Relation between α-hochschild Homology and α-hochschild Cohomology of a Quantum Algebra

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1 International Journal of Algebra, Vol. 6, 2012, no. 21, Relation between α-hochschild Homology and α-hochschild Cohomology of a Quantum Algebra Eleonora Cerati Departamento de Matematica - Facultad de Humanidades y Ciencias Universidad Nacional del Litoral Ciudad Universitaria-Paraje El Pozo s/n Santa Fe - Argentina eleonoracerati@gmail.com Ingrid Schwer Departamento de Matemtica - Facultad de Ingenieria Quimica Universidad Nacional del Litoral Santiago del Estero Santa Fe - Argentina ischwer@fiq.unl.edu.ar Abstract Using a quantum version of the Koszul complex and techniques defined by Wambst [12] which allow to build complexes quasi isomorphic to the standard Hochschild complex, we compute the Hochschild α homology of the algebra A = C{x 1,...,x n } generated over C by x 1,...,x n with relations x i x j = q ij x j x i (1 i, j n) for a linear automorphism α of A. For any multi-integer γ N n the Koszul complex K γ (A, A α ), where A α coincides with A as a vector space but the structure of right A- module is twisted by the morphism α, splits into a direct sum of subcomplexes K γ of elements of degree γ in the Koszul complex. Calling C = {γ N n / i, 1 i n, γ i =0orx γ x i = α(x i )x γ }, we prove that HH α (A) = γ C K γ. Since the algebra A is AS-Gorenstein, generalized Koszul and of finite global dimension, according to Berger and Marconnet [1], we apply Van den Bergh Duality Theorem [10] to compute α Hochschild cohomology. Mathematics Subject Classification: 16E40, 81R50 Keywords: α homology, α cohomology, quantum algebra

2 1026 E. Cerati and I. Schwer 1 Introduction In this work, we use techniques defined by Wambst [12], generalizing objects from commutative algebra, conveniently adapted for the computation of the cohomology of quantum algebras such as the quantum version of the Koszul complex. These methods are applied to the case of the multiparameter quantum affine space. We compute HHα(A) i for 0 i 4 where A = C{x 1,...,x n } is the algebra generated over C by x 1,...,x n with x i x j = q ij x j x i and α is a linear automorphism of A. Diverse results are obtained when we consider different types of automorphisms. To generalize the results obtained we compute the α Hochschild homology of A using Van den Bergh Duality Theorem [10]. In section 2, we recall some definitions and propositions about the multiparameter quantum affine space, its automorphisms and α- derivations, given in [3]. We compute the α Hochschild cohomology groups in degrees 2, 3 and 4 for different types of linear automorphisms using the definition given by Redondo and Solotar [6] and the computation of the α Hochschild cohomology in degrees 0 and 1 given in [3]. In section 3 we prove a theorem that allow us to compute the α Hochschild homology of the quantum algebra A using techniques defined by Wambst [12]. In section 4, we study the relation between the α Hochschild homology and the α Hochschild cohomology of A. Since A is a 2 homogeneous graded algebra, AS-Gorenstein and generalized Koszul, we apply Van den Bergh Duality Theorem [10] to compute the α Hochschild cohomology. We obtain that HHα(A) 0 =C and HHα(A) n = 0 for n>0, when α is of type 1. Forα of type 2, HHα n(a) =C Cxj x jn+1 n for n>0. This generalizes the obtained results for n =2, 3 and 4 in [2] and [3]. 2 Preliminaries 2.1 The multiparameter quantum affine space Let C be the complex field and Q =(q ij ) M n (C) with n 2, whose elements verify q ii = 1, q ij = q 1 ji if i j and q ij is not a root of unit. Let A = C{x 1,...,x n } be the algebra generated by x 1,...,x n with relations x i x j = q ij x j x i for 1 i, j n. In fact, A is the algebra of the quantum affine space of dimension n, that is a graded algebra with grading given by the total degree. The monomials x t = x t 1 1 x t x tn n, with t =(t 1,t 2,...,t n ) N n form a basis of A as C vector

