Cohomology of finite dimensional pointed Hopf algebras

COHOMOLOGY OF FINITE DIMENSIONAL POINTE D HOPF ALGEBRAS M. MASTNAK, J. PEVTSOV A, P . SCHA UENBUR G, AND S. WITHERSPOON Abstract. W e prove finite generation of the cohomolog y r ing o f any finite di- mensional po int ed Hopf algebra, having ab elian group o f gro uplike elemen ts, under some mild restrictions on the gr oup or der. The pr o of uses the r ecent classification by Andruskiewitsch and Schneider of suc h Hopf algebras. Exam- ples include all of Lusztig’s small quan tum g roups, whos e coho mology was first computed explicitly b y Ginzburg and Kumar, as well as many new p ointed Hopf algebras . W e also show that in g eneral the cohomo logy ring of a Hopf algebra in a braided ca tegory is br aided co mmutative. As a consequence we obtain some further information about the str ucture o f the cohomology r ing o f a finite di- mensional po inted Hopf algebra and its r elated Nichols alg ebra. 1. Intro duction Golo d [18 ], V enk o v [26], and Ev ens [14] pro v ed that the cohomo lo gy ring of a fi- nite g r oup with co efficien ts in a field of p o sitive c haracteristic is finitely generated. This fundamen tal result op ened the do o r to using geometric metho ds in the study of cohomolog y and mo dular represen tations of finite groups. This g eometric ap- proac h was pioneered by Quillen [25] and expanded b y Carlson [11], and Avrunin and Scott [5]. F riedlander and Suslin [16] v astly g eneralized the result of V enk o v and Ev ens, proving that the cohomolo g y ring of any finite group sc heme (equiv- alen tly , finite dim ensional co commutativ e Hopf algebra) o v er a field of p ositiv e c haracteristic is finitely generated. In a differen t direction, Ginzburg and Kumar [17] computed the cohomology ring of each of Lusztig’s small quan tum g r o ups u q ( g ), defined o v er C , under some restrictions on the parameters; it is the c o ordi- nate r ing of the nilp otent cone of the Lie algebra g , and th us is finitely generated. The result is parallel to an earlier result o f F riedlander a nd Parshall [1 5] who com- puted the cohomolog y ring of a restricted Lie algebra in p ositiv e c haracteristic. Recen tly , Bendel, Nak ano, Parshall, and Pillen [7] calculated the cohomolo g y of a small quantum group under significan tly loo sened restrictions on the parameters. The b roa d common feature of these works is that they in ve stigate cohomology of certain nonsemisimple finite dimensional Hopf a lg ebras. Hence, thes e results lead one to ask whether the c ohomolo gy ring of an y finite dimensional Hopf algebra is Date : J uly 29 , 20 09. 1 2 M. MASTNAK, J. PEVTSOV A, P . SCHAUENBURG, A ND S. WI THERSPOON finitely generated. A p ositiv e a nsw er w ould sim ultaneously generalize the kno wn results for co commutativ e Hopf algebras and f or small quantum gro ups. More generally , Etingof and Ostrik [13] conjectured finite g eneration o f cohomology in the context of finite tensor catego ries. In this pap er, w e b egin the task of pro ving this conjecture for more general classes of nonco comm utativ e Hopf algebras ov er a field of c haracteristic 0. W e pro v e finite generation of the cohomology ring o f an y finite dimensional p ointed Hopf algebra, with ab elian group of grouplik e elemen ts, unde r some mild re- strictions on the group o rder. P oin ted Hopf algebras are precisely those whose coradicals a re group algebras, and in turn these groups determine a large part of their s tructure. W e use the recen t classification of these Hopf algebras b y Andruskiewits ch and Schn eider [4]. Eac h has a presen tation b y g enerators and relations similar to those of the small quantum groups, y et t hey are m uc h more general. Due to this similarity , some of the t echniq ues of Ginzburg and Kumar [17] yield r esults in this general setting. How ev er some differences do arise, notably that cohomology ma y no long er v anish in o dd degrees, and that useful connections to Lie algebras ha v e no t b een dev eloped. Th us w e m ust tak e a somewhat differen t approac h in this pap er, whic h also yields new pro ofs of some of the results in [1 7]. Since we ha v e less information a v ailable in this general setting, w e prov e finite generation without computing the f ull structure of the cohomology r ing . How ev er w e do explicitly iden tify a subalgebra ov er whic h the cohomology is finite, and establish some results ab out its structure. These structure results follow in par t from our general result in Section 3, that the cohomology ring of a Hopf algebra in a bra ided category is a lw a ys braided graded comm utativ e. This g eneralizes the we ll-known result that the cohomology ring of a Hopf algebra is graded comm utativ e, one pro of of whic h follo ws from the existence of tw o definitions of its multiplication. W e generalize that pro of, g iving a braided v ersion of the Eck mann-Hilton argumen t, from whic h f ollo ws the braided graded comm utativit y result. W e a pply this to a Nic hols algebra in Section 5, thus obtaining some details ab out the structure of the finitely generated cohomology ring of the corresp onding p oin ted Hopf algebra. Of course one hopes fo r results in y et greater generalit y . Ho we v er the structure of finite dimensional noncomm utativ e, nonco comm utative , nonsemisimple Hopf algebras, other than those treated in this pap er, is lar gely unkno wn. There a re a v ery small n um b er of kno wn (nontrivial) finite dimensional p oin ted Ho pf algebras ha ving nonab elian groups o f gr o uplik e elemen ts (see for example [1, 2]). Ev en few er examples a re kno wn of nonp oin ted, nonsemis imple Hopf algebras [10]. T o pro v e finite generation of cohomolog y in gr eat er generalit y , it ma y b e necessary to find gen eral tec hniq ues, suc h as the em b edding of any finite group sc heme in to GL n used b y F riedlander and Suslin [16], rather tha n dep ending on structural kno wledge of the Hopf algebras as w e do he re. POINTED HOPF ALGEBRAS 3 Our proof of finite generation is a t w o-step redu ction to t ype A 1 × · · · × A 1 , in whic h case the corresp onding Nichols algebra is a quan tum complete intersec tion. F or these alg ebras, w e compute cohomolo gy explicitly via a resolution constructed in Section 4. This resolution is a da pted from [8, 20] where similar algebras were considered (but in p ositiv e c haracteristic). Eac h of our tw o reduction steps inv olv es a sp ectral sequence asso ciated to an a lgebra filtrat io n. In the first step ( Section 5) a Radford bipro duct of the form B ( V )# k Γ, for a group Γ and Nic hols algebra B ( V ), has a filtration for which the asso ciated gra ded algebra has t yp e A 1 × · · · × A 1 . This filtratio n is generalized from D e Concini and Kac [12]. W e iden tify some p ermanen t cycles and apply a lemma adapted fro m F riedlander and Suslin [1 6] to conclude finite g eneration. In the second reduction step (Section 6), any of Andruskiewits ch a nd Schneid er’s p ointed Hopf algebras u ( D , λ, µ ) has a filtration for whic h the asso ciated gra ded algebra is a Radford bipro duct, whose cohomology features in Section 5. Again we iden tify some p ermanen t cycles and conclude finite generation. A s a corollary w e show that the Ho c hsc hild cohomology ring of u ( D , λ, µ ) is also finitely generated. The first and last author s thank Ludw ig-Maximilians-Univ ersit¨ at M ¨ u nch en for its hospitality during the first stag es of this pro ject. The second a uthor thanks MSRI for it s hospitalit y and supp ort during the final stage of this w ork. The first aut ho r w as supp orted b y an NSER C p ostdo cto r a l fellow ship. The second au- thor w as partially supp orted b y the NSF gran ts D MS-062915 6 and DMS- 0 800940. The t hird a ut ho r was supp orted by D e utsche F orschungsge meinschaft through a Heisen berg F ellows hip. T he last author w as partially supp or ted b y NSF gran ts DMS-044347 6 and DMS-0800 8 32, a nd NSA gran t H98230 -07-1- 0038. The last author thanks D. J. Be nson for ve ry useful c onv ers ations. 2. De finitions and Preliminar y Resul ts Let k be a field, usually a ssumed to b e algebraically closed and of c haracteristic 0. All tensor pro ducts are o v er k unless otherwise indicated. Let Γ b e a finite group. Hopf algebra s in Y etter-Drinfeld categories. A Y et ter-Drinfe l d mo dule ov er k Γ is a Γ- graded v ector space V = ⊕ g ∈ Γ V g that is also a k Γ- mo dule for whic h g · V h = V g hg − 1 for all g , h ∈ Γ. The grading b y the group Γ is equiv alen t to a k Γ- como dule s tructure on V , that is a map δ : V → k Γ ⊗ V , defined b y δ ( v ) = g ⊗ v for all v ∈ V g . Let Γ Γ Y D denote the category of all Y etter-Dr inf eld mo dules o v er k Γ. The catego r y has a tensor pro duct: ( V ⊗ W ) g = ⊕ xy = g V x ⊗ W y for all g ∈ Γ , and Γ acts diagonally on V ⊗ W , that is, g ( v ⊗ w ) = g v ⊗ g w fo r all g ∈ Γ, v ∈ V , and w ∈ W . There is a br aiding c : V ⊗ W ∼ → W ⊗ V for all V , W ∈ Γ Γ Y D as follo ws: Let g ∈ Γ, v ∈ V g , and w ∈ W . T hen c ( v ⊗ w ) = g w ⊗ v . Th us Γ Γ Y D is a braided monoidal category . (F o r details on the category Γ Γ Y D , including the connection to Hopf algebras recalled b elow, see for example [4 ].) 4 M. MASTNAK, J. PEVTSOV A, P . SCHAUENBURG, A ND S. WI THERSPOON Let C b e an y braided monoidal category . F or simplicit y , w e will alw ays a ssume the tensor pro duct is strictly asso ciativ e with strict unit ob ject I . An algebr a in C is an ob ject R together with morphisms u : I → R and m : R ⊗ R → R in C , suc h that m is asso ciativ e in the sense that m ( m ⊗ 1 R ) = m (1 R ⊗ m ), a nd u is a unit in the sense that m ( u ⊗ 1 R ) = 1 R = m (1 R ⊗ u ). The definition of a coalgebra in C is similar, with the ar r o ws going in the opp o site direction. Th us, an algebra (resp. coalgebra) in Γ Γ Y D is simply an ordinary algebra ( resp. coalgebra) with m ultiplication (resp. comultiplic atio n) a graded and equiv ariant map. A br aide d Hopf algebr a in C is an a lg ebra as w ell as coalgebra in C suc h that its com ultiplication a nd counit are algebra morphisms, and suc h that the iden tit y morphism id : R → R has a con v olution in v erse s in C . When w e say that the comultiplication ∆ : R → R ⊗ R should b e an algebra morphism, the braiding c of C arises in the definition of the algebra structure of R ⊗ R (so in particular, a bra ided Hopf a lg ebra in Γ Γ Y D is not an ordinary Hopf algebra). More generally , if A, B are t w o algebras in C , their tensor pro duct A ⊗ B is defined to ha v e m ultiplication m A ⊗ B = ( m A ⊗ m B )(1 A ⊗ c ⊗ 1 B ). An example of a braided Hopf algebra in Γ Γ Y D is the Nichols alg ebr a B ( V ) asso ciated to a Y etter-D rinfeld mo dule V o v er k Γ; B ( V ) is the quotient of the tensor algebra T ( V ) by the largest homog eneous braided Hopf ideal generated b y homogeneous elemen ts of degree at least 2. F or details, see [3, 4]. In this pap er, w e need only t he structure o f B ( V ) in some cases as are explicitly describ ed b elo w. If R is a braided Hopf algebra in Γ Γ Y D , t hen its R ad f o r d bip r o duct (o r b osoniza- tion ) R # k Γ is a Hopf algebra in the usual sense (that is, a Hopf algebra in k k Y D ). As an algebra, R # k Γ is just a skew gr oup algebr a , that is, R # k Γ is a free R - mo dule with basis Γ on whic h m ultiplication is defined by ( r g )( sh ) = r ( g · s ) g h fo r all r , s ∈ R and g , h ∈ G . Com ultiplication is g iv en b y ∆( r g ) = r (1) ( r (2) ) ( − 1) g ⊗ ( r (2) ) (0) g , for all r ∈ R and g ∈ Γ, where ∆( r ) = P r (1) ⊗ r (2) in R as a Hopf algebra in Γ Γ Y D and δ ( r ) = P r ( − 1) ⊗ r (0) denotes the k Γ-como dule structure. The p oin ted Hopf algebras of Andruskiewitsch and Schneid er. A p ointe d Hopf algebra H is one for whic h all simple como dules are one dimensional. This is equiv alen t to the c ondition H 0 = k Γ w here Γ = G ( H ) is the group of grouplike elemen ts of H and H 0 is the coradical of H (the initial t erm in the coradical filtration). The Hopf algebras of Andruskiewitsc h and Sc hneider in [4] ar e p ointed, and are deformations of Radf o rd bipro ducts. They dep end on the follo wing data: Let θ b e a p ositive integer. Let ( a ij ) 1 ≤ i,j ≤ θ b e a Cartan matrix of finite typ e , that is the Dynkin diagram of ( a ij ) is a disjoint union of copies of some of the diagrams A • , B • , C • , D • , E 6 , E 7 , E 8 , F 4 , G 2 . In particular, a ii = 2 f or 1 ≤ i ≤ θ , a ij is a nonp ositiv e in teger fo r i 6 = j , and a ij = 0 implie s a j i = 0. Its D ynk i n diagr am is a graph with v ertices lab eled 1 , . . . , θ : The v ertices i and j are connecte d b y a ij a j i edges, and if | a ij | > | a j i | , there is an arro w p oin ting from j to i . POINTED HOPF ALGEBRAS 5 No w assume Γ is a b elian , and denote b y ˆ Γ its dua l group of c haracters. F or eac h i , 1 ≤ i ≤ θ , c hoose g i ∈ Γ and χ i ∈ ˆ Γ s uc h that χ i ( g i ) 6 = 1 and (2.0.1) χ j ( g i ) χ i ( g j ) = χ i ( g i ) a ij (the Cartan c ondition ) holds f or 1 ≤ i, j ≤ θ . Letting q ij = χ j ( g i ), t his b ecomes q ij q j i = q a ij ii . Call (2.0.2) D = (Γ , ( g i ) 1 ≤ i ≤ θ , ( χ i ) 1 ≤ i ≤ θ , ( a