On the Hochschild homology of elliptic Sklyanin algebras

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1 On the Hochschild homology of elliptic Sklyanin algebras Serge Roméo Tagne Pelap May 13, 2008 Abstract In this paper, we compute the Hochschild homology of elliptic Sklyanin algebras. These algebras are deformations of polynomial algebra with a Poisson bracket called the Sklyanin Poisson bracket. Keywords : Hochschild homology, quantum space, Poisson structures, Poisson homology. Introduction The family of algebras dened by Sklyanin in ( [20]), which today carries his name, is naturally associated with two parameters : an elliptic curve and a point on this curve. It is a family of associative algebras with 4 generators and six quadratic relations. These algebras are at deformations of polynomial algebras with four variables. The paper ( [17]) of Odesskii and Feigin gives a generalization of these algebras. Considering an elliptic curve E and a point η of this curve, the elliptic algebras or algebras of Feigin- Odesskii are the family of associative algebras Q n,k (E, η), k < n (which are mutually prime ), with n generators and the relations : r Z/nZ θ j i+r(k 1) (0) θ kr (η)θ j i r ( η) x j rx i+r = 0 (1) where θ α, α Z/nZ, are theta functions ( [17]). These algebras have the following properties ( [17]) : Q n,k (E, 0) = C[x 1,, x n ] Q n,n 1 (E, η) = C[x 1,, x n ] Q n,k (E, η) Q n,k (E, η) if kk 1(mod n). The algebras Q n,k (E, η) can be considered as a graded deformation of polynomial ring. The corresponding Poisson algebra is denoted by q n,k (E). Among these Poisson structures q n,k (E), there are only two (q 3,1 (E) and q 4,1 (E)) which are Jacobian Poisson structures. Considering n 2 polynomials P i in K n with coordinates x i, i = 1,..., n, where K is a eld of characteristic zero, for any polynomial λ K[x 1,..., x n ], the associated Jacobian Poisson is given by the bracket : {, } : K[x 1,..., x n ] K[x 1,..., x n ] K[x 1,..., x n ] 1

2 {f, g} = λ df dg dp 1... dp n 2 dx 1 dx 2... dx n, f, g K[x 1,..., x n ]. (2) Let us describe our basic Poisson algebra q 4,1 (E) in detail. Let P 1 = 1 2 (x2 0 + x 2 2) + kx 1 x 3, P 2 = 1 2 (x2 1 + x 2 3) + kx 0 x 2, where k C. The Poisson Sklyanin bracket is given on C[x 0, x 1, x 2, x 3 ] by the formula : {f, g} := df dg dp 1 dp 2 dx 0 dx 1 dx 2 dx 3. Then the brackets between the coordinate functions are dened by (mod 4) : {x i, x i+1 } = k 2 x i x i+1 x i+2 x i+3 ; {x i, x i+2 } = k(x 2 i+3 x 2 i+1), i = 0, 1, 2, 3. We will follow Sklyanin's version of this structure, considering the quadrics P 1, P 2 in C 4 and the complete intersection of these quadrics : P 1 = 1 2 (x2 1 + x x 2 3) (3) P 2 = 1 2 (x2 0 + J 1 x J 2 x J 3 x 2 3) (4) The bracket of this algebra q 4,1 (E) is given by the following formulas : {x 0, x i } = ( 1) i det( P k x l ), l 0, i; k = 1, 2; {x i, x j } = ( 1) i+j det( P k x l ), l i, j; k = 1, 2. In ( [26]), Michel Van den Bergh computes the Hochschild homology of Q 3,1 (E, η). The goal of this paper is to compute the Hochschild homology of Q 4,1 (E, η) using a method similar to that of Van den Bergh and Nicolas Marconnet ( [16]). The paper is organized as follows. We start by reviewing some general facts on the Hochschild homology, Poisson homology and, Koszul algebras : If A is an associative algebra over K, where K is a eld. One can dene the complex (C (A), b), where C n (A) = A (n+1), and the dierential is given by : n 1 b(a 0 a n ) = ( 1) n a n a 0 a n 1 + ( 1) i a 0 a i a i+1 a n. i=0 2