3 Relation between α-hochschild homology 1027 space. It is then verified that: x j x t = where x s = x s x sn n. 1 l<j n x t x s = q t l jl xt x t j 1 j 1 xt j+1 j x t j+1 j+1...xtn n 1 l<k n 2.2 The automorphisms of A q t ks l s k t l kl x s x t Let α : A A be a C automorphism of algebras. Since α(x i x in n )= [α(x 1 )] i 1... [α(x n )] in, α is uniquely determined by α(x i ), 1 i n. As it has been proved in [3], the automorphisms obtained are of type α(x i )=a i x i + f i where a i C {0} and gr(f i ) 2orf i =0. From now on, we will work with linear automorphisms, that is f i = The α centralizer The α centralizer of the generators of A is: Z α ({x s }) = {x j /x j x s = α(x s )x j,j Z n } = {x j / q j ks l j l s k kl x s x j = a(x s )x s x j, 1 l<k n}, where a(x s )=a s a sn n. Remark 2.1 Let C{x 1,...,x n } be the quantum affine space of dimension n, with q ij C and let q ij > 1, (1 i<j n). Then Z α ({x s } s Z n) if and only if there are integers j 1,j 2,...,j n such that α(x i )= k i qj k ki x i. In this case Z α ({x s } s Z n)= { x j 1 1 x j } x jn n. 2.4 Classification of linear automorphisms The above remark suggests the following classification of the linear automorphisms: the automorphisms α Aut C{x 1,..., x n }\{α : α(x i )= k i qj k ki x i} will be called type 1 - automorphisms. Those excluded before will be called type 2 - automorphisms.

4 1028 E. Cerati and I. Schwer 2.5 The α derivations An α-derivation of A is a C linear map d α : C{x 1,...,x n } C{x 1,...,x n }, such that d α (ab) =α(a)d α (b)+d α (a)b, a, b C{x 1,...,x n }. Proposition 2.2 (see [3]) If α is of type 1, then every α derivation is inner. If α is of type 2, then every α derivation is the sum of an inner derivation and an α central derivation. 3 The α Hochschild cohomology 3.1 The α Hochschild cohomology in degrees 0 and 1 Redondo and Solotar define in [6] the α Hochschild cohomology of the algebra A with coefficients in the A bimodule M as: HH n α (A, M) =Hn (A, M), where M is the A bimodule whose structure is given by a m b = α(a)mb. In particular, if A is an algebra and M is an A-bimodule, the α Hochschild cohomology of A in degrees zero and one are given by: HH 0 α(a, M) ={m M : ma = α(a)m} HH 1 α(a, M) =Der α (A, M)/DerInt α (A, M) For the quantum affine space, we have compute [3]: Proposition 3.1 If α is an automorphism of type 1 of C{x 1,...,x n }, then HH 0 α(c{x 1,...,x n })=HH 1 α(c{x 1,...,x n })=0. We have also characterized the cohomology groups in degrees zero and one for automorphisms of type 2 [3]. Proposition 3.2 Let C{x 1,...,x n } be the quantun affine n space, q ij > 1, (1 i<j n), and α an automorphism of type 2 of C{x 1,...,x n }. Then and HHα 0 (C{x 1,...,x n })=<Z α ({x s }),s Z n >= {λx j x jn,λ C} n HH 1 α(c{x 1,...,x n })={d α : d α is an α central derivation of C{x 1,...,x n }}.