ij ) 1 ≤ i,j ≤ θ ) a datum of finite Cartan typ e asso ciated to Γ and ( a ij ). The Hopf algebras of in terest will b e gene rated as algebras b y Γ and sym b ols x 1 , . . . , x θ . Let V b e the v ector space with basis x 1 , . . . , x θ . Then V has a structure of a Y etter-D rinfeld mo dule o v er k Γ: V g = Span k { x i | g i = g } and g ( x i ) = χ i ( g ) x i for 1 ≤ i ≤ θ and g ∈ Γ. This induces the structure of an algebra in Γ Γ Y D on the tensor algebra T ( V ). In particular Γ acts b y a uto morphisms o n T ( V ), and T ( V ) is a Γ- graded algebra in whic h x i 1 · · · x i s has degree g i 1 · · · g i s . The braiding c : T ( V ) ⊗ T ( V ) → T ( V ) ⊗ T ( V ) is induced b y c ( x i ⊗ y ) = g i ( y ) ⊗ x i . Moreo v er, T ( V ) can b e made a bra ided Hopf algebra in Γ Γ Y D if w e define com ultiplication as the unique alg ebra map ∆ : T ( V ) → T ( V ) ⊗ T ( V ) satisfying ∆( v ) = v ⊗ 1 + 1 ⊗ v for all v ∈ V . W e de fine the braided comm utators ad c ( x i )( y ) = [ x i , y ] c := x i y − g i ( y ) x i , for all y ∈ T ( V ), and similarly in quotien ts o f T ( V ) b y homogeneous ideals. Let Φ denote the ro ot system corresp o nding to ( a ij ), a nd let Π denote a fixed set of simple ro o ts. If α i , α j ∈ Π, write i ∼ j if the corresp onding v ertices in the D ynkin diagram of Φ are in the same connected comp onen t. Cho ose scalars λ = ( λ ij ) 1 ≤ i 0, and d i (Φ( a 1 , . . . , a θ )) = 0 if a i = 0. Extend eac h d i to an S -mo dule homomorphism. Note t hat d 2 i = 0 fo r eac h i since x N i i = 0 and σ i ( a i ) + σ i ( a i − 1) = N i . If i < j , we hav e d i d j (Φ( a 1 , . . . , a θ )) = d i Y ℓ 0 Y m<ℓ ( − 1) a m q − σ ℓ ( a ℓ +1) τ m ( a m ) mℓ ! α j x j − 1 ℓ Φ( a 1 , . . . , a ℓ + 1 , . . . , a θ ) , if a ℓ is ev en δ j,N ℓ − 1 Y m<ℓ ( − 1) a m q − σ ℓ ( a ℓ +1) τ m ( a m ) mℓ ! α j Φ( a 1 , . . . , a ℓ + 1 , . . . , a θ ) , if a ℓ is o dd where δ j > 0 = 1 if j > 0 a nd 0 if j = 0. Calculations sho w that for all i , 1 ≤ i ≤ θ , ( s i d i + d i s i )( α j x j i Φ( a 1 , . . . , a θ )) =  α j x j i Φ( a 1 , . . . , a θ ) , if j > 0 or a i > 0 0 , if j = 0 and a i = 0 POINTED HOPF ALGEBRAS 21 and s ℓ d i + d i s ℓ = 0 for all i, ℓ with i 6 = ℓ . F o r eac h x j 1 1 · · · x j θ θ Φ( a 1 , . . . , a θ ), let c = c j 1 ,...,j θ ,a 1 ,...,a θ b e the cardinality of the set of a ll i (1 ≤ i ≤ θ ) suc h that j i a i = 0. D efine s ( x j 1 1 · · · x j θ θ Φ( a 1 , . . . , a θ )) = 1 θ − c j 1 ,...,j θ ,a 1 ,...,a θ ( s 1 + · · · + s θ )( x j 1 1 · · · x j θ θ Φ( a 1 , . . . , a θ )) . Then s d + d s = id on eac h K n , n > 0. That is, K • is exact in p ositiv e degrees. T o sho w that K • is a resolution of k , put k in degree − 1, and let the corresp onding differen tial b e the counit map ε : S → k . It can b e sho wn directly t ha t K • then becomes exact at K 0 = S : The k ernel of ε is spanned o v er the field k b y the elemen ts x j 1 1 · · · x j θ θ Φ(0 , . . . , 0), 0 ≤ j i ≤ N i , with at least one j i 6 = 0. Let x j 1 1 · · · x j θ θ Φ(0 , . . . , 0) b e suc h an elemen t, and let i b e the smallest p ositiv e integer suc h that j i 6 = 0. Then d ( x j i − 1 i · · · x j θ θ Φ(0 , . . . , 1 , . . . , 0)) is a nonzero scalar m ultiple of x j i i · · · x j θ θ Φ(0 , . . . , 0). Th us k er( ε ) = im( d ) , and K • is a free resolution of k as an S -mo dule. Next w e will use K • to compute Ext ∗ S ( k , k ). Applying Hom S ( − , k ) to K • , the induced differen tial d ∗ is the zero map since x σ i ( a i ) i is alw ay s in the augmen tatio n ideal. Th us the cohomolo gy is the complex Hom S ( K • , k ), and in degree n this is a v ector space of dimension  n + θ − 1 θ − 1  . Now let ξ i ∈ Hom S ( K 2 , k ) b e the function dual to Φ(0 , . . . , 0 , 2 , 0 , . . . , 0) (the 2 in the i th place) and η i ∈ Hom S ( K 1 , k ) b e the function dual to Φ(0 , . . . , 0 , 1 , 0 , . . . , 0) ( t he 1 in the i th place). By abuse of notat io n, identify these f unctions with the corresp onding elemen ts in H 2 ( S, k ) and H 1 ( S, k ), resp ectiv ely . W e will sho w that the ξ i , η i generate H ∗ ( S, k ), and determine the relations among them. In o rder to do this w e will abuse not ation further and denote b y ξ i and η i the corresp onding c hain maps ξ i : K n → K n − 2 and η i : K n → K n − 1 defined by ξ i (Φ( a 1 , . . . , a θ )) = Y ℓ>i q N i τ ℓ ( a ℓ ) iℓ Φ( a 1 , . . . , a i − 2 , . . . , a θ ) η i (Φ( a 1 , . . . , a θ )) = Y ℓi ( − 1) a ℓ q τ ℓ ( a ℓ ) iℓ x σ i ( a i ) − 1 i Φ( a 1 , . . . , a i − 1 , . . . , a θ ) . Calculations sho w that these are indeed c hain maps. The ring structure of the subalgebra of H ∗ ( S, k ) generated by ξ i , η i is giv en by comp osition of these c hain maps. Direct calculation sho ws that the relations giv en in Theorem 4.1 b elo w hold. (Note that if N i 6 = 2 t he la st relation implies η 2 i = 0.) Alternativ ely , in case S = B ( V ), a Nic hols alg ebra defined in Section 2, w e ma y apply Corollary 3.13 to obtain the relations in Theorem 4 .1 b elow , p erforming calculations that are a sp ecial case o f the ones in the pro of of Theorem 5.4. Thus an y elemen t in the algebra generated b y t he ξ i and η i ma y b e written as a linear com bination of elemen ts of the form ξ b 1 1 · · · ξ b θ θ η c 1 1 · · · η c θ θ with b i ≥ 0 and c i ∈ { 0 , 1 } . Such an elemen t tak es Φ(2 b 1 + c 1 , . . . , 2 b θ + c θ ) to a nonzero scalar multiple o f Φ(0 , . . . , 0) 22 M. MASTNAK, J. PEVTSOV A, P . SCHAUENBURG, A ND S. WI THERSPOON and all other S -basis elemen ts of K P (2 b i + c i ) to 0. Since t he dimension of H n ( S, k ) is  n + θ − 1 θ − 1  , t his show s that the ξ b 1 1 · · · ξ b θ θ η c 1 1 · · · η c θ θ form a k - basis for H ∗ ( S, k ). W e ha v e prov en: Theorem 4.1. L et S b e the k -alg ebr a gener ate d by x 1 , . . . , x θ , subje ct to r elations (4.0.1). Then H ∗ ( S, k ) is gener ate d by ξ i , η i ( i = 1 , . . . , θ ) whe r e deg ξ i = 2 and deg η i = 1 , subje ct to the r elations (4.1.1) ξ i ξ j = q N i N j j i ξ j ξ i , η i ξ j = q N j j i ξ j η i , and η i η j = − q j i η j η i . Note that although the relatio ns (4.1.1) can b e obtained as a consequence of Corollary 3.13, in order to obtain the full statemen t of the theorem, we needed more information. Remark 4.2. W e obtain [1 7 , Prop. 2.3 .1] as a corollary: In this case we replace the generators x i of S b y the generators E α ( α ∈ ∆ + ) of Gr u + q , whose relatio ns are E α E β = q h α,β i E β E α ( α ≻ β ) , E ℓ α = 0 ( α ∈ ∆ + ) . Theorem 4 .1 then implies that H ∗ (Gr u + q ) is generated by ξ α , η α ( α ∈ ∆ + ) with relations ξ α ξ β = ξ β ξ α , η α ξ β = ξ β η α , η α η β = − q −h α,β i η β η α ( α ≻ β ) , η 2 α = 0 , whic h is pr ecisely [17, Prop. 2.3 .1 ]. No w assume a finite gro up Γ a cts on S b y auto morphisms for whic h x 1 , . . . , x θ are eigen v ectors. F or eac h i , 1 ≤ i ≤ θ , let χ i b e the c haracter on Γ for whic h g x i = χ i ( g ) x i . As the c haracteristic o f k is 0 , w e hav e Ext ∗ S # k Γ ( k , k ) ≃ Ext ∗ S ( k , k ) Γ , where the action of Γ at the c hain lev el on (4 .0 .2) is as usual in degree 0, but shifted in higher degrees so as to make the differen tials comm ute with the action of Γ. Sp ecifically , note that the following action of Γ o n K • comm utes with the differen tials: g · Φ( a 1 , . . . , a θ ) = θ Y ℓ =1 χ ℓ ( g ) τ ℓ ( a ℓ ) Φ( a 1 , . . . , a θ ) for all g ∈ Γ, and a 1 , . . . , a θ ≥ 0. Then the induced action of Γ on generators ξ i , η i of the cohomology ring H ∗ ( S, k ) is give n explicitly b y (4.2.1) g · ξ i = χ i ( g ) − N i ξ i and g · η i = χ i ( g ) − 1 η i . POINTED HOPF ALGEBRAS 23 5. Coradicall y graded finite dimensional pointed Hopf algebras Let D b e arbitrary data as in ( 2 .0.2), V the corresp onding Y etter-Drinfeld mo d- ule, and R = B ( V ) its Nichols algebra, and describ ed in Sec tion 2. By Lemma 2.4, there is a filtration on R for whic h S = Gr R is of type A 1 × · · · × A 1 , g iv en b y generators a nd relations of ty p e (4.0.1). Th us H ∗ ( S, k ) is giv en b y Theorem 4.1. As the filtratio n is finite, there is a conv ergen t sp ectral sequence asso ciated to the filtration (see [2 7, 5.4.1]) : (5.0.2) E p,q 1 = H p + q (Gr ( p ) R, k ) + 3 H p + q ( R, k ) . It follows that the E 1 -page of the sp ectral sequence is giv en b y Theorem 4.1 with grading corresp onding to t he filtration on R . W e will see that by (5 .0.6) and Lemma 5.1 b elow, the generators ξ i are in degrees ( p i , 2 − p i ), where (5.0.3) p i = N β 1 · · · N β i ( N β i · · · N β r h t( β i ) + 1) . Since the PBW basis elemen ts (2.3 .1) are eigen v ectors for Γ, the actio n o f Γ on R preserv es the filtration, and w e further g et a sp ectral sequence conv e rging to the cohomology of u ( D , 0 , 0 ) ≃ R # k Γ: (5.0.4) H p + q (Gr ( p ) R, k ) Γ + 3 H p + q ( R, k ) Γ ≃ H p + q ( R # k Γ , k ) . Moreo v er, if M is a finitely generated R # k Γ -mo dule, there is a sp ectral sequence con v erging to the cohomology of R with co efficien ts in M : (5.0.5) H p + q (Gr ( p ) R, M ) + 3 H p + q ( R, M ) , also compatible with the action of Γ. W e wish to apply Lemma 2.5 to the sp ectral sequence (5.0.4) for the filtered algebra R . In order to do so, we m ust find some p ermanent cycles. The Ho c hsc hild cohomology o f R in degree 2, with trivial co efficien ts, w as stud- ied in [24]: There is a linearly indep enden t set of 2-co cycles ξ α on R , indexed b y the p ositiv e ro ots α ([24, Theorem 6.1.3]). W e will use the notation ξ α in place of the notation f α used there. As show n in [24], these 2-co cycles ma y b e expressed as functions at the c hain lev el in the fo llo wing w ay . Let x a , x b denote arbitrary PBW basis elemen ts (2.3.1) of u ( D , 0 , 0). Let e x a , e x b denote corresp onding elemen ts in the infinite dimensional algebra U ( D , 0) a rising from the section o f the quotien t map U ( D , 0) → u ( D , 0 , 0) for whic h PBW basis elemen ts are sen t to PBW basis elemen ts. Then (5.0.6) ξ α ( x a ⊗ x b ) = c α where c α is the co efficien t of e x N α α in the pro duct e x a · e x b , and x a , x b range o v er a ll pairs of PBW basis elemen ts. A direct proo f that these are 2- co cycles is in [2 4, § 6.1]; alternativ ely the pro o f in Lemma 6.2 b elo w, that analogous functions f α in 24 M. MASTNAK, J. PEVTSOV A, P . SCHAUENBURG, A ND S. WI THERSPOON higher degrees are co cycles fo r u ( D , λ, 0), applies with minor mo difications to the ξ α in t his contex t. W e wish to relate these functions ξ α to elemen ts on the E 1 -page of the sp ectral sequence (5.0.2), found in the previous section. Recall t ha t β 1 , . . . , β r en umerate the p ositiv e r o ots, a nd the in teger p i is defined in (5.0.3 ). By (5.0.6 ) and Lemma 2.4, w e ha v e ξ β i ↓ F p i − 1 ( R ⊗ R ) = 0 but ξ β i ↓ F p i ( R ⊗ R ) 6 = 0 . Note that the filtration on R induces a filtra t io n o n R + , and th us a (decreasing) filtration on the complex C • defined in (2.4 .1), giv en b y F p C n = { f : ( R + ) ⊗ n → k | f ↓ F p − 1 (( R + ) ⊗ n ) = 0 } . W e conclude t ha t ξ β i ∈ F p i C 2 but ξ β i 6∈ F p i +1 C 2 . Denoting the corresp onding co cycle b y the same letter, w e further conclude that ξ β i ∈ im { H 2 ( F p i C • ) → H 2 ( C • ) } = F p i H 2 ( R, k ), but ξ β i 6∈ im { H 2 ( F p i +1 C • ) → H 2 ( C • ) } = F p i +1 H 2 ( R, k ). Hence, ξ β i can b e identified with the correspo nding non trivial homogeneous elemen t in the asso ciated graded complex: e ξ β i ∈ F p i H 2 ( R, k ) /F p i +1 H 2 ( R, k ) ≃ E p i , 2 − p i ∞ . Since ξ β i ∈ F p i C 2 but ξ β i 6∈ F p i +1 C 2 , it induces an elemen t ¯ ξ β i ∈ E p i , 2 − p i 0 = F p i C 2 /F p i +1 C 2 . Since ¯ ξ β i is induced b y a n actual co cycle in C • , it will b e in the k ernels of all t he differen tials of the spectral sequence. Hence, t he residue of ¯ ξ β i will b e in the E ∞ –term where it will b e iden tified with the non-zero elemen t e ξ β i since these classes are induced b y the same co cycle in C • . W e conclude that ¯ ξ β i ∈ E p i , 2 − p i 0 , and, corresp ondingly , its ima g e in E p i , 2 − p i 1 ≃ H 2 (Gr R, k ) whic h w e denote by the same sym b ol, is a p ermanen t cycle. Note that the results of [24] apply equally w ell to S = Gr R to yield similar co cycles ˆ ξ α via the formula (5.0.6), for eac h p o sitive ro ot α in type A 1 × · · · × A 1 . No w S = Gr R has generators corresp onding to the r o ot vec tors of R , a nd w e similarly identify elemen ts in cohomology . Thus eac h ˆ ξ α , α a p ositiv e ro ot in t yp e A 1 × · · · × A 1 , ma y b e relab eled ˆ ξ β i for some i (1 ≤ i ≤ r ), the indexing corresp onding to β 1 , . . . , β r ∈ Φ + . Comparing the v alues of ξ β i and ˆ ξ β i on basis elemen ts x a ⊗ x b of Gr R ⊗ Gr R w e conclude that they are the same f unction. Hence ˆ ξ β i ∈ E p i , 2 − p i 1 are p ermanen t cycles. W e wish to iden tify these elemen ts ˆ ξ β i ∈ H 2 (Gr R, k ) with the cohomology classes ξ i in H ∗ ( S, k ) constructed in Section 4, as w e may th us exploit the algebra structure of H ∗ ( S, k ) giv en in Theorem 4 .1. W e use the same sym b ols x β i to denote the corresp onding ro ot v ectors in R and in S = Gr R , as this should cause no confusion. Lemma 5.1. F or e ach i ( 1 ≤ i ≤ r ), the c ohomolo