { 1 , 2 } . Now, it is easy to see tha t a) If ς , δ 1 and δ 2 satisfy conditions (5), (9) and (10) in Subse c tion 2.2, then ς ( v 1 m w g ) = ˆ ς ( v 1 ) · · · ˆ ς ( v m ) χ ς ( g ) w g , δ 1 ( v 1 m w g ) = m X j =1 α ( v 1 ,j − 1 ) ˆ δ 1 ( v j ) ς ( v j +1 ,m w g ) + α ( v 1 m ) δ 1 ( g ) , δ 2 ( v 1 m w g ) = m X j =1 ς  α − 1 ( v 1 ,j − 1 )  ˆ δ 2 ( v j ) v j +1 ,m w g + ς  α − 1 ( v 1 m )  δ 2 ( g ) , where v hl = v h · · · v l . b) By Lemmas 3.2, 3.3 a nd 3.4, the maps deﬁned in a) satisfy conditions (1), (5), (8), (9) and (10) in Subsection 2 .2 if and only if items (1 ) and (2) o f the present theorem are fulﬁlled. So, in order to ﬁnish the pr o o f it s uﬃces to c heck that: c) Conditions (6) a nd (7) in Subsection 2.2 ar e satisﬁed if and only if items (3)–(5) of the present theor em are fulﬁlled, d) Conditions (2), (3) and (4) in Subsection 2.2 are satisﬁed if and only if item (6) of the present theor em is fulﬁlled. W e leav e this task to the reader .  W e ar e going now to co nsider several particular cases, with the pur pos e of o b- taining more precise results. This will allow us to g iv e some sp eciﬁc examples o f formal deformatio ns of a sso ciative a lgebras. 14 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI 3.1 First case Let ˆ α , χ α , α and s b e as in the discussion following Prop osition 3.1. Let ˆ δ 1 : V → A , ˆ δ 2 : V → A and ˆ ς : V → V be k -linear maps and let χ ς : G → k × be a map. Assume that the kernels of ˆ δ 1 and ˆ δ 2 hav e co dimension 1, ker ˆ δ 1 6 = ker ˆ δ 2 and there exist x i ∈ V \ ker ˆ δ i , such that ˆ δ i ( x i ) = P i w g i with P i ∈ S ( V ) and g i ∈ G . Without lo ss of g enerality we can assume that x 1 ∈ ker ˆ δ 2 and x 2 ∈ ker ˆ δ 1 (and we do it). F or g ∈ G and i ∈ { 1 , 2 } , let λ ig , ω i , ν i ∈ k b e the elements deﬁned by the following conditions: g x i − λ ig x i ∈ ker ˆ δ i , ˆ ς ( x i ) − ω i x i ∈ ker ˆ δ i and ˆ α ( x i ) − ν i x i ∈ ker ˆ δ i . Theorem 3.6. Ther e is an H q -mo dule algebr a structu r e on ( A, s ) , satisfying σ · v = ˆ ς ( v ) , σ · w g = χ ς ( g ) w g , D i · v = ˆ δ i ( v ) and D i · w g = 0 for al l v ∈ V , g ∈ G and i ∈ { 1 , 2 } , if and only if (1) ˆ ς is a bije ctive k [ G ] -line ar map and χ ς is a gr oup homomorphism, (2) ˆ ς ( v ) = g − 1 1 ˆ α ( v ) for al l v ∈ ker ˆ δ 1 and ˆ ς ( v ) = g 2 ˆ α ( v ) for al l v ∈ ker ˆ δ 2 , (3) g 1 and g 2 b elo ng to t he c enter of G , (4) ker ˆ δ 1 and ker ˆ δ 2 ar e G -submo dules of V , (5) g P 1 = λ 1 g χ − 1 α ( g ) χ ς ( g ) f − 1 ( g , g 1 ) f ( g 1 , g ) P 1 for al l g ∈ G , (6) g P 2 = λ 2 g χ α ( g ) χ − 1 ς ( g ) f − 1 ( g , g 2 ) f ( g 2 , g ) P 2 for al l g ∈ G , (7) ˆ α (ker ˆ δ i ) = ker ˆ δ i for i ∈ { 1 , 2 } , (8) P 1 ∈ ker δ 2 and P 2 ∈ ker δ 1 , wher e δ 1 and δ 2 ar e the maps deﬁne d by δ 1 ( v 1 m w g ) := m X j =1 α ( v 1 ,j − 1 ) ˆ δ 1 ( v j ) ς ( v j +1 ,m w g ) , δ 2 ( v 1 m w g ) := m X j =1 ς  α − 1 ( v 1 ,j − 1 )  ˆ δ 2 ( v j ) v j +1 ,m w g , in which v hl = v h · · · v l , (9) ς ( P i ) = q − 1 ω i χ − 1 ς ( g i ) P i and α ( P i ) = ν i χ − 1 α ( g i ) P i for i ∈ { 1 , 2 } , wher e ς is the map given by ς ( v 1 m w g ) = ˆ ς ( v 1 ) · · · ˆ ς ( v m ) χ ς ( g ) w g , (10) If q 6 = 1 and q l = 1 , then δ l 1 = δ l 2 = 0 . In order to prove this result we ﬁr st need to establish so me a uxiliary results. Lemma 3 .7. The fol lowing facts hold: (1) Condition (3.4 ) of L emma 3.3 is satisﬁe d if and only if g 1 ˆ ς ( v ) = ˆ α ( v ) for al l v ∈ ker ˆ δ 1 . (2) Condition (3.5) of Le mma 3.3 is satisﬁ e d if and only if g 2 v = ˆ ς  ˆ α − 1 ( v )  for al l v ∈ ker ˆ δ 2 . Pr o of. W e pr ove item 1 ) and we leave item 2 ), which is similar, to the rea der. W e m ust chec k that ˆ δ 1 ( v ) ˆ ς ( w ) + ˆ α ( v ) ˆ δ 1 ( w ) = ˆ δ 1 ( w ) ˆ ς ( v ) + ˆ α ( w ) ˆ δ 1 ( v ) for all v , w ∈ V (3.6) UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 15 if a nd only if ˆ ς 1 ( v ) = g − 1 1 ˆ α ( v ) for all v ∈ ker ˆ δ 1 . It is clear that we can supp ose that v , w ∈ { x 1 } ∪ ker ˆ δ 1 . When v , w ∈ ker ˆ δ 1 or v = w = x 1 the equality (3.6) is trivial. Assume v = x 1 and w ∈ ker ˆ δ 1 . Then, ˆ δ 1 ( v ) ˆ ς ( w ) + ˆ α ( v ) ˆ δ 1 ( w ) = P 1 w g 1 ˆ ς ( w ) = P 1 g 1 ˆ ς ( w ) w g 1 and ˆ δ 1 ( w ) ˆ ς ( v ) + ˆ α ( w ) ˆ δ 1 ( v ) = ˆ α ( w ) P 1 w g 1 = P 1 ˆ α ( w ) w g 1 So, in this case, the result is tr ue. Ca se v ∈ ker ˆ δ 1 and w = x 1 can b e trea ted in a similar wa y .  Lemma 3 .8. The fol lowing facts hold: (1) Items (1)(a) and (1)(b) of L emma 3.4 ar e satisﬁe d if and only if (a) ker ˆ δ 1 is a G -submo dule of V , (b) g 1 b elo ngs to t he c enter of G , (c) g P 1 = λ 1 g χ − 1 α ( g ) χ ς ( g ) f − 1 ( g , g 1 ) f ( g 1 , g ) P 1 , for al l g ∈ G . (2) Items (2)(a) and (2)(b) of L emma 3.4 ar e satisﬁe d if and only if (a) ker ˆ δ 2 is a G -submo dule of V , (b) g 2 b elo ngs to t he c enter of G , (c) g P 2 = λ 2 g χ α ( g ) χ − 1 ς ( g ) f − 1 ( g , g 2 ) f ( g 2 , g ) P 2 , for al l g ∈ G . Pr o of. W e prove item 1) and we leav e item 2) to the reader . S ince δ 1 = 0 , it is suﬃcient to prove that ˆ δ 1 ( g v ) χ ς ( g ) w g = χ α ( g ) w g ˆ δ 1 ( v ) for all v ∈ V and g ∈ G , (3.7) if and only if co nditions (1)(a), (1 )(b) and (1)(c) are satisﬁed. W e can as sume that v ∈ { x 1 } ∪ ker ˆ δ 1 . When v ∈ ker ˆ δ 1 , then equality (3.7) is true if a nd only if g v ∈ ker ˆ δ 1 . Now, since ˆ δ 1 ( g x 1 ) χ ς ( g ) w g = λ 1 g P 1 w g 1 χ ς ( g ) w g = λ 1 g P 1 χ ς ( g ) f ( g 1 , g ) w g 1 g and χ α ( g ) w g ˆ δ 1 ( x 1 ) = χ α ( g ) w g P 1 w g 1 = χ α ( g ) g P 1 f ( g , g 1 ) w gg 1 , equality (3.7) is true for v = x 1 and g ∈ G if a nd only if conditions (1)(b) and (1)(c) are satisﬁed.  