5 Relation between α-hochschild homology The α Hochschild cohomology in degrees 2, 3 and Case n =2 If A = C{x 1,x 2 }, the α Hochschild cohomology in degree 2 is computed using the Koszul quantum complex defined by Wambst, that is quasi isomorphic to the Hochschild complex, to which the funtor Hom C (, S Q (V ) α ) is applied, where V is the vector space generated by x 1,x 2,...,x n, Q is the anti-symmetric multiplicatively matrix previously defined and S Q (V ) α is the algebra of the multiparameter quantum affine space with coefficients in the crossed product by its group of linear automorphisms. Since C = 0 Q (V ) and V = 1 Q (V ), we obtain the complex: 0 Hom C (C,S Q (V ) α ) d 0 Hom C (V,S Q (V ) α ) d 1 d 1 HomC ( 2 Q(V ),S Q (V ) α ) d 2 0 Given φ : V S Q (V ) α, such that φ = φ 1 +φ 2 with φ 1 (x 1 )= i,j a ijx i 1x j 2, φ 1 (x 2 )=φ 2 (x 1 )=0,φ 2 (x 2 )= i,j a ij xi 1 xj 2 and α(x 1 )=a 1 x 1, α(x 2 )=a 2 x 2,it is straightforward to prove that d 1 φ 1 (1.x 1 x 2 )= ( j (q 12 a 2 1)a 0j x j (q21 i 1 a 2 1)a ij x i 1 xj+1 2 ) i>0,j and d 1 φ 2 (1.x 1 x 2 )= i (a 1 q 12 )a i0x i 1 + i,j>0 (a 1 q 1 j 12 )a ijx i+1 1 x j 2. So, the image of d 1 is spanned by polynomials in x 1,inx 2 and polynomials where x 1 x 2 appears. As HHα 2(A) =S Q(V ) α /Imd 1, if α is of type 1: a 1 q j 2 21, a 2 q j 1 12, then HHα 2 (A) =C. If α is of type 2, in general a 1 = q j 2 21, a 2 = q j Therefore: d 1 φ 2 (1.x 1 x 2 )= i (q j 2 21 q 12)a i0 xi (q j 2 21 q1 j 12 )a ij xi+1 1 x j 2. i,j>0 d 1 φ 1 (1.x 1 x 2 )= ( j (q j )a 0j x j (q21 i 1 q j )a ij x i 1x j+1 2 ). i>0,j In general, if j 1 1,j 2 1, constants and monomials of the form x j do not appear in Imd 1, therefore: HH 2 α(a) =C Cx j

6 1030 E. Cerati and I. Schwer 1. If j 1 = 1,j 2 = 1, a 1 = q 1 21, a 2 = q 1 12, then constants, polynomials in x 1 and polynomials in x 2 do not appear in Imd 1, therefore: HH 2 α (A) =C i>0 Cx i 1 j>0 Cx j 2 2. If j 1 1 and j 2 = 1, a 1 = q21 1 and a 2 = q j 1 12, then constants and polynomials in x 1 do not appear in Imd 1, therefore: HH 2 α (A) =C i>0 Cx i If j 1 = 1,j 2 1, a 1 = q j 2 21 and a 2 = q 1 12, then constants and polynomials in x 2 do not appear in Imd 1, therefore: HH 2 α(a) =C j>0 Cx j 2 Remark 3.3 Naidu, Shroff and Witherspoon work [4] with a finite group G acting on A = S q (V ) by graded algebra automorphisms. If G acts diagonally on the basis {x 1,x 2,...,x N } of the vectorial space V, there exit scalars λ g,i K such that g x i = λ g,i x i, for all g G, i {1,...,N}. They describe in the example 4.2 the Hochschild cohomology of S q (V ) G, for a fix g G. They consider two special cases. In the first one, q 12 is not a root of unity and λ g,1, λ g,2 are not both equal to 1. They have: HH 0 g = {0} ; HH 1 g = {0} and HHg 2 = span K {(1 g) x 1 x 2 }. In the second case, q 12 is simultaneously a primitive lth root of unity, a l 1 th root of λ g,1 and a l 2 th root of λ 1 g,2. We are interesting in the first case when q 12 is not a root of unity and g G is a fix element. g acts diagonaly on the basis {x 1,x 2 } of V in a similar way as the automorphism α : A A, such that α(x i )=a i x i, of type 1. For this type of automorphisms, the α centralizer is null. We obtaine the following results: and HH 0 α (C{x 1,x 2 })=HH 1 α (C{x 1,x 2 })=0 HH 2 α (C{x 1,x 2 })=C, wich are isomorphic to the results obtained by Naidu, Shroff and Witherspoon.