gy classes ξ i and ˆ ξ β i c oincide as eleme nts of H 2 (Gr R, k ) . Pr o of. Let K • b e the c hain complex defined in Section 4, a pro jectiv e resolution of the trivial Gr R -mo dule k . Elemen ts ξ i ∈ H 2 (Gr R, k ) a nd η i ∈ H 1 (Gr R, k ) w ere defined via the complex K • . W e wish to iden tify ξ i with elemen ts of the c hain complex C • defined in (2.4.1), where A = R . T o this end w e define maps F 1 , F 2 POINTED HOPF ALGEBRAS 25 making the following diagram comm ute, where S = G r R : · · · → K 2 d − → K 1 d − → K 0 ε − → k → 0 ↓ F 2 ↓ F 1 k k · · · → S ⊗ ( S + ) ⊗ 2 ∂ 2 − → S ⊗ S + ∂ 1 − → S ε − → k → 0 In this diagr a m, the maps d are giv en in (4 .0.3), and ∂ i in ( 2.4.2). Let Φ( · · · 1 i · · · ) denote the ba sis elemen t of K 1 ha ving a 1 in the i th p osition, and 0 in all other p ositions, Φ( · · · 1 i · · · 1 j · · · ) (resp ectiv ely Φ( · · · 2 i · · · )) the basis elemen t of K 2 ha ving a 1 in the i th and j th p ositions ( i 6 = j ), and 0 in all o ther p o sitions (resp ectiv ely a 2 in the i th p osition and 0 in all other p ositions). Let F 1 (Φ( · · · 1 i · · · )) = 1 ⊗ x β i , F 2 (Φ( · · · 2 i · · · )) = N i − 2 X a i =0 x a i β i ⊗ x β i ⊗ x N i − a i − 1 β i , F 2 (Φ( · · · 1 i · · · 1 j · · · )) = 1 ⊗ x β i ⊗ x β j − q ij ⊗ x β j ⊗ x β i . Direct computations show t ha t the t w o non trivial squares in the dia g ram ab o v e comm ute. So F 1 , F 2 extend to maps F i : K i → S ⊗ ( S + ) ⊗ i , i ≥ 1, providing a c hain map F • : K • → S ⊗ ( S + ) ⊗• , thus inducing isomorphisms on cohomology . W e no w ve rify that the maps F 1 , F 2 mak e the desired iden tifications. By ξ β i w e mean the function on the reduced bar complex, ξ β i (1 ⊗ x a ⊗ x b ) := ξ β i ( x a ⊗ x b ), defined in (5.0.6). Then F ∗ 2 ( ξ β i )(Φ( · · · 2 i · · · )) = ξ β i ( F 2 (Φ( · · · 2 i · · · ))) = ξ β i ( N i − 2 X a i =0 x a i β i ⊗ x β i ⊗ x N i − a i − 1 β i ) = N i − 2 X a i =0 ε ( x a i β i ) ξ β i (1 ⊗ x β i ⊗ x N i − a i − 1 β i ) = ξ β i ( x β i ⊗ x N i − 1 β i ) = 1 . F urther, it may b e c hec k ed similarly that F ∗ 2 ( ξ β i )(Φ( · · · 1 i · · · 1 j · · · )) = 0 for all i, j and F ∗ 2 ( ξ β i )(Φ( · · · 2 j · · · )) = 0 for all j 6 = i . Therefore F ∗ 2 ( ξ β i ) is the dual function to Φ( · · · 2 i · · · ), whic h is precisely ξ i .  Similarly , we iden tify the elemen ts η i defined in Section 4 with functions at the c hain lev el in cohomology: Define (5.1.1) η α ( x a ) =  1 , if x a = x α 0 , otherwise . The f unctions η α represen t a basis of H 1 ( S, k ) ≃ Hom k ( S + / ( S + ) 2 , k ). Similarly functions corresp o nding t o the simple ro ots β i only represen t a basis of H 1 ( R, k ) ≃ 26 M. MASTNAK, J. PEVTSOV A, P . SCHAUENBURG, A ND S. WI THERSPOON Hom k ( R + / ( R + ) 2 , k ). A computation show s that F ∗ 1 ( η β i )(Φ( · · · 1 j · · · )) = δ ij , so that F ∗ 1 ( η β i ) is the dual function to Φ( · · · 1 i · · · ). Therefore η i and η β i coincide as elemen ts o f H 1 ( S, k ). F or eac h α ∈ Φ + , let M α b e any p ositive integer for whic h χ M α α = ε . (F or example, let M α b e the order of χ α .) Note that ξ M α α is Γ- inv ariant: By (4.2.1), g · ξ M α α = χ − M α N α α ( g ) ξ M α α = ξ M α α . Recall the notation u ( D , 0 , 0) ≃ R # k Γ, where R = B ( V ) is the Nic hols algebra. Lemma 5.2. The c ohomolo gy algebr a H ∗ ( u ( D , 0 , 0) , k ) is finitely gener ate d over the sub algebr a gener ate d by al l ξ M α α ( α ∈ Φ + ). Pr o of. Let E ∗ , ∗ 1 + 3 H ∗ ( R, k ) b e the spectral sequenc e (5.0.2), and let B ∗ , ∗ b e the bigraded subalgebra o f E ∗ , ∗ 1 generated by the elemen ts ξ i . By Lemma 5.1 and t he discussion prior to it, B ∗ , ∗ consists of p ermanen t cycles. Since ξ i is ¯ ξ β i b y Lemma 5.1, it is in bidegree ( p i , 2 − p i ). Let A ∗ , ∗ b e the subalgebra of B ∗ , ∗ generated b y ξ M α α ( α ∈ Φ + ). Then A ∗ , ∗ also consists of p ermanen t cycles. Observ e that A ∗ , ∗ is a subalgebra of H ∗ ((Gr R ) # k Γ , k ), whic h is graded comm utativ e since (Gr R ) # k Γ is a Hopf algebra. Hence, A ∗ , ∗ is comm utativ e a s it is concen trated in ev en (total) degrees. Finally , A ∗ , ∗ is No etherian since it is a p olynomial algebra in the ξ M α α . W e conclude that the bigra ded commutativ e algebra A ∗ , ∗ satisfies the h yp otheses o f Lemma 2.5. By Theorem 4.1, the algebra E ∗ , ∗ 1 ≃ H ∗ (Gr R, k ) is generated b y ξ i and η i where the generators η i are nilp otent. Hence, E ∗ , ∗ 1 is a finitely generated module ov er B ∗ , ∗ . The latter is clearly a finitely generated mo dule o v er A ∗ , ∗ . Hence E ∗ , ∗ 1 is a finitely generated mo dule o v er A ∗ , ∗ . Lemma 2.5 implies that H ∗ ( R, k ) is a No etherian T ot( A ∗ , ∗ )-mo dule; moreov er, the action of Γ on H ∗ ( R, k ) is compatible with the action o n A ∗ , ∗ since the sp ectral sequence is compatible with the action of Γ. Therefore, H ∗ ( R # k Γ , k ) ≃ H ∗ ( R, k ) Γ is a No etherian T ot( A ∗ , ∗ )-mo dule. Since T ot( A ∗ , ∗ ) is finitely generated, w e conclude that H ∗ ( R # k Γ , k ) is finitely generated.  W e immediately hav e the following theorem. The second statemen t of the the- orem follows b y a simple application of the second statemen t of Lemma 2.5. Theorem 5.3. The algebr a H ∗ ( u ( D , 0 , 0) , k ) is finitely ge ner ate d. If M is a fi nitely gener ate d u ( D , 0 , 0) -mo dule, then H ∗ ( u ( D , 0 , 0) , M ) is a finitely ge ner ate d mo dule over H ∗ ( u ( D , 0 , 0) , k ) . Thanks to Coro llary 3.1 3 , w e hav e some information ab out the algebra structure of the cohomolog y ring of the Nic hols algebra R = B ( V ): Recall that q β α = χ α ( g β ) (see (2.1.2) ). Compare the followin g result with the graded case, Theorem 4.1. Theorem 5.4. The fol lowing r elations hold in H ∗ ( R, k ) fo r a l l ξ α , ξ β ( α, β ∈ Φ + ) and η α , η β ( α, β ∈ Π ): ξ α ξ β = q N α N β β α ξ β ξ α , η α ξ β = q N β β α ξ β η α , and η α η β = − q β α η β η α . POINTED HOPF ALGEBRAS 27 Pr o of. As w e shall see, this is a consequence of Corollary 3.13, since R is a braided Hopf algebra in Γ Γ Y D . Note first that as a function on the Y etter-Dr inf eld mo dule R ∈ Γ Γ Y D the co cycle ξ α is Γ- ho mogeneous of degree g − N α α and spans a one-dimensional Γ- mo dule with c haracter χ − N α α . T o see this in f ull detail, let us rephrase t he definition (5.0.6) of ξ α as follows . One can write U ( D , 0) as a Radford bipro duct U ( D , 0 ) = ˆ R # k Γ with a braided Hopf alg ebra ˆ R ∈ Γ Γ Y D suc h tha t R is a quotien t of ˆ R . Let s : R → ˆ R b e the section of the surjection ˆ R → R that maps PBW basis elemen ts to PBW basis elemen ts. Note that s is a map in Γ Γ Y D . Finally , let p α : ˆ R → k b e the function pro jecting r ∈ ˆ R to the co efficien t in r of the PBW basis elemen t ˜ x N α α . Then ξ α is b y definition the pullbac k of p α under R ⊗ R s ⊗ s − − → ˆ R ⊗ ˆ R m − → ˆ R . As a consequenc e, since p α is clearly Γ-homog eneous of degree g − N α α and satisfies g · p α = χ − N α α ( g ) p α (cf. (4.2.1) ), the statemen ts on ξ α follo w. On t he other hand, it is easy to see from definition (5.1.1) tha t η α has degree g − 1 α and spans a one-dimensional Γ-mo dule with character χ − 1 α . Let us denote the opp osite o f m ultiplication in H ∗ ( R, k ) b y ζ ◦ θ := θ ζ . According to Corollary 3.1 3, the opp osite m ultiplication is braided graded comm utativ e. In particular ξ α ξ β = ξ β ◦ ξ α =  g − N β β · ξ α  ◦ ξ β = χ − N α α  g − N β β  ξ α ◦ ξ β = χ α ( g β ) N β N α ξ α ◦ ξ β = q N β N α β α ξ α ◦ ξ β = q N β N α β α ξ β ξ α and η α ξ β = ξ β η α =  g − N β β · η α  ◦ ξ β = χ − 1 α  g − N β β  η α ◦ ξ β = q N β β α ξ β η α as w ell as finally η α η β = η β ◦ η α = −  g − 1 β · η α  ◦ η β = − χ − 1 α  g − 1 β  η α ◦ η β = − q β α η β η α .  W e giv e a corollary in a sp ecial case. It generalizes [17, Theorem 2.5(i)]. Recall that Π denotes a set o f simple ro ots in the ro o t system Φ. Corollary 5.5. Assume ther e ar e n o Γ -inva ria nts in H ∗ (Gr R, k ) of the form ξ b 1 β 1 · · · ξ b r β r η c 1 β 1 · · · η c r β r for which c 1 + · · · + c r is o dd. Then H ∗ ( u ( D , 0 , 0) , k ) ≃ k h ξ α , η β | α ∈ Φ + , β ∈ Π i Γ , with the r elations of The or em 5. 4 , deg ( η β ) = 1 , deg( ξ α ) = 2 . If ther e ar e no Γ -invariants with c 1 + · · · + c r 6 = 0 , then H ∗ ( R # k Γ , k ) ≃ k [ ξ α | α ∈ Φ + ] Γ . 28 M. MASTNAK, J. PEVTSOV A, P . SCHAUENBURG, A ND S. WI THERSPOON Pr o of. The hypothesis of the first statemen t implies that H i (Gr R, k ) Γ = 0 for all o dd integers i . Th us on the E 1 -page, ev ery other diagonal is 0 . This implies E 1 = E ∞ . It follows that, a s a v ector space, H ∗ ( u ( D , 0 , 0) , k ) is exactly as stated. The algebra structure is a consequence of Theorem 5 .4 and the fact that the cohomology of a Hopf algebra is graded comm utative. The h yp othesis of the last statemen t implies further that the Γ-inv ariant subalgebra of H ∗ (Gr R, k ) is spanned b y elemen ts of the form ξ b 1 β 1 · · · ξ b r β r . By graded commutativit y of the cohomology ring of a Hopf algebra and the relations of Theorem 5.4, H ∗ ( R # k Γ , k ) ma y b e iden tified with the Γ-in v arian t subalgebra of a p olynomial ring in v ariables ξ α , with corresp onding Γ - action.  Remark 5.6. Assume the hypotheses of Corollary 5.5 , and that q αα 6 = − 1 for a ll α ∈ Π. Then η 2 α = 0 for all α ∈ Π, and it follo ws that the maximal ideal sp ectrum of H ∗ ( u ( D , 0 , 0) , k ) is Sp ec k [ ξ α | α ∈ Φ + ] Γ ≃ Sp ec k [ ξ α | α ∈ Φ + ] / Γ. W e giv e an example to sho w that cohomology may in fact be nonzero in o dd degrees, in contrast to that o f the small quan tum groups of [17]. This complicates an y determination of the explicit structure of cohomology in general. The simplest example o ccurs in t yp e A 1 × A 1 × A 1 , where there can exist a nonzero cycle in degree 3. Example 5.7. Let Γ = Z /ℓ Z with g enerator g . Let q b e a primitiv e ℓ th ro ot of unity . Let g 1 = g 2 = g 3 = g , and choose χ 1 , χ 2 , χ 3 so that the matrix ( q ij ) = ( χ j ( g i )) is   q q − 1 1 q q − 2 q 1 q − 1 q   . Let u ( D , 0 , 0) = R # k Γ b e the p ointe d Hopf alg ebra of type A 1 × A 1 × A 1 defined b y this dat a . Le t η 1 , η 2 , η 3 represen t elemen ts of H 1 ( R, k ) as defined b y (5.1.1). The action of Γ is describ ed in (4.2.1): g i · η j = q − 1 ij η j . Since the pro duct of all en tries in any g iv en ro w o f the matrix ( q ij ) is 1, w e conclude that η 1 η 2 η 3 is in v arian t under Γ. Hence, it is a non trivial co cycle in H 3 ( u ( D , 0 , 0) , k ). W e also give an example in type A 2 × A 1 to illustrate, in par ticular, that the metho ds employ ed in [17] do not transfer to our more general setting. That is, for an arbit r a ry (cora dically gra ded) pointed Hopf alg ebra, the first pag e of the sp ectral sequence (5.0.4) can hav e non trivial elemen ts in o dd degrees. In the sp ecial case of a small quan tum group (with some restrictions on the o r der of the ro ot of unity ), it is shown in [1 7] that this do es not happ en. Example 5.8. Let Γ = Z /ℓ Z × Z /ℓ Z , with generators g 1 , g 2 . Let q b e a primitiv e ℓ th ro ot of unit y ( ℓ o dd), and let D b e of ty p e A 2 × A 1 so that the Cartan matrix POINTED HOPF ALGEBRAS 29 is   2 − 1 0 − 1 2 0 0 0 2   . Let χ 1 , χ 2 b e as for u q ( sl 3 ) + , t hat is χ 1 ( g 1 ) = q 2 , χ 1 ( g 2 ) = q − 1 , χ 2 ( g 1 ) = q − 1 , χ 2 ( g 2 ) = q 2 . No w let g 3 := g 1 g 2 and χ 3 := χ − 1 1 χ − 1 2 . Then χ 3 ( g 3 ) = q − 2 6 = 1 and the Cartan condition holds, for example χ 3 ( g 1 ) χ 1 ( g 3 ) = q − 1 q = 1 = χ 1 ( g 1 ) a 13 since a 13 = 0. Let R = B ( V ), the Nic hols algebra defined fro m this data. The ro ot v ector corresp onding to the nonsimple p ositive ro ot is x 12 = [ x 1 , x 2 ] c = x 1 x 2 − q − 1 x 2 x 1 . The relations among the ro ot v ectors other than x 3 are no w x 2 x 1 = q x 1 x 2 − q x 12 , x 12 x 1 = q − 1 x 1 x 12 , x 2 x 12 = q − 1 x 12 x 2 . The asso ciated graded a lg ebra, whic h is of t yp e A 1 × A 1 × A 1 × A 1 , thus has relations ( excluding tho se inv olving x 3 that do not c hange): x 2 x 1 = q x 1 x 2 , x 12 x 1 = q − 1 x 1 x 12 , x 2 x 12 = q − 1 x 12 x 2 . Th us the cohomology of the asso ciated graded algebra Gr R # k Γ is the subalgebra of Γ- in v arian ts of an a lgebra with g enerators ξ 1 , ξ 12 , ξ 2 , ξ 3 , η 1 , η 12 , η 2 , η 3 (see Theorem 4.1). The elemen t η 1 η 2 η 3 is Γ-inv ariant since g · η i = χ i ( g ) − 1 η i for all i , and χ 3 = χ − 1 1 χ − 1 2 . Therefore in the sp ectral sequenc e relating the cohomology of G r R # k Γ to that of R # k Γ, there are some o dd degree elemen ts on the E 1 page. 6. Finite dimensional pointed Hopf algebras Let u ( D , λ, µ ) b e one of the finite dimensional p oin ted Hopf alg ebras from the Andruskiewits ch -Schne ider classification [4], as describ ed in Section 2. With re- sp ect to the coradical filtration, its asso ciated gr a ded algebra is Gr u ( D , λ, µ ) ≃ u ( D , 0 , 0) ≃ R # k Γ where R = B ( V ) is a Nic hols alg ebra, also describ ed in Section 2. By Theorem 5 .3 , H ∗ ( u ( D , 0 , 0) , k ) is finitely g enerated. Belo w w e apply the sp ectral sequence fo r a filtered algebra, employ ing this new c hoice of filtra tion: (6.0.1) E p,q 1 = H p + q ( u ( D , 0 , 0) , k ) + 3 H p + q ( u ( D , λ, µ ) , k ) . As a consequence o f t he following lemma, we ma y assume without lo ss of gen- eralit y that all r o ot v ector relations (2.1.4) are trivial. 