Pr o of of The or em 3.6. First note that item (1) coincide with item (1) of Theo- rem 3 .5 and that, by Lemmas 3.7 and 3 .8, item (2) of Theorem 3 .5 is eq uiv ale nt to items (2)–(6). Item (4) of Theorem 3.5 and tw o o f the equalities in item (6) of the same theorem, ar e tr iv ially satisﬁed b ecause δ 1 = δ 2 = 0. Since ˆ δ i  ˆ α ( x i )  = ν i ˆ δ i ( x i ) = ν P i w g i and α  ˆ δ i ( x i )  = α ( P i w g i ) = α ( P i ) χ α ( g i ) w g i , item (3) of Theor em 3.5 is true if and only if item (7 ) and the se c ond equality in item (9) hold. Since ˆ α is k [ G ]-linear , item (5) of Theorem 3.5 is an immediate con- sequence of item (2) of T heo rem 3.6. Finally we consider the no n- trivial eq ualities in item (6) of Theorem 3.5. It is easy to see that ˆ δ i  ˆ ς ( x i )  = q ς  ˆ δ i ( x i )  if a nd only if the ﬁrst equa lit y in item (9) holds. O n the other ha nd ˆ δ i  ˆ ς ( v )  = q ς  ˆ δ i ( v )  for all v ∈ ker ˆ δ i if a nd only if ˆ ς (ker ˆ δ i ) ⊆ ker ˆ δ i , whic h fo llo ws from items (2), (4) and (7). The equality δ 2  ˆ δ 1 ( v )  = δ 1  ˆ δ 2 ( v )  is tr ivially satisﬁed for v ∈ ker ˆ δ 1 ∩ ker ˆ δ 2 , and for v ∈ { x 1 , x 2 } it is equiv alent to item (8). Lastly , the remaining equality coincides with item (10).  R emark 3.9 . The following facts hold: 16 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI - Since ˆ α and ˆ ς are bijective k [ G ]-linear maps, fr om item (2) Theorem 3.6 it follows that g − 1 1 v = g 2 v for all v ∈ ker ˆ δ 1 ∩ ker ˆ δ 2 . (3.8) - Since x 1 ∈ ker ˆ δ 2 and k er ˆ δ 2 is G -stable, g x 1 − λ 1 g x 1 ∈ ker ˆ δ 1 ∩ ker ˆ δ 2 . Similarly g x 2 − λ 1 g x 2 ∈ ker ˆ δ 1 ∩ ker ˆ δ 2 . - Since ker ˆ δ i is a G -submo dule o f V and the k -linear map V / / V v  / / g v is an isomor phis m fo r each g ∈ G , it is impos sible that g x i ∈ ker ˆ δ i . Con- sequently , λ ig ∈ k × for ea c h g ∈ G . Moreov er, using ag ain that ker ˆ δ i is a G -submo dule of V , it is easy to s ee that the map g 7→ λ ig is a group homomorphism. Items (1), (2), (4), (7) and the fact that ˆ α is bijective imply that also ω 1 , ω 2 , ν 1 , ν 2 ∈ k × . - Since ˆ ς ( x 1 ) = ˆ α ( g 2 x 1 ) ≡ λ 1 g 2 ˆ α ( x 1 ) ≡ λ 1 g 2 ν 1 x 1 (mo d k er ˆ δ 1 ) , we hav e ω 1 = λ 1 g 2 ν 1 . A similar ar g umen t shows that ν 2 = λ 2 g 1 ω 2 . Corollary 3.10 . Assume that the c onditio ns ab ove The or em 3.6 ar e fu lﬁ l le d and that ther e ex ists an H q -mo dule algebr a structu r e on ( A, s ) satisfying σ · v = ˆ ς ( v ) , σ · w g = χ ς ( g ) w g , D i · v = ˆ δ i ( v ) and D i · w g = 0 for al l v ∈ V , g ∈ G and i ∈ { 1 , 2 } . If P 1 ∈ S (ker ˆ δ 1 ) and P 2 ∈ S (ker ˆ δ 2 ) , then λ 1 g 1 λ 1 g 2 = q and λ 2 g 1 λ 2 g 2 = q − 1 . Mor e over g 0 := g 1 g 2 has determinant 1 as an op er a tor on V . Pr o of. By items (9), (2) a nd (5) of Theor em 3.6, q − 1 ω 1 χ − 1 ς ( g 1 ) P 1 = ς ( P 1 ) = g − 1 1 ˆ α ( P 1 ) = ν 1 χ − 1 α ( g 1 ) g − 1 1 P 1 = ν 1 λ − 1 1 g 1 χ − 1 ς ( g 1 ) P 1 . Hence λ 1 g 1 λ 1 g 2 = q as we wan t, since ω 1 = ν 1 λ 1 g 2 . The pro of that λ 2 g 1 λ 2 g 2 = q − 1 is similar . It rema ins to check that det( g 0 ) = 1. Since k er ˆ δ 1 and ker ˆ δ 2 are G - inv ariant, we hav e g x 1 ∈ ker ˆ δ 2 and g x 2 ∈ ker ˆ δ 1 for all g ∈ G , and so g 0 x 1 ∈ λ 1 g 1 λ 1 g 2 x 1 + W and g 0 x 2 ∈ λ 2 g 1 λ 2 g 2 x 1 + W, where W = ker ˆ δ 1 ∩ ker ˆ δ 2 . Moreover, by Rema rk 3 .9 we know that g 0 acts as the ident ity ma p o n W and hence det( g 0 ) = λ 1 g 1 λ 1 g 2 λ 2 g 1 λ 2 g 2 = 1.  R emark 3.11 . A particular case is the H q -mo dule algebra A considered in [W, Section 4], in w hich P 1 = 1, g 1 = 1 and ˆ α is the identit y ma p. Our P 2 , g 2 and f corr espo nd in [W] to s , g and α , resp ectively . Our computatio ns s how that the condition that h ( s ) = x 1 ( h ) x 2 ( h ) α ( g , h ) α − 1 ( h, g ) s , which app ears a s informed b y the cohomolo gy of ﬁnite gr oups in [W], is in fact necessar y for the existence o f the H q -mo dule alge br a structure of A , and it do es not dep end on cohomolo gical consideratio ns. In particular we need this condition for any group G , ﬁnite or not. Similarly the conditions that g is central and det( g ) = 1 are necessary even for inﬁnite groups . UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 17 Let G , V , f : G × G → k × and A b e as at the b eginning o f this section. Let ˆ α : V → V b e a bijective k [ G ]-linear map, χ α : G → k × a gro up homomor phism, α : A → A the algebra automor phism induced by ˆ α and χ α , and s the go o d tra ns- po sition asso ciated with α . Le t a) V 1 6 = V 2 subspaces of co dimension 1 of V such that V 1 and V 2 are ˆ α -stable G -submo dules of V , b) g 1 and g 2 central elements o f G such that g − 1 1 v = g 2 v for a ll v ∈ V 1 ∩ V 2 , c) χ ς : G → k × a group homomorphis m and ˆ ς : V → V the map deﬁned by ˆ ς ( v ) := ( ˆ α ( g − 1 1 v ) if v ∈ V 1 , ˆ α ( g 2 v ) if v ∈ V 2 , d) x 1 ∈ V 2 \ V 1 , x 2 ∈ V 1 \ V 2 , P 1 ∈ S ( V 1 ), P 2 ∈ S ( V 2 ) and ˆ δ 1 , ˆ δ 2 : V → A the maps deﬁned by ker ˆ δ i := V i and ˆ δ i ( x i ) := P i w g i . F or g ∈ G a nd i ∈ { 1 , 2 } , let λ ig , ν i , ω i ∈ k × be the elements deﬁned by the conditions g x i − λ ig x i ∈ V i , ˆ α ( x i ) − ν i x i ∈ V i and ˆ ς ( x i ) − ω i x i ∈ V i . The following re sult is a sort of a reformulation of The o rem 3.6, more appropria te to co nstruct explicit examples. The only new hypothesis that we need is that P i ∈ S ( V i ). Corollary 3. 