7 Relation between α-hochschild homology Case n = 3 When A = C{x 1,x 2,x 3 } and if α is of type 1, we obtain that HHα(A) 2 =0 and HHα 3 (A) =C. If α is of type 2, there are j 1,j 2,j 3 Z such that a 1 = q j 2 21 qj 3 31, a 2 = q j 1 12 qj 2 32, a 3 = q j 1 13q j Here, we consider those automorphisms for which the exponents j 1,j 2,j 3 don t make the corresponding powers of q 13, q 21 and q 23 equal. In general, HHα 3(A) =C Cxj x j if j 1,j 2,j 3 are not equal to 1. Next we analyze some particular cases: 1. If only j 3 = j 2 = 1, we have: HH 3 α(a) =C i>0 Cx i On the other hand, if j 1 = j 3 = 1, we get: HH 3 α (A) =C j>0 Cx j If j 1 = j 2 = 1: HH 3 α(a) =C k>0 Cx k If j 1 = j 2 = j 3 = 1: HHα(A) 3 =C Cx i 1 Cx j 2 Cx k 3. i>0 j>0 k> Case n =4 For A = C{x 1,x 2,x 3,x 4 }, we first consider, as above, the case such that there are no m Z 4, m 0, with (q 32 ) m 3 =(q 21 ) m 1, (q 32 ) m 3 =(q 41 ) m 4. If α is of type 1, HHα(A) 2 =HHα(A) 3 = 0 and HHα(A) 4 =C. When α is of type 2, there are j 1,j 2,j 3,j 4 Z such that α(x 1 )=q j 2 α(x 2 )=q j 1 12 qj 3 32 qj 4 42 x 2, α(x 3 )=q j 1 13 qj 2 23 qj 4 43 x 3 and α(x 4 )=q j 1 14 qj 2 24 qj 3 HH 4 α(a) =C Cx j x j x j 4+1 j 4 1. But if 1. j 2 = j 3 = j 4 = 1, then HH 4 α (A) =C Cx i j 1 = j 3 = j 4 = 1, then HH 4 α (A) =C Cx j j 1 = j 2 = j 4 = 1,then HH 4 α(a) =C Cx k 3. 21q j 3 31q j 4 41x 1, 34 x 4; then 4 if j 1 1, j 2 1, j 3 1 and

8 1032 E. Cerati and I. Schwer 4. j 1 = j 2 = j 3 = 1, then HH 4 α (A) =C Cx l j 1 = j 2 = j 3 = j 4 = 1, then HH 4 α (A) =C Cx i 1 Cx j 2 Cx k 3 Cx l For any value of j 1 and j 2 (different from 1) and j 3 = j 4 = 1,then HHα 4(A) =C Cxj For any value of j 1 and j 3 (different from 1) and j 2 = j 4 = 1,then HHα(A) 4 =C Cx j x j α-hochschild homology of the multiparameter quantum affine space We use some techniques defined by Wambst [12] for the computation of the α-hochschild homology of the quantum algebra A α. In this case we need to define: C = {γ N n / for every i, 1 i n, γ i =0orx γ x i = α(x i )x γ }. Theorem 4.1 Let A = C{x 1,...,x n } be the C-algebra with relations x i x j = q ij x j x i, 1 i, j n and the linear automorphism α of A with α(x i )=a i x i, 1 i n. Then the α-hochschild homology groups of the multiparameter quantum algebra A on C are given by: (HH α ) m (A) = Cx γ x β for all m N. β {0, 1} n β = m γ N n β + γ C Proof According to Wambst s definition of Koszul complexes [12], the α-hochschild homology groups of A are isomorphic to the homology groups of the complex K (A, A α ), where A α is as before. The differential of the complex is: d(x γ x i1... x ip )= p k 1 p ( 1) k+1 [ q isik x γ x ik q ik i s α(x ik )x γ ] k=1 s=1 s=k+1 x i1... x ik... x ip

9 Relation between α-hochschild homology 1033 on a homogeneous element of S Λ, where S = S Q (V ) and Λ = Λ Q (V ), that is to say, the algebra of the multiparameter quantum affine space V and the analog of the exterior algebra, linearly generated by the family {x ε x εn n,ε i {0, 1}}, respectively. The homogeneous element x i1... x ip of Λ, is indicated by x β, with β the multi-integer such that β ik =1ifk varies from 1 to p and β j = 0 in any other case. Then the differential d is: d(x γ x β )= n Ω(γ,β,i)x γ+[i] x β [i] i=1 where the value of Ω(γ,β,i) is null if x γ+β x i = α(x i )x γ+β or if β i = 0 and, in any other case, it is ε(β,i)( k 1 n s=1 qβs sk h=k+1 qγ h hk n s=k+1 qβs ks a P k 1 k l=1 qγ l kl ), with i 1 ε(β,i) =( 1) s=1 βs, γ N n and β {0, 1} n, i =1,...,n. The complex K (A, A α ) is graded with deg(x γ x β ) = γ + β. Given δ N n, let K δ be the subcomplex of elements of degree δ. It is possible to consider these subcomplexes since the image through d of an homogeneous element of degree δ is also of degree δ. If δ C, the restriction of d to K δ is null. If δ / C, it will be shown that K δ is acyclic using an homotopy h. So HH α (A) = δ C K δ. The homotopy h is defined as: h(x γ x β )= 1 γ + β n ω(γ,β,i)x γ [i] x β+[i] i=1 where ω(γ,β,i) is null if x γ+β x i = α(x i )x γ+β,ifβ i = 1 or if γ i = 0 and in any other case it is equal to [Ω(γ [i],β+[i],i)] 1, with γ N n and β {0, 1} n, such that γ + β / C and i =1,...,n. On the other hand, for γ N n, γ denotes the number of integers i, 1 i n, such that α(x i )x γ x γ x i and γ i 0. It is well defined, since if γ/ C, then γ 0. Then, h is an endomorphism of degree 1 of K γ. We will prove that it verifies h d + d h = id. = For this, we shall compute h d+ d h on an homogeneous element x γ x β : 1 γ + β ( h d + d h)(x γ x β )= n [Ω(γ,β,i)ω(γ+[i],β [i],i)+ω(γ,β,i)ω(γ [i],β+[i],i)]x γ x β + i=1 1 + [ω(γ,β,j)ω(γ [j],β+[j],i)+ω(γ,β,i)ω(γ +[i],β [i],j)] γ + β i j

10 1034 E. Cerati and I. Schwer When i j: x γ [j]+[i] x β+[j] [i] ω(γ,β,j)ω(γ [j],β+[j],i)+ω(γ,β,i)ω(γ +[i],β [i],j)=0. Indeed, let Γ(i, j) =ω(γ,β,j)ω(γ [j],β+[j],i) and Γ (i, j) =Ω(γ,β,i)ω(γ+ [i],β [i],j). We will see that either Γ(i, j) =Γ (i, j) = 0 or they are both not null. If Γ(i, j) = 0, then ω(γ,β,j)=0orω(γ [j],β+[j],i) = 0. For ω(γ,β,j) to be zero one must have either x γ+β x j = α(x j )x γ+β,orβ j =1,orγ j =0. In the first case, x γ+[i]+β [i] x j = x j x γ+[i]+β [i] and therefore ω(γ +[i],β [i],j)=0 and Γ (i, j) = 0. In the second case, β j = 1 and then (β [i]) j = β j = 1, which implies that Γ (i, j) = 0. In the third one γ j =0,so(γ +[i]) j = 1 and then ω(γ +[i],β [i],j) = 0, so Γ (i, j) =0. For Ω(γ [j],β +[j],i) to be zero one must have either x γ [j]+β+[j] x i = α(x i )x γ+β, or β j+1 = 0. In the first case, x γ [j]+β+[j] x i = α(x i )x γ+β and therefore Ω(γ,β,i) = 0 and Γ (i, j) = 0. In the second case, β j + 1 = 0 and then β j =(β [i]) j = 1, which implies that ω(γ +[i],β [i],j) = 0 and therefore Γ (i, j) =0. It may be prove in a similar way that Γ (i, j) = 0 implies that Γ(i, j) =0. We study the cases i<jand i>j. Let us suppose that i<j. In this case Γ(i, j) = ω(γ,β,j)ω(γ [j],β +[j],i) = [Ω(γ [j],β +[j],j)] 1 Ω(γ [j],β +[j],i) = (ε(β [j],j)) 1 ( j n s=1 qβs sk r=j+1 q γr jr ε(β +[j],i)( i n s=1 qβs si r=i+1 q γr ir Γ (i, j) = Ω(γ,β,i)ω(γ +[i],β [i],j) = ε(β,i)( i n s=1 qβs si r=i+1 q γr ir n s=i+1 qβs (ε(β +[i] [j],j)) 1 ( j s=1 qβs sj q ji n s=j+1 qβs js a j 1 j r=1 q γr rj q ji ) 1 Evidently, i s=1 qβs si j s=1 qβs sj q ji = Besides, if i<j: n r=i+1 q γr ir n r=j+1 q γr jr i n s=1 qβs si r=i+1 q γr ir n r=j+1 q γr jr j s=1 qβs sj q ji q ij n s=i+1 qβs n s=j+1 qβs js a j j 1 r=1 q γr rj ) 1 q ij n s=i+1 qβs is q ija r i 1 r=1 q γr ri ) is a i n r=j+1 q γr jr n s=j+1 qβs js a j i 1 r=1 q γr ri ) is q ija r i 1 n s=i+1 qβs is a i 1 i n s=j+1 qβs js a j r=1 q γr ri j 1 = r=1 q γr rj q ji r=1 q γr ri j 1. r=1 q γr rj q ji ε(β [j],j)ε(β +[j],j)=( 1) β 1+β β j 1 ( 1) β 1+β β i 1