3 The homology of this complex is denoted by HH (A), and is called the Hochschild homology of A with coecients in A. On the other hand, considering a commutative k-algebra R, we say that R is a Poisson algebra if R is endowed with an antisymmetric biderivation {, } : R R R such that (R, {, }) is a Lie algebra. In other words, R is endowed with a K-bilinear map {, } : R R R such that : {a, bc} = {a, b}c + b{a, c} {a, b} = {b, a} {a, {b, c}} + {b, {c, a}} + {c, {a, b}} = 0 (Leibniz's rule); (antisymetry); (Jacobi's identity). where a, b, c R. The canonical or Poisson homology was introduced independently by Brylinski ( [3]) (as an important tool in computations of Hochschild and cyclic homology), and by Koszul and Gelfand-Dorfman (inspired by their algebraic approach to the study of bi-hamiltonian structures). Given a Poisson algebra (R, {, }). This homology is dened as the homology of a differential complex degree -1 {Ω (R), }, where the ( 1) dierential is given on the decomposable dierential forms as k (f 0 df 1... df k ) = ( 1) i+1 {f 0, f i }df 1... df i... df k + 1 i<j k 1 i k ( 1) i+j f 0 d{f i, f j } df 1... df i... df j... df k and acts from Ω k (R) to Ω k 1 (R). In this rst section, we also introduce some generality about Koszul algebras. The next part is devoted to generalities on associative elliptic algebra Q 4,1 (E, η), also called the Sklyanin algebra. More explicitly, considering V a four dimensional C-vector space with S 0, S 1, S 2, S 3 as a basis. The Sklyanin algebra algebra Q 4,1 (E, η) = T (V )/(I 2 ) where (I 2 ) is a bilateral ideal generated by the subspace I 2 V V with basis : f 0i = [S 0, S i ] α i (S j S k + S k S j ); f jk = [S j, S k ] (S 0 S i + S i S 0 ), α 1, α 2, α 1 C, and (i, j, k) is a cyclic permutation of (1, 2, 3). Smith and Stanford proved in ( [22]), this algebra is a Koszul algebra. We use that to compute two of Hochschild : q(π) = 3(S 2 S 3 f 01 + S 3 S 1 f 02 + S 1 S 2 f 03 + S 0 S 1 f 23 ) Q 4,1 (E, η) 4 ; q( ) = 1 q(π) Q 4,1 (E, η) 5. 3

4 The last and the main part of this paper is devoted to the computation of the Hochschild homology. We obtain the following result : Theorem 0.1. HH 4 (Q 4,1 (E, η)) is a free C[ P 1, P 2 ]-module of rank 1 generated by the homogeneous element of degree 4. HH 3 (Q 4,1 (E, η)) is a free C[ P 1, P 2 ]-module of rank 1 generated by the homogeneous element Π of degree 4. HH 2 (Q 4,1 (E, η)) is a free C[ P 1, P 2 ]-module of rank 6 generated by homogeneous elements of respective degrees 3, 3, 3, 3, 4, 4. HH 1 (Q 4,1 (E, η)) is a free C[ P 1, P 2 ]-module of rank 13 generated by homogeneous elements of respective degrees 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4. HH 0 (Q 4,1 (E, η)) is a free C[ P 1, P 2 ]-module of rank 7 generated by homogeneous elements of respective degrees 0, 1, 1, 1, 1, 2, 2. Here C[ P 1, P 2 ] is the center of algebra Q 4,1 (E, η). Corollary 0.1. As C-vector spaces, the homological groups HH i (Q 4,1 (E, η)) have the following Poincaré series : P (P (HH 0 (Q 4,1 (E, η)), t) = 2t2 +4t+1 P (HH 1 (Q 4,1 (E, η)), t) = t4 +4t 3 +4t 2 +4t P (HH 2 (Q 4,1 (E, η)), t) = 2t4 +4t 3 P (HH 3 (A h ), t) = t 4 P (HH 4 (Q 4,1 (E, η)), t) = t 4 (1 t 2 ) 2. Acknowledgements. The author is grateful to Jean-Claude Thomas for useful comments and discussions. This work is a part of my thesis prepared at the University of Angers. I would like to take this opportunity to thank my advisors, Vladimir Roubtsov and Bitjong Ndombol, for suggesting that I tackle this interesting problem and I also thank them for being available during this projet. This work was partially supported by the Programme SARIMA. 4