12. Ther e is an H q -mo dule algebr a structu r e on ( A, s ) , satisfying σ · v = ˆ ς ( v ) , σ · w g = χ ς ( g ) w g , D i · v = ˆ δ i ( v ) and D i · w g = 0 for al l v ∈ V , g ∈ G and i ∈ { 1 , 2 } , if and only if (1) q = λ 1 g 1 λ 1 g 2 and q − 1 = λ 2 g 1 λ 2 g 2 , (2) g P 1 = λ 1 g χ − 1 α ( g ) χ ς ( g ) f − 1 ( g , g 1 ) f ( g 1 , g ) P 1 , (3) g P 2 = λ 2 g χ α ( g ) χ − 1 ς ( g ) f − 1 ( g , g 2 ) f ( g 2 , g ) P 2 , (4) α ( P i ) = ν i χ − 1 α ( g i ) P i , (5) P 1 ∈ ker δ 2 and P 2 ∈ ker δ 1 , wher e δ 1 , δ 2 : A → A ar e the maps deﬁne d in item (8) of The or em 3.6, (6) If q 6 = 1 and q l = 1 , then δ l 1 = δ l 2 = 0 . Pr o of. ⇐ ) By a), b), c) and d), it is o b vious that items (1), (2), (3), (4 ) and (7) of Theorem 3.6 ar e satisﬁed. Moreover items (2), (3), (5) a nd (6) are items (5), (6), (8) and (10) of Theorem 3.6 . So, we o nly must to chec k that item (9) of Theo r em 3.6 is satisﬁed. But the second equality in this item is exactly the o ne re quired in item (4) of the present coro llary , and we ar e go ing to chec k that the ﬁrst o ne is true with q = λ 1 g 1 λ 1 g 2 . Arg uing a s in Remark 3.9, a nd using item (2) with g = g 1 , item (1) and item (4), we obtain q − 1 ω 1 χ − 1 ς ( g 1 ) P 1 = q − 1 λ 1 g 2 ν 1 χ − 1 ς ( g 1 ) P 1 = q − 1 λ 1 g 1 λ 1 g 2 ν 1 χ − 1 α ( g 1 ) g − 1 1 P 1 = ν 1 χ − 1 α ( g 1 ) g − 1 1 P 1 = g − 1 1 α ( P 1 ) = ς ( P 1 ) , 18 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI where the la st equa lit y is tr ue since P 1 ∈ S ( V 1 ). Again arguing a s in Remark 3 .9, and using item (3) with g = g 2 , item (1) and item (4), we o btain q − 1 ω 2 χ − 1 ς ( g 2 ) P 2 = q − 1 λ − 1 2 g 1 ν 2 χ − 1 ς ( g 2 ) P 2 = q − 1 λ − 1 2 g 1 λ − 1 2 g 2 ν 2 χ − 1 α ( g 2 ) g 2 P 2 = ν 2 χ − 1 α ( g 2 ) g 2 P 2 = g 2 α ( P 2 ) = ς ( P 2 ) , where the last equa lit y is true since P 2 ∈ S ( V 2 ). ⇒ ) Items (2 ), (3), (5 ) and (6) a re items (5), (6), (8) and (1) o f Theor em 3.6, a nd item (4) is the ﬁrs t e q ualit y in item (9) of that theo rem. Finally item (1) follows from Corolla ry 3.10.  The following result shows that if x 1 and x 2 are eigenv ectors o f the ma ps v 7→ g 1 v and v 7→ g 2 v , then item (5) in the statement of Coro llary 3.12 ca n b e easily tested and item (6) ca n b e r emo ved from the hypothesis. Prop osition 3.13. Assume that c onditions a), b), c) and d) ab ove Cor ol lary 3.12 ar e fulﬁl le d . L et δ 1 and δ 2 b e the maps intro duc e d in item (8) of The or em 3.6. If λ 1 g 1 λ 1 g 2 = q , λ 2 g 1 λ 2 g 2 = q − 1 and g i x j = λ j g i x j for 1 ≤ i , j ≤ 2 , then: (1) δ l 1 = δ l 2 = 0 , whenever q 6 = 1 and q l = 1 . (2) If q = 1 or it is n ot a r o ot of unity, t hen P 1 ∈ ker δ 2 and P 2 ∈ ker δ 1 if and only if P 1 , P 2 ∈ S ( V 1 ∩ V 2 ) . (3) If q 6 = 1 is a primitive l -r o ot of unity, then P 1 ∈ ker δ 2 and P 2 ∈ ker δ 1 if and only if P 1 ∈ S  k x l 2 ⊕ ( V 1 ∩ V 2 )  and P 2 ∈ S  k x l 1 ⊕ ( V 1 ∩ V 2 )  . Pr o of. The pr opo sition is a direct conseque nc e o f the following formulas: δ s 1 ( x r 1 1 · · · x r n n w g ) = ( cα s  x r 1 − s 1 x r 2 2 · · · x r n n  w g s 1 g for s ≤ r 1 , 0 otherwise, and δ s 2 ( x r 1 1 · · · x r n n w g ) = ( dx r 2 − s 2 g s 2  x r 1 1 x r 3 3 · · · x r n n  w g s 2 g for s ≤ r 2 , 0 otherwise, where α s denotes the s -fold co mp osition o f α , c = χ s ς ( g ) χ s ( s − 1) / 2 ς ( g 1 ) χ s ( s − 1) / 2 α ( g 1 ) s − 1 Y k =0 ( r 1 − k ) q ! s − 1 Y k =0 f ( g 1 , g k 1 g ) ! α s − 1 ( P s 1 ) , d = λ sr 2 − s ( s +1) / 2 2 g 2 s − 1 Y k =0 ( r 2 − k ) q ! s − 1 Y k =0 f ( g 2 , g k 2 g ) ! s − 1 Y k =0 g k 2 P 2 ! . W e will pr ov e the formula for δ s 1 and we will leave the other one to the rea der. W e beg in with the case s = 1. Since x 2 , . . . , x n ∈ ker ˆ δ 1 and ˆ δ 1 ( x 1 ) = P 1 w g 1 , fro m the deﬁnition of δ 1 it follows tha t δ 1 ( x r 1 1 · · · x r n n w g ) = r 1 − 1 X j =0 α ( x j 1 ) P 1 w g 1 ς ( x r 1 − j − 1 1 x r 2 2 · · · x r n n w g ) . UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 19 Thu s, using the deﬁnition of ς , item c) a bove Cor ollary 3.12, the fact that α is G -linear and the hypo thesis, w e obta in δ 1 ( x r 1 1 · · · x r n n w g ) = r 1 − 1 X j =0 α ( x j 1 ) P 1 w g 1 α ( g 2 x r 1 − j − 1 1 ) g − 1 1 α ( x r 2 2 · · · x r n n ) χ ς ( g ) w g = r 1 − 1 X j =0 α ( x j 1 ) P 1 α ( g 1 g 2 x r 1 − j − 1 1 ) α ( x r 2 2 · · · x r n n ) χ ς ( g ) f ( g 1 , g ) w g 1 g = χ ς ( g )( r 1 ) q f ( g 1 , g ) P 1 α ( x r 1 − 1 1 x r 2 2 · · · x r n n ) w g 1 g . Assume that s ≤ r 1 and that the for m ula for δ s 1 holds. Since c dep ends on s , r 1 and g , it will b e conv enient fo r us to use the more pr ecise notatio n c s,r 1 ( g ) for c . F rom items (3) and (5) o f T he o rem 3.5 and item (9) of Theo rem 2.4, It follows ea sily that α   δ 1 = δ 1   α on S ( V ). Using this fact, item (9) of Theorem 2.4 a nd the inductive hypothesis, we obtain δ s +1 1 ( x r 1 1 · · · x r n n w g ) = α  c sr 1 ( g )  α s  δ 1 ( x r 1 − s 1 x r 2 2 · · · x r n n )  ς ( w g s 1 g ) . If s = r 1 , then δ 1 ( x r 1 − s 1 x r 2 2 · · · x r n n ) = 0. Otherwise, δ s +1 1 ( x r 1 1 · · · x r n n w g ) = cα s  α ( x r 1 − s − 1 1 x r 2 2 · · · x r n n ) w g 1  ς ( w g s 1 g ) = cα s +1 ( x r 1 − s − 1 1 x r 2 2 · · · x r n n ) χ s α ( g 1 ) χ s ς ( g 1 ) χ ς ( g ) f ( g 1 , g s 1 g ) w g s +1 1 g , where c = α  c s,r 1 ( g )  α s  c 1 ,r 1 − s (1)  . The formula for δ s +1 1 follows immediately from this fact.  Example 3.14 . Let G = h g i b e an or der r cy c lic group, ξ an element of k × and f ξ : G ⊗ G → k the c ocy c le deﬁned by f ξ ( g u , g v ) := ( 1 if u + v < r , ξ otherwise. Of course, if r = ∞ , then for any ξ this is the trivial co cycle. Let V b e a vector space endow ed with an a c tion of G and let A b e the cro ssed pro duct A = S ( V )# f ξ G . Let { x 1 , . . . , x n } b e a basis of V . Let us V 1 and V 2 denote the subspaces of V genera ted by { x 2 , . . . , x n } and { x 1 , x 3 , . . . , x n } resp ectively . Let ˆ α : V → V b e a bijective k [ G ]-linear map. Assume tha t V 1 and V 2 are ˆ α -s table G -submo dules of V and that there exist λ 1 , λ 2 ∈ k × such that g x 1 = λ 1 x 1 and g x 2 = λ 2 x 2 . Let m 1 , m 2 ∈ Z . Assume that g m 1 + m 2 v = v for all v ∈ V 1 ∩ V 2 (if r < ∞ we can take 0 ≤ m 1 , m 2 < r ). Let ˆ ς : V → V b e the ma p deﬁned by ˆ ς ( v ) := ( ˆ α ( g − m 1 v ) if v ∈ V 1 , ˆ α ( g m 2 v ) if v ∈ V 2 , and let χ α , χ ς : G → k × be tw o morphisms. Co nsider the a utomorphism of alg ebras α : A → A given by α ( v ) := ˆ α ( v ) for v ∈ V and α ( w g ) = χ α ( g ) w g , and deﬁne ˆ δ 1 , ˆ δ 2 : V → A by ˆ δ 1 ( x 2 ) = · · · = ˆ δ 1 ( x n ) := 0 , ˆ δ 1 ( x 1 ) := P 1 w g m 1 , ˆ δ 2 ( x 1 ) = ˆ δ 2 ( x 3 ) = · · · = ˆ δ 1 ( x n ) := 0 , ˆ δ 2 ( x 2 ) := P 2 w g m 2 , where P 1 ∈ S ( V 1 ) \ { 0 } and P 2 ∈ S ( V 2 ) \ { 0 } . Let s b e the transp osition o f H q with A a sso ciated with