11 Relation between α-hochschild homology 1035 and ε(β,i)ε(β +[i] [j],j)=( 1) β 1+β β i 1 ( 1) β (β i +1)+...+β j 1. Both expressions have opposite signs. The case i>jis analogous. As a consequence: n (ω(γ,β,i)ω(γ [i],β+[i],i)+ω(γ,β,i)ω(γ +[i],β [i],i)) = γ + β. i=1 Since γ/ C, γ +β 0. We denote ω 1 = ω(γ,β,i), Ω 1 =Ω(γ [i],β+[i],i), ω 2 = ω(γ +[i],β [i],i) and Ω 2 =Ω(γ,β,i). If γ i = 0 and β i 0, then ω 1 =Ω 2 and if γ i 0 and β i = 0, then ω 1 Ω 1 = 1 and ω 2 Ω 2 =0. Ifγ i 0 and β i 0, then ω 1 Ω 1 = 0 and ω 2 Ω 2 =1. From the previous equality, we obtain the following: ( h d + d h)(x γ x β )= γ + β γ + β xγ x β = x γ x β. Corollary 4.2 For α of type 1, the α centralizer is empty, so C = {(0, 0,...,0)}.Therefore: (HH α ) 0 (A) =C and (HH α ) m (A) =0,m>0. Remark 4.3 Richard [8] studies a family of iterated Ore extensions called twisted polynomial algebras S Λ n,r(k), which are parametrized by an integer n 1, an integer r, with 0 r n and a multiplicatively antisymmetric matrix Λ=(λ ij ) 1 i,j n M n (K ). He computes their homologies and obtain that the Hochschild homology of S Λ n,r(k) is null in degrees greater than n + r. He observes that for r =0, the algebra S Λ n,0 (k) is the quantum affine space O Λ(k n ). If we restrict our results for the case in which the automorphism α (of type 1) is the identity, we reobtain his results. Corollary 4.4 For α of type 2, the α-centralizer is {x j 1 1,...,x jn n }, then C = {(0, 0,...,0), (j 1,j 2,...,j n )}. As a consequence, (HH α ) 0 (A) = C Cx j x jn n 1 (HH α ) 1 (A) = n i=1 Cxj x j i 1 i (HH α ) 2 (A) = i<k Cxj x j i 1 i (HH α ) 3 (A) = i<k<h Cxj x j i 1 i And finally:...x jn n...x j k 1 k x i...x j k 1 k...x jn n x ix k...x j h 1 h...x jn n (HH α ) n (A) =Cx j 1 1 x j x jn 1 n x 1 x 2...x n. x ix k x h