30 M. MASTNAK, J. PEVTSOV A, P . SCHAUENBURG, A ND S. WI THERSPOON Lemma 6.1. (i) F o r al l D , λ , µ , ther e is an isomorp h ism of gr ade d alge b r as, H ∗ ( u ( D , λ, µ ) , k ) ≃ H ∗ ( u ( D , λ, 0) , k ) . (ii) L et M b e a finitely gener ate d u ( D , λ, µ ) - m o dule. T her e exists a finitely gener ate d u ( D , λ, 0) -mo dule f M and an is o morphism of H ∗ ( u ( D , λ, µ ) , k ) -mo dules H ∗ ( u ( D , λ, µ ) , M ) ≃ H ∗ ( u ( D , λ, 0) , f M ) , wher e the action of H ∗ ( u ( D , λ, µ ) , k ) on H ∗ ( u ( D , λ, 0) , f M ) is via the isomorphism of gr ade d algebr as in (i). Pr o of. Define the subset I ⊂ Γ as follows: g ∈ I if a nd only if there exists a ro o t v ector relation x N α α − u α ( µ ) (see (2 .1 .4)), suc h that the elemen t u α ( µ ) of k Γ has a nonzero co efficien t of g when written a s a linear combination of group elemen ts. If x N α α − u α ( µ ) is a nontrivial ro ot v ector relation, then necessarily x N α α comm utes with elemen ts of Γ, implying that χ N α α = ε . From this and (2.3.2), w e see that x N α α is cen tral in u ( D , λ, µ ). Since x N α α = u α ( µ ), this elemen t of the group ring k Γ m ust also b e cen tral. Since Γ acts diagonally , eac h g roup elemen t in v olv ed in u α ( µ ) is necessarily cen tral in u ( D , λ, µ ) as w ell. Let Z = k h I i , a subalgebra o f u ( D , λ, µ ), and let u = u ( D , λ, µ ) / ( g − 1 | g ∈ I ). W e hav e a sequence o f algebras (see [17, § 5.2]): Z → u ( D , λ, µ ) → u ( D , λ, µ ) / / Z ≃ u. Hence, there is a m ultiplicativ e sp ectral sequence H p ( u, H q ( Z , k )) = ⇒ H p + q ( u ( D , λ, µ ) , k ) . Since the c haracteristic o f k do es not divide the order of the group, w e hav e H q > 0 ( Z , k ) = 0. Th us the sp ectral sequence collapses, and w e get an isomorphism of gra ded algebras (6.1.1) H ∗ ( u, k ) ≃ H ∗ ( u ( D , λ, µ ) , k ) . Similarly , if M is a ny u ( D , λ, µ ) -mo dule, then we ha v e a sp ectral sequence of H ∗ ( u, k )-mo dules H p ( u, H q ( Z , M )) = ⇒ H p + q ( u ( D , λ, µ ) , M ) . The sp ectral sequence collapses, and we get a n isomorphism (6.1.2) H ∗ ( u, M Z ) ≃ H ∗ ( u ( D , λ, µ ) , M ) whic h resp ects the action of H ∗ ( u ( D , λ, µ ) , k ), where H ∗ ( u ( D , λ, µ ) , k ) acts on the left side via the isomorphism (6.1.1). Note that Z = k h I i is also a cen tral subalgebra of u ( D , λ, 0 ) . Arguing exactly as ab ov e, we get an isomorphism of g raded a lgebras H ∗ ( u, k ) ≃ H ∗ ( u ( D , λ, 0) , k ) , whic h implies (i). POINTED HOPF ALGEBRAS 31 Let M b e a u ( D , λ, µ )- mo dule. Let f M b e a u ( D , λ, 0)- mo dule whic h we get b y inflating the u -mo dule M Z via the pro jec tion u ( D , λ, 0) / / / / u . Since ( f M ) Z ≃ M Z b y construction, w e get an isomorphism of u ( D , λ, 0 ) -mo dules H ∗ ( u, M Z ) ≃ H ∗ ( u ( D , λ, 0) , f M ) using another spectral sequ ence argumen t. Combin ing with the isomorphism (6.1.2), w e g et (ii).  By Lemma 6.1, it suffices to work with the cohomology o f u ( D , λ, 0), in whic h all the ro ot v ectors are nilp oten t. In this case w e define some p ermanen t cycles : As b efore, for eac h α ∈ Φ + , let M α b e an y p ositiv e inte ger for which χ M α α = ε (for example, tak e M α to b e the order of χ α ). If α = α i is simple, then χ α ( g α ) has order N α , a nd so N α divides M α . W e previously iden tified a n elemen t ξ α of H 2 ( R, k ), where R = B ( V ) . No w B ( V ) is no longer a subalgebra of u ( D , λ, 0) in general, due to the p oten tial existence of nontriv ial linking relations, but we will sho w that still there is an elemen t analogous to ξ M α α in H 2 M α ( u ( D , λ, 0) , k ). F or simplicit y , let U = U ( D , λ ) and u = u ( D , λ, 0) ≃ U ( D , λ ) / ( x N α α | α ∈ Φ + ) (see Section 2). Let U + , u + denote the augmen tation ideals of U , u . W e will use a similar construction as in Section 4 of [24], defining functions as elemen ts of the bar complex (2.4.1). F or eac h α ∈ Φ + , define a k -linear function e f α : ( U + ) 2 M α → k by first letting r 1 , . . . , r 2 M α b e PBW basis elemen ts (2.3.1) and requiring e f α ( r 1 ⊗ · · · ⊗ r 2 M α ) = γ 12 γ 34 · · · γ 2 M α − 1 , 2 M α where γ ij is t he co efficien t of x N α α in the pro duct r i r j as a linear com bination of PBW basis elemen ts. Now define e f α to b e 0 whenev er a tensor fa ctor is in k Γ ∩ k er ε , and e f α ( r 1 g 1 ⊗ · · · ⊗ r 2 M α g 2 M α ) = e f α ( r 1 ⊗ g 1 r 2 ⊗ · · · ⊗ g 1 ··· g 2 M α − 1 r 2 M α ) for a ll g 1 , . . . , g 2 M α ∈ Γ. It follo ws from t he definition of e f α and the fact that χ M α α = ε that e f α is Γ- in v arian t. W e will show that e f α factors through the quotient u + of U + to giv e a map f α : ( u + ) ⊗ 2 M α → k . Precisely , it suffices to sho w that e f α ( r 1 ⊗ · · · ⊗ r 2 M α ) = 0 whenev er one of r 1 , . . . , r 2 M α is in the k ernel of the quotien t map π : U ( D , λ ) → u ( D , λ, 0). Supp ose r i ∈ k er π , that is r i = x a 1 β 1 · · · x a r β r and for some j , a j ≥ N β j . Since x N β j β j is braided-cen tral, r i is a scalar mu ltiple of x N β j β j x b 1 β 1 · · · x b r β r for some b 1 , . . . , b r . No w e f α ( r 1 ⊗ · · · ⊗ r 2 M α ) is the pro duct of the co efficien ts of x N α α in r 1 r 2 , . . . , r 2 M α − 1 r 2 M α . How ev er, the co efficien t of x N α α in each of r i − 1 r i and r i r i +1 is 0: If α = β i , then since r i − 1 , r i +1 ∈ U + , this pro duct cannot 32 M. MASTNAK, J. PEVTSOV A, P . SCHAUENBURG, A ND S. WI THERSPOON ha v e a nonzero co efficien t fo r x N α α . If α 6 = β i , the same is true since x N β j β j is a f actor of r i − 1 r i and of r i r i +1 . Therefore e f α factors to giv e a linear map f α : ( u + ) 2 M α → k . In calculations, w e define f α via e f α and a c hoice of section of the quotien t ma p π : U → u . Lemma 6.2. F or e ach α ∈ Φ + , f α is a c o cycle. The f α ( α ∈ Φ + ) r epr es ent a line arly i n dep endent subset of H ∗ ( u ( D , λ, 0) , k ) . Pr o of. W e first v erify that e f α is a co cycle on U : Let r 0 , . . . , r 2 M α ∈ U + , o f p o sitiv e degree. Then d ( e f α )( r 0 ⊗ · · · ⊗ r 2 M α ) = 2 M α − 1 X i =0 ( − 1) i +1 e f α ( r 0 ⊗ · · · ⊗ r i r i +1 ⊗ · · · ⊗ r 2 M α ) . By definition of e f α , no te that the first t w o terms cancel: e f α ( r 0 r 1 ⊗ r 2 ⊗ · · · ⊗ r 2 M α ) = e f α ( r 0 ⊗ r 1 r 2 ⊗ · · · ⊗ r 2 M α ) , and similarly for all other terms, so d ( e f α )( r 0 ⊗ · · · ⊗ r 2 M α ) = 0. A similar calculatio n sho ws that d ( e f α )( r 