α . There is an H q -mo dule a lgebra structure over ( A, s ) sa tisfying σ · v = ˆ ς ( v ) , σ · w g = χ ς ( g ) w g , D i · v = ˆ δ i ( v ) and D i · w g = 0 for a ll v ∈ V , if and only if 20 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI (1) q = λ m 1 + m 2 1 and q − 1 = λ m 1 + m 2 2 , (2) g P 1 = λ 1 χ − 1 α ( g ) χ ς ( g ) P 1 and g P 2 = λ 2 χ α ( g ) χ − 1 ς ( g ) P 2 , (3) α ( P 1 ) = ν 1 χ − m 1 α ( g ) P 1 and α ( P 2 ) = ν 2 χ − m 2 α ( g ) P 2 , (4) If q = 1 or q is not a r o o t of unity , then P 1 , P 2 ∈ k [ x 3 , . . . , x n ], (5) If q 6 = 1 is a primitive l -ro ot of unit y , then P 1 ∈ k [ x l 2 , x 3 , . . . , x n ] and P 2 ∈ k [ x l 1 , x 3 , . . . , x n ] . Consequently , in order to obtain explicit examples of braided H q -mo dule alge br a structures on an a lg ebra A of the shap e S ( V )# f ξ G , wher e V is a k -vector space with basis { x 1 , . . . , x n } and G = h g i is a c y clic gr oup of order r ≤ ∞ , w e pro ceed as follows: First: W e deﬁne a n a ction o f G on V . F or this we c ho ose - a k -linea r auto morphism γ of V 12 := h x 3 , . . . , x n i , whose order divides r if r < ∞ , - λ 1 , λ 2 ∈ k × such that λ r 1 = λ r 2 = 1 if r < ∞ , and we set g x i :=      λ 1 x 1 if i = 1, λ 2 x 2 if i = 2, γ ( x i ) if i ≥ 3. Second: W e construct the algebra A . F or this we cho ose ξ ∈ k × and we deﬁne A = S ( V )# f ξ G , where f ξ is the co cycle asso cia te with ξ . Third: W e e ndo w A with a k -a lgebra automor phism α . F or this we take ν 1 , ν 2 , η ∈ k × such that η r = 1 if r < ∞ , a k - linear automor phism α ′ of V 12 and v 1 , v 2 ∈ V 12 , and we deﬁne α ( w g ) := η w g and α ( x i ) :=      ν 1 x 1 + v 1 if i = 1 , ν 2 x 2 + v 2 if i = 2 , α ′ ( x i ) if i ≥ 3. F ourth: W e choose m 1 , m 2 ∈ Z and ζ ∈ k × such that γ m 1 + m 2 = id , ( λ 1 λ 2 ) m 1 + m 2 = 1 and ζ r = 1 if r < ∞ , and we deﬁne ς ( w g ) := ζ w g and ς ( x i ) :=      λ m 2 1 ( ν 1 x 1 + v 1 ) if i = 1 , λ − m 1 2 ( ν 2 x 2 + v 2 ) if i = 2, α ′  γ m 2 ( x i )  if i ≥ 3 . Fifth: we set q := λ m 1 + m 2 1 and we cho ose P 1 , P 2 ∈ S ( V ) \ { 0 } such that - if q is not a ro ot of unity , then P 1 , P 2 ∈ k [ x 3 , . . . , x n ], - if q is a pr imitiv e l -ro ot of unit y , then P 1 ∈ k [ x l 2 , x 3 , . . . , x n ] and P 2 ∈ k [ x l 1 , x 3 , . . . , x n ] , - g P 1 = λ 1 η − 1 ζ P 1 and g P 2 = λ 2 η ζ − 1 P 2 , - α ( P 1 ) = ν 1 η − m 1 P 1 and α ( P 2 ) = ν 2 η − m 2 P 2 . Now, by the discussion a t the b eginning of this example, there is an H q -mo dule algebra structure on ( A, s ), where s : H q ⊗ A → A ⊗ H q is the go o d transp osition asso ciated with α , such that σ · x j = ς ( x j ) , σ · w g = ζ w g , D i · w g = 0 and D i ( x j ) = ( 0 if i 6 = j , P i w g mi if i = j , UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 21 where i ∈ { 1 , 2 } and j ∈ { 1 , . . . , n } . R emark 3.15 . If P 1 (0) 6 = 0 and P 2 (0) 6 = 0, then the conditions in the ﬁr st step are fulﬁlled if a nd o nly if λ 1 λ 2 = 1, η = λ 1 ζ , ν 1 = η m 1 , ν 2 = η m 2 , P 1 and P 2 are G -inv ariants, α ( P 1 ) = P 1 and α ( P 2 ) = P 2 . 3.2 Second case. Let ˆ α , χ α , α and s be a s in the discussion following Prop osition 3 .1, let χ ς : G → k × be a map and let ˆ δ 1 : V → A , ˆ δ 2 : V → A and ˆ ς : V → V b e k -linear maps s uc h that ker ˆ δ 1 6 = ker ˆ δ 2 are subspaces of co dimension 1 of V . Here we ar e g oing to consider a more genera l situation that the one studied in the pr e vious subsection. Assume that for each i ∈ { 1 , 2 } there ex ist - an element x i ∈ V \ ker ( ˆ δ i ), - diﬀerent elements g i 1 , . . . , g in i of G , - p olynomials P ( i ) g i 1 , . . . , P ( i ) g in i ∈ S ( V ) \ { 0 } , such that ˆ δ i ( x i ) = n i X j =1 P ( i ) g ij w g ij . (The reas o n for the notation P ( i ) g ij instead of the more simple P ij will b ecame clear in items (5) and (6) of the following theo rem). Without los s of g enerality we ca n assume that x 1 ∈ ker ˆ δ 2 and x 2 ∈ ker ˆ δ 1 (and we do it). F or g ∈ G and i ∈ { 1 , 2 } , let λ ig , ω i , ν i ∈ k b e the elements deﬁned by the following co nditions: g x i − λ ig x i ∈ ker ˆ δ i , ˆ ς ( x i ) − ω i x i ∈ ker ˆ δ i and ˆ α ( x i ) − ν i x i ∈ ker ˆ δ i . Lemma 3 .16. The fol lowing facts hold: (1) Condition (3.4 ) of L emma 3.3 is s at isﬁe d if and only if g 1 j ˆ ς ( v ) = ˆ α ( v ) for al l j ≤ n 1 and v ∈ ker ˆ δ 1 . (2) Condition (3.5) of L emma 3.3 is s atisﬁe d if and only if g 2 j v = ˆ ς  ˆ α − 1 ( v )  for al l j ≤ n 2 and v ∈ ker ˆ δ 2 . Pr o of. Mimic the pro of of Lemma 3.7.  Lemma 3 .17. The fol lowing facts hold: (1) Items (1)(a) and (1)(b) of L emma 3.4 ar e satisﬁe d if and only if (a) ker ˆ δ 1 is a G -submo dule of V , (b) { g 1 j : 1 ≤ j ≤ n 1 } is a un ion of c onjugacy classes of G , (c) g P (1) g 1 j = λ 1 g χ − 1 α ( g ) χ ς ( g ) f − 1 ( g , g 1 j ) f ( g g 1 j g − 1 , g ) P (1) gg 1 j g − 1 for j ≤ n 1 . (2) Items (2)(a) and (2)(b) of L emma 3.4 ar e satisﬁe d if and only if (a) ker ˆ δ 2 is a G -submo dule of V , (b) { g 2 j : 1 ≤ j ≤ n 2 } is a un ion of c onjugacy classes of G , (c) g P (2) g 2 j = λ 2 g χ α ( g ) χ − 1 ς ( g ) f − 1 ( g , g 2 j ) f ( g g 2 j g − 1 , g ) P (2) gg 2 j g − 1 for j ≤ n 2 . Pr o of. Mimic the pro of of Lemma 3.8.  