12 1036 E. Cerati and I. Schwer 5 Relation between α-hochschild homology and α-hochschild cohomology of A Several authors, as Michel Van den Bergh, Roland Berger and Nicolas Marconnet, have studied the relationship between Hochschild homology and cohomology for different kind of rings and modules. Berger and Marconnet [1] proved, based on Van den Bergh s duality theorem, that for an algebra A, N-homogeneous that is AS-Gorestein and of finite global dimension D for any bimodule M, it is verified that HH i (A, M) = HH D i (A, ε D+1 φ M) Theorem 5.1 Let A = C{x 1,...,x n } be the C algebra with the relations x i x j = q ij x j x i for 1 i<j n and α the linear automorphism of A with α(x i )=a i x i. Then the relations between the α Hochschild homology and the α Hochschild cohomology of A are (HH α ) i (A) = (HH ϕ 1 α) (n i) (A) and (HH ϕα ) i (A) = (HH α ) (n i) (A) The algebra A = C{x 1,...,x n }, belongs to the class of the N-homogeneous graded algebra, when N = 2. It is also known [9] that this algebra has {x 1,...,x n } as PBW base, which is an enough condition for A to be a Koszul homogeneous algebra,just as Priddy has proved [5] it. Berger and Marconnet [1], in their theorem 4.5, prove that for a Koszul homogeneous algebra, the Hochschild dimension and global dimension are equal. In our case, the Hochschild dimension of A is n and therefore the gldima = n is obviously finite. Besides, A is AS-regular and consequently is AS-Gorestein. Van den Bergh shows in his proposition 2 [10], the Frobenius automorphism of A!, which he calls φ!, and the automorphism φ of A which is the adjoint of φ!. On the other hand, ε is the automorphism of A which is the product by ( 1) m on the homogeneous component of degree m of A. For the case N = 2, the algebra of Yoneda of A is always symmetric graded and so results φ = ε n+1 and then ε n+1 φ =1 A. ε n+1 φm in this case is: ε n+1 φa α = U A A α where U = Ext i A e(a, Ae ) by proposition 2 of [10], that is 0 if i n ([1], section 6) and is ε n+1 φa α,itistosay ϕ A α,ifi = n. ϕ is an automorphism of A defined by ϕ =1 A if α is of type 1 and ϕ(x i )= k i q ikx i if α is of type 2.

13 Relation between α-hochschild homology 1037 Then: and (HH α ) i (A) = (HH ϕ 1 α) (n i) (A) (HH ϕα ) i (A) = (HH α ) (n i) (A). Remark 5.2 Computing (HH ϕ 1 α) n (A), we obtain for n =2: (HH ϕ 1 α) 0 (A) =C Cx j for n =3: (HH ϕ 1 α) 2 (A) =Cx j 1 1 x j 2 2 x 1 x 2 ; (HH ϕ 1 α) 0 (A) =C Cx j x j (HH ϕ 1 α) 3 (A) =Cx j 1 1 x j 2 2 x j 3 3 x 1 x 2 x 3 that are the same results that we obtained for direct computations. References [1] R. Berger and N. Marconnet, Koszul and Gorenstein properties for homogeneous algebras. Algebr. Represent. Theory 9 (2006), [2] E. Cerati and I. Schwer, La α-cohomología de Hochschild del espacio afín cuántico multiparámetro. LVII Reunión de Comunicaciones Científicas de la Unión Matemática Argentina. Córdoba, [3] E. Cerati and I. Schwer, Derivations and automorphisms in the twisted polynomial algebra. Algebras Groups Geom. 17 (2000), [4] D. Naidu, P.Shroff and S. Witherspoon, Hochschild cohomology of group extensions of quantum symmetric algebras. Proc. Amer. Math. Soc. 139 (2011), [5] S. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), [6] M. J. Redondo and A. Solotar. α-derivations II: the non-commutative case. Bol. Acad. Nac. de Cienc. (Córdoba) 65 (2000),

14 1038 E. Cerati and I. Schwer [7] L. Richard, Hochschild homology and cohomology of some classical and quantum noncommutative polynomial algebras. J. Pure Appl. Algebra 187 (2004), [8] L. Richard, Equivalence rationnelle et homologie de Hochschild pour certaines algèbres polynomiales classiques et quantiques. Thèse.Universit Blaise Pascal(2002), France. [9] S. P. Smith, Some finite-dimensional algebras related to elliptic curves. Representation theory of algebras and related topics (Mexico City, 1994), ,CMS Conf. Proc., 19, Amer. Math. Soc., Providence, RI, [10] M. van den Bergh, A relation between Hochschild homology and cohomology for Gorenstein rings.proc. Amer. Math. Soc. 126 (1998), [11] M. van den Bergh, Erratum to: A relation between Hochschild homology and cohomology for Gorenstein rings. Proc. Amer. Math. Soc. 126 (1998), no. 5, ]. Proc. Amer. Math. Soc. 130 (2002), [12] M. Wambst, Complexes de Koszul quantiques. Ann. Inst. Fourier (Grenoble) 43 (1993), Received: April, 2012