0 g 0 ⊗ · · · ⊗ r 2 M α − 1 g 2 M α − 1 ) = 0 for all g 0 , . . . , g 2 M α − 1 ∈ Γ. If there is an elemen t of k Γ ∩ k er ε in one o f the fa cto r s, we obta in 0 as well by the definition of e f α , a similar calculation. No w we v erify t ha t f α is a co cycle on the quotient u of U : Let r 0 , . . . , r 2 M α ∈ u + . Again we ha ve d ( f α )( r 0 ⊗ · · · ⊗ r 2 M α ) = 2 M α − 1 X i =0 ( − 1) i +1 f α ( r 0 ⊗ · · · ⊗ r i r i +1 ⊗ · · · ⊗ r 2 M α ) . W e will sho w that f α ( r 0 r 1 ⊗ r 2 ⊗ · · · ⊗ r 2 M α ) = f α ( r 0 ⊗ r 1 r 2 ⊗ · · · ⊗ r 2 M α ) , and similarly for the other terms. Let e r i denote the elemen t of U corresp onding to r i under a c hosen section of the quotient map π : U → u . Note that e r 0 · e r 1 = g r 0 r 1 + y and e r 1 · e r 2 = g r 1 r 2 + z for some y , z ∈ k er π . So f α ( r 0 r 1 ⊗ r 2 ⊗ · · · ⊗ r 2 M α ) = e f α ( g r 0 r 1 ⊗ e r 2 ⊗ · · · ⊗ e r 2 M α ) = e f α (( e r 0 · e r 1 − y ) ⊗ e r 2 ⊗ · · · ⊗ e r 2 M α ) = e f α ( e r 0 · e r 1 ⊗ e r 2 ⊗ · · · ⊗ e r 2 M α ) = e f α ( e r 0 ⊗ e r 1 · e r 2 ⊗ · · · ⊗ e r 2 M α ) = e f α ( e r 0 ⊗ ( e r 1 · e r 2 + z ) ⊗ · · · ⊗ e r 2 M α ) = e f α ( e r 0 ⊗ g r 1 r 2 ⊗ · · · ⊗ e r 2 M α ) = f α ( r 0 ⊗ r 1 r 2 ⊗ · · · ⊗ r 2 M α ) . Other computations fo r this case are similar to those for U . Th us f α is a co cycle on u . POINTED HOPF ALGEBRAS 33 W e prov e that in a giv en degree, the f α in that degree represen t a linearly indep enden t set in cohomology: Supp o se P α c α f α = ∂ h for some scalars c α and linear ma p h . Then for each α , c α = ( X c α f α )( x α ⊗ x N α − 1 α ⊗ · · · ⊗ x α ⊗ x N α − 1 α ) = ∂ h ( x α ⊗ x N α − 1 α ⊗ · · · ⊗ x α ⊗ x N α − 1 α ) = − h ( x N α α ⊗ x α ⊗ · · · ⊗ x N α − 1 α ) + · · · − h ( x α ⊗ · · · ⊗ x N α − 1 α ⊗ x N α α ) = 0 since x N α α = 0 in u = u ( D , λ, 0).  These functions f α corresp ond t o their coun terparts ξ M α α defined on u ( D , 0 , 0), in the E 1 -page of the sp ectral sequence (6 .0 .1), by observing what they do as functions at the lev el o f ch ain complexes (2.4 .1). Theorem 6.3. The algebr a H ∗ ( u ( D , λ, µ ) , k ) is fi nitely gener ate d. If M is a finitely gener ate d u ( D , λ, µ ) -mo dule, then H ∗ ( u ( D , λ, µ ) , M ) is a finitely gener ate d mo dule over H ∗ ( u ( D , λ, µ ) , k ) . Pr o of. By Lemma 6 .1, it suffices to pro v e the statemen ts in the case µ = 0. W e ha v e E ∗ , ∗ 1 ≃ H ∗ ( u ( D , 0 , 0) , k ), where u ( D , 0 , 0) ≃ R # k Γ, R = B ( V ). By Lemma 5.2, E ∗ , ∗ 1 is finitely generated ov er its subalgebra that is generated b y all ξ M α α . By Lemma 6.2 and the ab ov e remarks, eac h ξ M α α is a p ermanent cycle , corresp onding to the co cycle f α on u ( D , λ, µ ). The rest o f the pro o f is an a pplication of Lemma 2.5, where A ∗ , ∗ is the subalgebra of E ∗ , ∗ 1 generated b y the ξ M α α ( α ∈ Φ + ), similar to the pro of of Lemma 5.2 and Theorem 5.3.  Corollary 6.4. The Ho chschild c ohomolo gy ring H ∗ ( u ( D , λ, µ ) , u ( D , λ, µ )) is finitely gener ate d. Pr o of. Apply Theorem 6.3 to the finitely generated u ( D , λ , µ )-mo dule u ( D , λ, µ ), under the adjoint a ctio n. By [17, Prop. 5.6], this is isomorphic to the Ho c hsc hild cohomology ring of u ( D , λ, µ ).  In t he sp ecial case of a small quan tum group, w e obtain the follow ing finite generation result (cf. [7, Thm. 1.3.4]). Corollary 6.5. L et u q ( g ) b e a quantize d r estricte d enveloping algebr a such that the or der ℓ of q is o dd and p ri m e to 3 if g is o f typ e G 2 . Then H ∗ ( u q ( g ) , k ) is a finitely g e ner ate d algebr a. Mor e over, fo r any finitely gener ate d u q ( g ) -mo dule M , H ∗ ( u q ( g ) , M ) is a fin itely gen er ate d mo dule over H ∗ ( u q ( g ) , k ) . Remark 6.6. The r estrictions on ℓ corresp ond to the assumptions (2.1.1). How- ev er our tec hniques and results are more general: The restrictions are used in t he classification of Andruskie witsc h and Schn eider, but not in our argumen ts. W e need only the filtratio n lemma of D e Concini and Kac [12, Lemma 1.7 ] as gener- alized to our setting (Lemma 2.4) to guarantee existence o f the needed sp ectral 34 M. MASTNAK, J. PEVTSOV A, P . SCHAUENBURG, A ND S. WI THERSPOON sequence s. Our results should hold for all small quan tum groups u q ( g ) havin g suc h a filtratio n, including those at ev en ro ots of unit y q for whic h q 2 d 6 = 1 (see [6] for the general theory at ev en ro ots of unit y). W e illustrate t he connection b etw ee n o ur results and those in [17] with a small example. Our structure results are not a s strong, ho w ev er our finite generation result is m uc h more g eneral. Example 6.7. As an alg ebra, u q ( sl 2 ) is g enerated by E , F , K , with relations K ℓ = 1, E ℓ = 0, F ℓ = 0, K E K − 1 = q 2 E , K F K − 1 = q − 1 F , and (6.7.1) E F − F E = K − K − 1 q − q − 1 , where q is a primitive ℓ th r o ot of 1, ℓ > 2. Consider the cora dical filtration on u q ( sl 2 ), in whic h deg ( K ) = 0 , deg( E ) = deg( F ) = 1. Note that Gr u q ( sl 2 ) is generated by E , F , K , with a ll relations b eing the same except that (6.7.1 ) is replaced b y E F − F E = 0. This is a n a lgebra of the ty p e featured in Section 4: H ∗ (Gr u q ( sl 2 ) , k ) ≃ k [ ξ 1 , ξ 2 , η 1 η 2 ] / (( η 1 η 2 ) 2 ), since these ar e the in v arian ts, under the action of Γ = h K i , of the cohomology of the subalgebra of Gr u q ( sl 2 ) generated b y E , F . By [17], H ∗ ( u q ( sl 2 ) , k ) ≃ k [ α , β , γ ] / ( α β + γ 2 ) , the co o rdinate algebra of the nilp otent cone of sl 2 . Iden tify α ∼ ξ 1 , β ∼ ξ 2 , γ ∼ η 1 η 2 : Then as maps, deg ( α ) = ℓ , deg( β ) = ℓ , deg( γ ) = 2, so αβ + γ 2 = 0 will imply γ 2 = 0 in the asso ciated graded algebra, as exp ected. Reference s [1] N. Andr uskiewitsch and F. F antino, “On p o inted Hopf a lgebras asso cia ted with alter nat- ing and dihedr al g roups,” a rxiv:math/0 70255 9 . [2] N. Andruskiew its ch and M. Gra ˜ na , “ Braided Hopf algebra s o ver no nab elian finite groups,” math.QA/980 2074. [3] N. Andruskiewitsch and H.-J. 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WI THERSPOON Dep ar tment of Ma thema tics and Computer Science, Saint Mar y’s University, Halif ax, NS B3H 3C3, Canada E-mail addr ess : mmast nak@c s.smu.ca Dep ar tment o f Ma thema tics, University o f W ashington, Sea ttle, W A 98195, USA E-mail addr ess : julia @math .washingto n.edu Ma thema tisches Institut der Universit ¨ at M ¨ unchen, Theresienstr. 39, 80333 M ¨ unchen, Germany E-mail addr ess : schau enbur g@math.lmu .de Dep ar tment o f Ma thema tics, Texas A&M University, College St a tion, TX 77843, USA E-mail addr ess : sjw@m ath.t amu.edu