Theorem 3.18. Ther e is an H q -mo dule algebr a structu r e on ( A, s ) , satisfying σ · v = ˆ ς ( v ) , σ · w g = χ ς ( g ) w g , D i · v = ˆ δ i ( v ) and D i · w g = 0 for al l v ∈ V , g ∈ G and i ∈ { 1 , 2 } , if and only if 22 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI (1) ˆ ς is a bije ctive k [ G ] -line ar map and χ ς is a gr oup homomorphism, (2) ˆ ς ( v ) = g − 1 1 j ˆ α ( v ) for j ≤ n 1 and al l v ∈ ker ˆ δ 1 , and ˆ ς ( v ) = g 2 j ˆ α ( v ) for j ≤ n 2 and al l v ∈ ker ˆ δ 2 , (3) { g ij : 1 ≤ j ≤ n i } is a union of c onjugacy classes of G for i ∈ { 1 , 2 } , (4) ker ˆ δ 1 and ker ˆ δ 2 ar e G -submo dules of V , (5) g P (1) g 1 j = λ 1 g χ − 1 α ( g ) χ ς ( g ) f − 1 ( g , g 1 j ) f ( g g 1 j g − 1 , g ) P (1) gg 1 j g − 1 for j ≤ n 1 , (6) g P (2) g 2 j = λ 2 g χ α ( g ) χ − 1 ς ( g ) f − 1 ( g , g 2 j ) f ( g g 2 j g − 1 , g ) P (2) gg 2 j g − 1 for j ≤ n 2 , (7) ˆ α (ker ˆ δ i ) = ker ˆ δ i for i ∈ { 1 , 2 } , (8) P n 1 j =1 P (1) g 1 j w g 1 j ∈ ker δ 2 and P n 2 j =1 P (2) g 2 j w g 2 j ∈ ker δ 1 , wher e δ 1 and δ 2 ar e the maps deﬁne d by δ 1 ( v 1 m w g ) := m X j =1 α ( v 1 ,j − 1 ) ˆ δ 1 ( v j ) ς ( v j +1 ,m w g ) , δ 2 ( v 1 m w g ) := m X j =1 ς  α − 1 ( v 1 ,j − 1 )  ˆ δ 2 ( v j ) v j +1 ,m w g , in which v hl = v h · · · v l , (9) ς ( P ( i ) g ij ) = q − 1 ω i χ − 1 ς ( g ij ) P ( i ) g ij and α ( P ( i ) g ij ) = ν i χ − 1 α ( g ij ) P ( i ) g ij for i ∈ { 1 , 2 } and j ≤ n i , wher e ς is t he map given by ς ( v 1 m w g ) := ˆ ς ( v 1 ) · · · ˆ ς ( v m ) χ ς ( g ) w g . (10) If q 6 = 1 and q l = 1 , then δ l 1 = δ l 2 = 0 . Pr o of. Mimic the pro of of Theorem 3.6, but using Lemmas 3.16 and 3.1 7 instead of Lemmas 3.7 and 3.8, resp ectively .  R emark 3.19 . Since α and ς a re bijective k [ G ]-linear maps, fro m item (2) it follows that g 1 j v = g 1 h v for 1 ≤ j, h ≤ n 1 and all v ∈ ker ˆ δ 1 , (3.9) g 2 j v = g 2 h v for 1 ≤ j, h ≤ n 2 and all v ∈ ker ˆ δ 2 , (3.10) g − 1 1 j v = g 2 h v for 1 ≤ j ≤ n 1 , 1 ≤ h ≤ n 2 and all v ∈ ker ˆ δ 1 ∩ ker ˆ δ 2 . (3.11) On the other hand, arguing as in Remark 3.9 we can chec k that - g x i − λ 1 g x i ∈ ker ˆ δ 1 ∩ ker ˆ δ 2 for all g ∈ G , - λ ig ∈ k × for all g ∈ G , - the maps g 7→ λ ig are morphisms, - ω 1 , ω 2 , ν 1 , ν 2 ∈ k × . Finally , since ˆ ς ( x 1 ) = ˆ α ( g 2 j x 1 ) ≡ λ 1 g 2 j ˆ α ( x 1 ) (mo d k er ˆ δ 1 ) , we have ω 1 = λ 1 g 2 j ν 1 for j ≤ n 2 . Similarly , ν 2 = λ 2 g 1 j ω 2 for j ≤ n 1 . Consequently , λ 1 g 21 = · · · = λ 1 g 2 n 2 and λ 2 g 11 = · · · = λ 2 g 1 n 1 , which also follows from (3.9) and (3 .10). UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 23 Corollary 3.20 . Assume that the c onditions at the b e ginning of the pr esent sub- se ction ar e fulﬁl le d and that ther e ex ists an H q -mo dule algebr a structu r e on ( A, s ) , satisfying σ · v = ˆ ς ( v ) , σ · w g = χ ς ( g ) w g , D i · v = ˆ δ i ( v ) and D i · w g = 0 for al l v ∈ V , g ∈ G and i ∈ { 1 , 2 } . If P (1) g 1 j ∈ S (ker ˆ δ 1 ) and P (2) g 2 h ∈ S (ker ˆ δ 2 ) for al l j ≤ n 1 and h ≤ n 2 , then λ 1 g 1 j λ 1 g 2 h = q and λ 2 g 1 j λ 2 g 2 h = q − 1 . Mor e over g 1 j g 2 h has determinant 1 as an op e r ator on V . Pr o of. This result genera liz e s Coro llary 3.1 0 , and its pro of is similar .  Let G , V , f : G × G → k × , A , ˆ α : V → V , χ α : G → k × , α : A → A and s b e a s below of Remar k 3.11. Assume we have a) subspaces V 1 6 = V 2 of co dimension 1 of V such tha t V 1 and V 2 are ˆ α -stable G -submo dules of V , a nd vectors x 1 ∈ V 2 \ V 1 and x 2 ∈ V 1 \ V 2 , b) diﬀerent elements g i 1 , . . . , g in i of G , where i ∈ { 1 , 2 } , such that: – { g 11 , . . . , g 1 n 1 } and { g 21 , . . . g 2 n 2 } are unions o f conjugacy class es of G , – g 1 j v = g 1 h v for 1 ≤ j, h ≤ n 1 and all v ∈ V 1 , – g 2 j v = g 2 h v for 1 ≤ j, h ≤ n 2 and all v ∈ V 2 , – g − 1 1 j v = g 2 h v for 1 ≤ j ≤ n 1 , 1 ≤ h ≤ n 2 and all v ∈ V 1 ∩ V 2 . c) a morphism χ ς : G → k × d) Nonzero p olynomials P (1) g 1 j ∈ S ( V 1 ) a nd P (2) g 2 h ∈ S ( V 2 ), where 1 ≤ j ≤ n 1 and 1 ≤ h ≤ n 2 . Let ˆ ς : V → V and ˆ δ 1 , ˆ δ 2 : V → A b e the maps deﬁned by ˆ ς ( v ) := ( ˆ α ( g − 1 11 v ) if v ∈ V 1 , ˆ α ( g 21 v ) if v ∈ V 2 , ker ˆ δ i := V i and ˆ δ i ( x i ) := n i X j =1 P ( i ) g ij w g ij . F or g ∈ G and i ∈ { 1 , 2 } , let λ ig , ν i ∈ k × be the elements deﬁned by the following conditions: g x i − λ ig x i ∈ V i and α ( x i ) − ν i x i ∈ V i . Note that, by item b), λ 2 g 11 = · · · = λ 2 g 1 n 1 and λ 1 g 21 = · · · = λ 1 g 2 n 2 . Corollary 3. 21. Ther e is an H q -mo dule algebr a structu r e on ( A, s ) , satisfying σ · v = ˆ ς ( v ) , σ · w g = χ ς ( g ) w g , D h · v = ˆ δ h ( v ) and D h · w g = 0 , for al l v ∈ V , g ∈ G and i ∈ { 1 , 2 } , if and only if for al l j ≤ n 1 and h ≤ n 2 the fol lowing facts hold: (1) q = λ 1 g 1 j λ 1 g 21 and q − 1 = λ 2 g 11 λ 2 g 2 h , (2) g P (1) g 1 j = λ 1 g χ − 1 α ( g ) χ ς ( g ) f − 1 ( g , g 1 j ) f ( g g 1 j g − 1 , g ) P (1) gg 1 j g − 1 , (3) g P (2) g 2 h = λ 2 g χ α ( g ) χ − 1 ς ( g ) f − 1 ( g , g 2 h ) f ( g g 2 h g − 1 , g ) P (2) gg 2 h g − 1 , (4) α ( P (1) g 1 j ) = ν 1 χ − 1 α ( g 1 j ) P (1) g 1 j and α ( P (2) g 2 h ) = ν 2 χ − 1 α ( g 2 h ) P (2) g 2 h , (5) P n 1 j =1 P (1) g 1 j w g 1 j ∈ ker δ 2 and P n 2 h =1 P (2) g 2 h w g 2 h ∈ ker δ 1 , wher e δ 1 , δ 2 : A → A ar e the maps deﬁne d in item (8) of The or em 3.18 (6) If q 6 = 1 and q l = 1 , then δ l 1 = δ l 2 = 0 . Pr o of. It is s imila r to the pro of o f Corollar y 3.1 2, using Theorem 3.18 instead of Theor em 3 .6. The pro of that ς is G -linear r equires a dditionally the fact that gg ij g − 1 v = g ij v for 1 ≤ i ≤ 2 and 1 ≤ j ≤ n i , which is true by b).  24 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI R emark 3.22 . Assume that the hyp o thesis of Corollar y 3.21 ar e fulﬁlled. Then, as it was no te ab ov e this corolla ry , λ 2 g 11 = · · · = λ 2 g 1 n 1 and λ 1 g 21 = · · · = λ 1 g 2 n 2 . Moreov er, by item (1) it is clea r that λ 1 g 11 = · · · = λ 1 g 1 n 1 and λ 2 g 21 = · · · = λ 2 g 2 n 2 . Prop osition 3.23. L et G , V , f , A , α , V 1 , V 2 , g 11 , . . . , g 1 n 1 , g 21 , . . . , g 2 n 2 , ˆ ς , χ ς , ˆ δ 1 , ˆ δ 2 , x 1 , x 2 , ν 1 , ν 2 , λ 1 g and λ 2 g , wher e g ∈ G , b e as in t he discussion ab ove Cor o l lary 3.21. Assume that λ 2 g 11 = · · · = λ 2 g 1 n 1 , λ 1 g 21 = · · · = λ 1 g 2 n 2 , λ 1 g 11 = · · · = λ 1 g 1 n 1 , λ 2 g 21 = · · · = λ 2 g 2 n 2 , and that c onditions a), b), c) and d) ab ove that c or ol lary ar e fulﬁl le d. If λ 1 g 11 λ 1 g 21 = q , λ 2 g 11 λ 2 g 21 = q − 1 and g ih x j = λ j g ih x j , for 1 ≤ i , j ≤ 2 and 1 ≤ h ≤ n i , then: (1) δ l 1 = δ l 2 = 0 , whenever q 6 = 1 and q l = 1 . (2) If q = 1 or q is not a r o ot of unity, t hen P (1) g 1 j ∈ ker δ 2 and P (2) g 2 h ∈ ker δ 1 if and only if P (1) g 1 j , P (2) g 2 h ∈ S ( V 1 ∩ V 2 ) . (3) If q 6 = 1 is a primitive l -r o ot of unity, then P (1) g 1 j ∈ ker δ 2 and P (2) g 2 h ∈ ker δ 1 if and only if P (1) g 1 j ∈ S  k x l 2 ⊕ ( V 1 ∩ V 2 )  and P (2) g 2 h ∈ S  k x l 1 ⊕ ( V 1 ∩ V 2 )  . Pr o of. Let x r = x r 1 1 · · · x r n n . Using the hypo thesis it is easy to chec k by induction on s that δ s 1 ( x r w g ) =      X h ∈ I s n 1 c h c ′ h α s ( x r 1 − s 1 x r 2 2 · · · x r n n ) w g 1 h s g 1 h s − 1 ··· g 1 h 1 g for s ≤ r 1 , 0 otherwise, and δ s 2 ( x r w g ) =      X h ∈ I s n 2 d h d ′ h x r 2 − s 2 g s 21  x r 1 1 x r 3 3 · · · x r n n  w g 2 h s g 2 h s − 1 ··· g 2 h 1 g for s ≤ r 2 , 0 otherwise, where I s n i = I n i × · · · × I n i | {z } s times , with I n i = { 1 , . . . , n i } , α s denotes the s -fold co mp osition o f α , c h = χ s ς ( g ) s − 1 Y k =1 χ s − k ς ( g 1 h k ) s Y k =2 χ k − 1 α ( g 1 h k ) , c ′ h = s − 1 Y k =0 ( r 1 − k ) q ! s Y k =1 f ( g 1 h k , g 1 h k − 1 · · · g 1 h 1 g ) ! s Y k =1 α s − 1 ( P (1) g 1 h k ) ! , d h = λ sr 2 − s ( s +1) / 2 2 g 21 , d ′ h = s − 1 Y k =0 ( r 2 − k ) q ! s Y k =1 f ( g 2 h k , g 2 h k − 1 · · · g 2 h 1 g ) ! s − 1 Y k =0 g k 21 P (2) g 2 h s − k ! . The result follows ea sily from these formulas.  UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 25 Example 3.24. Let D u be the Dihedral group D u := h s, t | s 2 , t u , stst i . Then D u acts on k [ X 1 , X 2 ] via s X 1 = − X 1 , s X 2 = − X 2 , t X 1 = X 1 and t X 2 = X 2 . Let A = k [ X 1 , X 2 ]# D u . W e hav e - Assume u is even. Then, there is a n H 1 -mo dule a lgebra structur e on A , such that σ · X 1 = X 1 , σ · X 2 = X 2 , σ · w t i = w t i , σ · w t i s = − w t i s , D 1 · X 1 = w t + w t − 1 , D 1 · X 2 = 0 , D 1 · w t i = 0 , D 1 · w t i s = 0 , D 2 · X 1 = 0 , D 2 · X 2 = w t u/ 2 , D 2 · w t i = 0 , D 2 · w t i s = 0 . - There is an H − 1 -mo dule algebra structure on A , s uc h that σ · X 1 = X 1 , σ · X 2 = − X 2 , σ · w t i = w t i , σ · w t i s = − w t i s , D 1 · X 1 = u − 1 X i =0 w t i s , D 1 · X 2 = 0 , D 1 · w t i = 0 , D 1 · w t i s = 0 , D 2 · X 1 = 0 , D 2 · X 2 = w t + w t − 1 , D 2 · w t i = 0 , D 2 · w t i s = 0 . - Assume u is even. Let α : A → A be the k -a lg ebra map deﬁned by α ( Qw t i ) := Qw t i and α ( Qw t i s ) := − Q w t i s , and le t s : H 1 ⊗ A → A ⊗ H 1 be the transp osition asso ciated with α . There is an H 1 -mo dule algebra structure on A , s uc h that σ · X 1 = X 1 , σ · X 2 = X 2 , σ · w t i = w t i , σ · w t i s = w t i s , D 1 · X 1 = w t + w t − 1 , D 1 · X 2 = 0 , D 1 · w t i = 0 , D 1 · w t i s = 0 , D 2 · X 1 = 0 , D 2 · X 2 = w t u/ 2 , D 2 · w t i = 0 , D 2 · w t i s = 0 . 4 Non triviality of the deformations Let A = S ( V )# f G be as in Section 3. By Theo rem 1 .1 6 we know that each H q -mo dule algebra ( A, s ), with s a go o d transp osition, pro duces to a for mal defor- mation A F of A , which is constructed using the UDF F = exp q ( tD 1 ⊗ D 2 ). The aim of this sec tio n is to prove that if ( A, s ) sa tisﬁes the conditions re quired in Co rol- lary 3.21 a nd P (1) g 1 j , P (2) g 2 h ∈ S ( V 1 ∩ V 2 ) for 1 ≤ j ≤ n 1 and 1 ≤ h ≤ n 2 , then A F is non trivial. W e will prov e this showing that its inﬁnitesimal Φ( a ⊗ b ) = δ 1  α − 1 ( a )  δ 2 ( b ) , is no t a co bounda ry . F or this we use a co mplex X ∗ ( A ), giving the Ho c hschild cohomolog y of A , which is simpler than the canonica l one. 4.1 A simple resolution Given a s y mmetric k -a lgebra S := S ( V ), we consider the diﬀerential gra ded algebr a ( Y ∗ , ∂ ∗ ) genera ted by element s y v and z v , of ze r o degree, and v , of degree o ne, wher e v ∈ V , s ub ject to the relations z λv + w = λz v + z w , y λv + w = λy v + y w , v + w = λv + w, y v y w = y w y v , y v z w = z w y v , z v z w = z w z v , v y w = y w v , v z w = z w v , v 2 = 0 , 26 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI where λ ∈ k and v , w ∈ V , and with diﬀere ntial ∂ deﬁned by ∂ ( v ) := ρ v , where ρ v = z v − y v . Note that S is a subalgebra of Y ∗ via the embedding that takes v to y v for all v ∈ V . This pro duces a n structur e of left S -mo dule on Y ∗ . Similarly we co nsider Y ∗ as a right S -mo dule via the embedding of S in Y ∗ that takes v to z v for all v ∈ V . Prop osition 4.1. L et e µ : Y 0 → S b e t he algebr a map deﬁne d by e µ ( y v ) = e µ ( z v ) := v for al l v ∈ V . The S -bimo du le c omplex S Y 0 e µ o o Y 1 ∂ 1 o o Y 2 ∂ 2 o o Y 3 ∂ 3 o o Y 4 ∂ 4 o o Y 5 ∂ 5 o o . . . ∂ 6 o o (4.12) is c ontr a ctible as a left S - mo d ule c omple x. Pr o of. Let { x 1 , . . . , x n } b e a basis of V . W e will write y i , z i , ρ i and v i instead of y x i , z x i , ρ x i and v x i , resp ectively . A c o n tracting homotopy ς 0 : S → Y 0 and ς r +1 : Y r → Y r +1 ( r ≥ 0), of (4.12) is given b y ς (1) := 1 , ς  ρ m 1 i 1 v δ 1 i 1 · · · ρ m l i l v δ l i l  := ( ( − 1) s ρ m 1 i 1 v δ 1 i 1 · · · ρ m l − 1 i l − 1 v δ l − 1 i l − 1 ρ m l − 1 i l v i l if δ l = 0, 0 if δ l = 1, where we assume that i 1 < · · · < i l , δ 1 + · · · + δ l = s a nd m l + δ l > 0. In fact, a direct computation s ho ws that - e µ   σ − 1 (1) = e µ (1 ) = 1, - ς   e µ (1) = ς (1) = 1 and ∂   ς (1) = ∂ (0) = 0, - If x = x ′ ρ m l i l , where m l > 0 and x ′ = ρ m 1 i 1 · · · ρ m l − 1 i l − 1 with i 1 < · · · < i l , then ς   e µ ( x ) = ς (0) = 0 a nd ∂   ς ( x ) = ∂ ( x ′ ρ m l − 1 i l v i l ) = x , - Let x = x ′ ρ m l i l v δ l i l , where m l + δ l > 0 and x ′ = ρ m 1 i 1 v δ 1 i 1 · · · ρ m l − 1 i l − 1 v δ l − 1 i l − 1 with i 1 < · · · < i l and δ 1 + · · · + δ l = s > 0. If δ l = 0, then ς   ∂ ( x ) = ς  ∂ ( x ′ ) ρ m l i l  = ( − 1) s − 1 ∂ ( x ′ ) ρ m l − 1 i l v i l , ∂   ς ( x ) = ∂  ( − 1) s x ′ ρ m l − 1 i l v i l  = ( − 1) s ∂ ( x ′ ) ρ m l − 1 i l v i l + x , and if δ l = 1, then ς   ∂ ( x ) = ς  ∂ ( x ′ ) ρ m l i l v i l + ( − 1 ) s − 1 x ′ ρ m l +1 i l  = x , ∂   ς ( x ) = ∂ (0) = 0 . The result follows immediately from all these facts.  Let G b e a gro up acting on V . W e cons ider S a s a k [ G ]-mo dule algebra via the action induced by the o ne of G on V . Let f : k [ G ] × k [ G ] → k × be a nor mal co cycle and let A = S # f k [ G ] b e the asso cia ted cr ossed pr oduct. In the seque l we will use the following Notation 4.2. W e let k [ G ] denote k [ G ] /k . Mor eov er: - Given g 1 , . . . , g s ∈ k [ G ] and 1 ≤ i < j ≤ s , we s e t g ij := g i ⊗ · · · ⊗ g j . - Given v 1 , . . . , v r ∈ V and 1 ≤ i < j ≤ r , we set v ij := v i · · · v j . UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 27 F or all r, s ≥ 0, let Z s = ( A ⊗ k [ G ] ⊗ s ) ⊗ S A and X r s = ( A ⊗ k [ G ] ⊗ s ) ⊗ S Y r ⊗ S A, where we consider A ⊗ k [ G ] ⊗ s as a right S -mo dule via ( a 0 w g 0 ⊗ g 1 s ) · a = a 0 g 0 ··· g s aw g 0 ⊗ g 1 s . The X r s ’s and the Z s ’s are A -bimo dules in a ca no nical wa y . Note that Z s ≃ A ⊗ k [ G ] ⊗ s ⊗ k [ G ] and X r s ≃ A ⊗ k [ G ] ⊗ s ⊗ Λ r V ⊗ A. In particular, X r s is a fr e e A -bimo dule. Co nsider the diag ram of A -bimo dules and A -bimo dule maps . . . − δ 2   Z 2 − δ 2   X 02 µ 2 o o X 12 d 0 12 o o . . . d 0 22 o o Z 1 − δ 1   X 01 µ 1 o o X 11 d 0 11 o o . . . d 0 21 o o Z 0 X 00 µ 0 o o X 10 d 0 10 o o . . . , d 0 20 o o where - each δ s is deﬁned by δ (1 ⊗ g 1 s ⊗ S 1) := w g 1 ⊗ g 2 s ⊗ S 1 + s − 1 X i +1 ( − 1) i f ( g i , g i +1 ) ⊗ g 1 ,i − 1 ⊗ g i g i +1 ⊗ g i +2 ,s ⊗ S 1 + ( − 1 ) s 1 ⊗ g 1 ,s − 1 ⊗ S w g s , - for each s ≥ 0, the complex ( X ∗ s , d ∗ s ) is ( − 1 ) s times ( Y ∗ , ∂ ∗ ), tensor ed ov er S , o n the right with A and on the left with A ⊗ k [ G ] ⊗ s , - for each s ≥ 0, the map µ s is deﬁned by µ (1 ⊗ g 1 s ⊗ 1 ) := 1 ⊗ g 1 s ⊗ S 1 . Each row in this diag ram is contractible as a left A -mo dule. A contracting homo - topy ς 0 0 s : Z s → X 0 s and ς 0 r +1 ,s : X r s → X r +1 ,s ( r ≥ 0 ), is given by ς 0 (1 ⊗ g 1 s ⊗ S 1) := 1 ⊗ g 1 s ⊗ 1 , ς 0 (1 ⊗ g 1 s ⊗ S P ⊗ S 1) := ( − 1) s 1 ⊗ g 1 s ⊗ S ς ( P ) ⊗ S 1 . F or r ≥ 0 and 1 ≤ l ≤ s , we deﬁne A -bimo dule maps d l r s : X r s → X r + l − 1 ,s − l , recursively on l a nd r , b y: d l ( x ) :=          ς 0   δ   µ ( x ) if l = 1 and r = 0, − ς 0   d 1   d 0 ( x ) if l = 1 and r > 0, − P l − 1 j =1 ς 0   d l − j   d j ( x ) if 1 < l a nd r = 0, − P l − 1 j =0 ς 0   d l − j   d j ( x ) if 1 < l a nd r > 0, for x = 1 ⊗ g 1 s ⊗ v 1 r ⊗ 1 . 28 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI Theorem 4.3. Ther e is a r esolution of A as an A - bimo dule A X 0 − µ o o X 1 d 1 o o X 2 d 2 o o X 3 d 3 o o X 4 d 4 o o . . . , d 5 o o wher e µ : X 00 → A is the mu ltiplic ation m ap, X n = M r + s = n X r s and d n = n X l =1 d l 0 n + n X r =1 n − r X l =0 d l r,n − r . Pr o of. See [G- G2, Appendix A].  Prop osition 4.4. The maps d l vanish for al l l ≥ 2 . Mor e over d 1 (1 ⊗ g 1 s ⊗ v 1 r ⊗ 1 ) = w g 1 ⊗ g 2 s ⊗ v 1 r ⊗ 1 + s − 1 X i =1 ( − 1) i f ( g i , g i +1 ) ⊗ g 1 ,i − 1 ⊗ g i g i +1 ⊗ g i +2 ,s ⊗ v 1 r ⊗ 1 + ( − 1 ) s 1 ⊗ g 1 ,s − 1 ⊗ g s v 1 · · · g s v r ⊗ w g s . In p artic ular, ( X ∗ , d ∗ ) is the total c omplex of the double c omplex . . . d 1 03   . . . d 1 13   . . . d 1 23   X 02 d 1 02   X 12 d 0 12 o o d 1 12   X 22 d 0 22 o o d 1 22   . . . d 0 32 o o X 01 d 1 01   X 11 d 0 11 o o d 1 11   X 21 d 0 21 o o d 1 21   . . . d 0 31 o o X 00 X 10 d 0 10 o o X 20 d 0 20 o o . . . , d 0 30 o o Pr o of. The co mputation of d 1 r s can b e o btained easily b y induction on r , using that d 1 ( x ) = ς 0   δ   µ ( x ) for x = 1 ⊗ g 1 s ⊗ 1 , and d 1 ( x ) = − ς 0   d 1   d 0 ( x ) for r ≥ 1 and x = 1 ⊗ g 1 s ⊗ v 1 r ⊗ 1 . The ass ertion for d l r s , with l ≥ 2, follows b y induction on l and r , using the re c ursive deﬁnition of d l r s .  4.2 A comparison ma p Let A = A/k . In this subsection we introduce a nd study a co mparison map from ( X ∗ , d ∗ ) to the canonica l norma lized Ho ch schild reso lution ( A ⊗ A ∗ ⊗ A, b ′ ∗ ). It is well known that there is an A -bimo dule ho motopy equiv alence θ ∗ : ( X ∗ , d ∗ ) → ( A ⊗ A ∗ ⊗ A, b ′ ∗ ) such that θ 0 = id A ⊗ A . It can b e re cursively deﬁned by θ 0 := id A ⊗ A and θ ( x ) := ( − 1 ) r + s θ   d ( x ) ⊗ 1 for x = 1 ⊗ g 1 s ⊗ v 1 r ⊗ 1 with r + s ≥ 1. Next we give a closed fo r m ula for θ ∗ . In o rder to es tablish this result we need to int ro duce a new notatio n. W e recur sively deﬁne ( w g 1 ⊗ · · · ⊗ w g s ) ∗ ( P 1 ⊗ · · · ⊗ P r ) by - ( w g 1 ⊗ · · · ⊗ w g s ) ∗ ( Q 1 ⊗ · · · ⊗ Q r ) := ( Q 1 ⊗ · · · ⊗ Q r ) if s = 0 , UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 29 - ( w g 1 ⊗ · · · ⊗ w g s ) ∗ ( Q 1 ⊗ · · · ⊗ Q r ) := ( w g 1 ⊗ · · · ⊗ w g s ) if r = 0 , - If r, s ≥ 1, then ( w g 1 ⊗ · · · ⊗ w g s ) ∗ ( Q 1 ⊗ · · · ⊗ Q r ) equals r X i =0 ( − 1) i ( w g 1 ⊗ · · · ⊗ w g s − 1 ) ∗ ( g s Q 1 ⊗ · · · ⊗ g s Q i ) ⊗ w g s ⊗ Q i +1 ⊗ · · · ⊗ Q r . Prop osition 4.5. We have: θ  1 ⊗ g 1 s ⊗ v 1 r ⊗ 1  = ( − 1) r X τ ∈ S r sg( τ ) ⊗  w g 1 ⊗ · · · ⊗ w g s  ∗ v τ (1 r ) ⊗ 1 , wher e S r is the symmetric gr oup in r elements and v τ (1 r ) = v τ (1) ⊗ · · · ⊗ v τ ( r ) . Pr o of. W e pro ceed b y induction on n = r + s . The case n = 0 is obvious. Suppo se that r + s = n and the result is v alid for θ n − 1 . By the recur sive deﬁnition of θ and Theorem 4.3, θ (1 ⊗ g 1 s ⊗ v 1 r ⊗ 1 ) = ( − 1) n θ   d (1 ⊗ g 1 s ⊗ v 1 r ⊗ 1 ) ⊗ 1 = ( − 1) n θ   ( d 0 + d 1 )(1 ⊗ g 1 s ⊗ v 1 r ⊗ 1 ) ⊗ 1 = r X i =1 ( − 1) i + r θ ( g 1 ··· g s v i ⊗ g 1 s ⊗ v 1 ,i − 1 v i +1 ,r ⊗ 1 ) ⊗ 1 − r X i =1 ( − 1) i + r θ (1 ⊗ g 1 s ⊗ v 1 ,i − 1 v i +1 ,r ⊗ v i ) ⊗ 1 + ( − 1 ) n θ ( w g 1 ⊗ g 2 s ⊗ v 1 r ⊗ 1 ) ⊗ 1 + s − 1 X i =1 ( − 1) n + i θ (1 ⊗ g 1 ,i − 1 ⊗ g i g i +1 ⊗ g i +1 ,s ⊗ v 1 r ⊗ 1 ) ⊗ 1 + ( − 1 ) r θ (1 ⊗ g 1 ,s − 1 ⊗ g 1 ,s − 1 ⊗ g s v 1 · · · g s v r ⊗ w g s ) ⊗ 1 . The desired r esult follows now from the inductive hypothesis.  4.3 The Ho chs c hild cohomolog y Let M b e an A -bimo dule and A e the env eloping a lg ebra o f A . Applying the functor Hom A e ( − , M ) to ( X ∗∗ , d 0 ∗∗ , d 1 ∗∗ ) and using the identiﬁcations Hom A e ( X r s , M ) ≃ Hom k ( k [ G ] ⊗ s ⊗ Λ r V , M ) we obtain the double complex . . . . . . . . . X 02 d 03 1 O O d 12 0 / / X 12 d 13 1 O O d 22 0 / / X 22 d 32 0 / / d 23 1 O O . . . X 01 d 02 1 O O d 11 0 / / X 11 d 12 1 O O d 21 0 / / X 21 d 31 0 / / d 22 1 O O . . . X 00 d 01 1 O O d 10 0 / / X 10 d 11 1 O O d 20 0 / / X 20 d 21 1 O O d 30 0 / / . . . , where X r s = Hom k ( k [ G ] ⊗ s ⊗ Λ r V , M ) , 30 JORG E A. GUCCIONE, JUAN J. GUCCIONE, AND CHRISTIAN V ALQUI d 0 ( ϕ )( g 1 s ⊗ v 1 ,r +1 ) = r +1 X i =1 ( − 1) s + i +1 ϕ ( g 1 s ⊗ v 1 ,i − 1 v i +1 ,r +1 ) v i + r +1 X i =1 ( − 1) s + i g 1 ··· g s v i ϕ ( g 1 s ⊗ v 1 ,i − 1 v i +1 ,r +1 ) , d 1 ( ϕ )( g 1 ,s +1 ⊗ v 1 r ) = w g 1 ϕ ( g 2 ,s +1 ⊗ v 1 r ) + s X i =1 ( − 1) i f ( g i , g i +1 ) ϕ ( g 1 ,i − 1 ⊗ g i g i +1 ⊗ g i +1 ,s +1 ⊗ v 1 r ) + ( − 1 ) s +1 ϕ ( g 1 s ⊗ g s +1 v 1 · · · g s +1 v r ) w g s +1 , whose total complex X ∗ ( M ) gives the Hochschild cohomo lo gy H ∗ ( A, M ) of A with co eﬃcien ts in M . The comparis on map θ ∗ induces a quas i-isomorphism θ ∗ :  Hom k ( A ∗ , M ) , b ∗  → X ∗ ( M ) . It is immediate that θ ( ϕ )( g 1 s ⊗ v 1 r ) = ( − 1 ) r X τ ∈ S r sg( τ ) ϕ  ( w g 1 ⊗ · · · ⊗ w g s ) ∗ v τ (1 r )  . where S r is the sy mmetric g roup in r elements and v τ (1 r ) = v τ (1) ⊗ · · · ⊗ v τ ( r ) . F rom now on we ta k e M = A a nd we write HH ∗ ( A ) instead of H ∗ ( A, A ). 4.4 Pro of of the main result W e are ready to pr ov e that the co cycle Φ is non trivial. F or this it is suﬃcient to show tha t θ (Φ) is not a cob oundary . Let x 1 , . . . , x n , P (1) g 11 , . . . , P (1) g 1 n 1 , P (2) g 21 , . . . , P (2) g 2 n 2 , g 11 , . . . g 1 n 1 and g 21 , . . . g 2 n 2 be as in Co rollary 3.21. A dir e c t computation, using the formulas for δ 1 and δ 2 obtained in the pro of of Pro p ositio n 3.23, shows that θ (Φ)( g ⊗ v ) = 0 and θ (Φ)( g ⊗ h ) = 0 . for g , h ∈ G and v ∈ V , a nd that θ (Φ)( x 1 x 2 ) = n 1 X j =1 n 2 X h =1 χ − 1 α ( g 1 j ) f ( g 1 j , g 2 h ) α − 1 ( P (1) g 1 j ) g 1 j P (2) g 2 h w g 1 j g 2 h and θ (Φ)( x i x j ) = 0 for 1 ≤ i < j ≤ n with ( i, j ) 6 = (1 , 2 ). W e next prove that θ (Φ) is not a cob oundary . Let ϕ 0 ∈ X 01 and ϕ 1 ∈ X 10 . By deﬁnition d 1 ( ϕ 0 )( g ⊗ h ) = w g ϕ 0 ( h ) − f ( g , h ) ϕ 0 ( g h ) + ϕ 0 ( g ) w h , d 0 ( ϕ 0 )( g ⊗ v ) = g v ϕ 0 ( g ) − ϕ 0 ( g ) v , d 1 ( ϕ 1 )( g ⊗ v ) = w g ϕ 1 ( v ) − ϕ 1 ( g v ) w g , d 0 ( ϕ 1 )( v 1 v 2 ) = ϕ 1 ( v 2 ) v 1 − v 1 ϕ 1 ( v 2 ) + v 2 ϕ 1 ( v 1 ) − ϕ 1 ( v 1 ) v 2 , and so θ (Φ) is a cob oundary if and only if there exist ϕ 0 and ϕ 1 such that w g ϕ 0 ( h ) − f ( g , h ) ϕ 0 ( g h ) + ϕ 0 ( g ) w h = 0 for all g , h ∈ G , g v ϕ 0 ( g ) − ϕ 0 ( g ) v + w g ϕ 1 ( v ) − ϕ 1 ( g v ) w g = 0 for all g ∈ G and v ∈ V , [ ϕ 1 ( x j ) , x i ] + [ x j , ϕ 1 ( x i )] = 0 for all i < j with ( i, j ) 6 = (1 , 2), UNIVERSAL DEFORMA TION FORMULAS AND BRAIDED MODULE ALGEBRAS 31 where, as usual, [ a, b ] = ab − b a , a nd [ ϕ 1 ( x 2 ) , x 1 ] + [ x 2 , ϕ 1 ( x 1 )] = n 1 X j =1 n 2 X h =1 χ − 1 α ( g 1 j ) f ( g 1 j , g 2 h ) α − 1 ( P (1) g 1 j ) g 1 j P (2) g 2 h w g 1 j g 2 h . But, since w g x j = g x j w g , w g 1 j g 2 h x 1 = f ( g 1 j , g 2 h ) − 1 w g 1 j w g 2 h x 1 = q x 1 and w g 1 j g 2 h x 2 = q − 1 x 2 , if ϕ 1 ( x 1 ) = X g ∈ G Q (1) g w g and ϕ 1 ( x 2 ) = X g ∈ G Q (2) g w g , then necessar ily X g ∈ Υ ( q − 1)  x 1 Q (2) g + q − 1 x 2 Q (1) g  w g = n 1 X j =1 n 2 X h =1 D j h α − 1 ( P (1) g 1 j ) g 1 j P (2) g 2 h w g 1 j g 2 h , where D j h = χ − 1 α ( g 1 j ) f ( g 1 j , g 2 h ) and Υ = { g 1 j g 2 h : 1 ≤ j ≤ n 1 and 1 ≤ h ≤ n 2 } , which is imp ossible b ecause α − 1 ( P (1) g 1 j ) g 1 j P (2) g 2 h ∈ k [ x 3 , . . . , x n ] \ { 0 } . References [A-S] N. Andruskiewitsch and H. J. Schne ider, Hopf algebr as of or der p 2 and br aide d Hopf algebr as of or der p , Journal of Algebra, vol 199 (1998) 430–454. [B-K-L- T ] Y. Bespalo v, T. K erler, V . Lyubashenk o and V. T urae v, Inte gr a ls for br aide d Hopf algebr as, Journal of Pure and Applied Algebra, vol 148 (2000) 113–164. [D] Y . Doi, Hopf mo dules in Y et ter Drinfeld c ate gories, Communications in Algebra, v ol 26 (1998) 3057–3070. [F-M-S] D. Fishman, S. Montgome ry and H. J. Sc hneider, F r o b enius e xtensions of sub algebr as of Hopf algebr as, T ransact ions of the Am erican M athemat ical So ciet y , vol 349 (1997) 4857–489 5. [G-G1] Jorge A. Guccione and Juan J. Guccione, The ory of br aide d H opf Cr osse d Pr o ducts, Journal of Algebra, vol 261 (2003) 54–101. [G-G2] Jorge A. Guccione and Juan J. Guccione, Ho chschild (c o)homo lo gy of Hopf cr osse d pr o ducts, K-theory , V ol 25, (2002) 138–169. [G-Z] Giaquinto and James J. Zang, Bialgebr a actions, twists, and universal deformation formulas, Journal of Pure and Appli ed A l gebra, vol 128 (1998) 133–151. [L1] V. Lyubashenk o Mo dular tr anformations for tensor c ate gories, Journal of Pure and Applied Algebra, vol 98 (1995) 279–327. [So] Y. Sommerh¨ auser, Inte gr als for br aide d Hopf algebra s, preprint. [T1] M. T akeuc hi, Survey of br aide d Hopf algebr as, Contemporary Mathematics, vol 267 (2000) 301–323. [T2] M. T ake uchi, Finite Hopf algebr as in br aide d tensor ca te gories, Journal of Pure and Applied Algebra, vol 138 (1999) 59–82. [W] S. Withersp onn, Skew derivations and deformations of a family of gr oup cr osse d pr o d- ucts, Communications in algebra, vol 34 (2006) 4187–4206 . Dep ar t amento de Ma tem ´ atica, F acul t ad de Ciencias Exact as y Na turales, P abell ´ on 1, Ciudad Universit aria, (1428) Buenos Aires, Argentina. E-mail addr ess : vander@dm .uba.ar Dep ar t amento de Ma tem ´ atica, F acul t ad de Ciencias Exact as y Na turales, P abell ´ on 1, Ciudad Universit aria, (1428) Buenos Aires, Argentina. E-mail addr ess : jjgucci@d m.uba.ar Pontificia Univ ersidad Ca t ´ olica del Per ´ u - Instituto de Ma tem ´ atica y Ciencias Afi- nes, Secci ´ on Mat em ´ aticas, PUCP, A v. Universit aria 18 01, Sa n Migu el, Lima 32, Per ´ u. E-mail addr ess : cvalqui@p ucp.edu.pe

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