5 1 Preliminary facts 1.1 Hochschild homology Let A be an associative algebra over K, where K is a eld. One can dene the complex (C (A), b) by : C n (A) = A (n+1) ; n 1 b(a 0 a n ) = ( 1) n a n a 0 a n 1 + ( 1) i a 0 a i a i+1 a n The Hochschild homology of A with coecients in A is the homology of the complex (C (A), b). This homology is denoted by HH (A). We denote by B Connes's coboundary C(A) C(A) dened as follows : B(a 0 a n ) = i=0 n n ( 1) ni 1 a i a n a 0 a i 1 +( 1) n ( 1) ni a i a n a 0 a i 1 1. i=0 We have b B + B b = Poisson homology A Poisson bracket on a commutative algebra R is an antisymmetric biderivation {, } : R R R such that (R, {, }) is a Lie algebra. Then (R, {, }) is called a Poisson algebra. Let us consider n 2 polynomials Q i in K[x 1,, x n ], where K is a eld of characteristic zero. For any polynomial λ K[x 1,..., x n ], we can dene a bilinear dierential operation : i=0 by the formula {, } : K[x 1,..., x n ] K[x 1,..., x n ] K[x 1,..., x n ] {f, g} = λ df dg dq 1... dq n 2 dx 1 dx 2... dx n, f, g K[x 1,..., x n ] (5) This operation gives a Poisson algebra structure on K[x 1,..., x n ]. The polynomials Q i, i = 1,..., n 2 are Casimir functions for the brackets (5) and any Poisson structure in K n with n 2 generic Casimirs Q i is written in this form. Every Poisson structure of this form is called a Jacobian Poisson structure (JPS). The case n = 4 in (5) corresponds to the classical (generalized) Sklyanin quadratic Poisson algebra. The real Sklyanin algebra is associated with the following two quadrics in K 4 : Q 1 = x x x 2 3 (6) Q 2 = x J 1 x J 2 x J 3 x 2 3 (7) Let {, } : R R R be a Poisson bracket on an algebra R. The Poisson boundary operator, also called the Brylinski or Koszul dierential and denoted by : Ω (R) Ω 1 (R) 5

6 is given by : k (F 0 df 1... df k ) = + 1 i<j k 1 i k ( 1) i+1 {F 0, F i }df 1... df i... df k ( 1) i+j F 0 d{f i, F j } df 1... df i... df j... df k where F 0,..., F k R. One can check, by a direct computation, that k is well-dened and that it is a boundary operator, k k+1 = 0. The homology of the complex (Ω (R), ), denoted by P H (R, ), is called the Poisson homology associated with the Poisson bracket {, }. In ( [26]) Michel Van den Bergh computes the Poisson homology of q 3,1 (E) which is the jacobian Poisson structure given by the polynomial Q = 1 3 (x3 1 + x3 2 + x3 3 ) + kx 1x 2 x 3. He obtains the following result : Theorem 1.1. The homological groups P H i (R, ), i = 0, 1, 2, 3, are free K[Q]-modules of ranks 8, 8, 1, 1. P H 2 (R, ) is generated by x 1 dx 2 dx 3 + x 2 dx 3 dx 1 + x 3 dx 1 dx 2 and P H 3 (R, ) is generated by dx 1 dx 2 dx 3. P H 0 (R, ) and P H 1 (R, ) have the following Poincaré series : (1 + t)3 P (P H 0 (R, ), t) = 1 t 3 P (P H 1 (R, ), t) = (1 + t)3 1 t 3 1 Using the similar method as Van den Bergh, Nicolas Marconnet computes the Poisson homology of a cubic Jacobian Poisson structure on the polynomial algebra K[x 1, x 2, x 3 ] given by a polynomial φ = 1 2 x2 3 + q 1 4 x q 1 4 x q 1 2 x2 1 x2 2. He obtains the following result : Theorem 1.2. ( [16]) The homological groups P H i (R, ), i = 0, 1, 2, 3, are free K[φ]- modules of ranks 9, 9, 1, 1. P H i (R, ) have the following Poincaré series : P (P H 0 (R, ), t) = t4 +2t 3 +3t 2 +2t+1 1 t 4 ; P (P H 1 (R, ), t) = 2t4 +2t 3 +3t 2 +2t 1 t 4 P (P H 2 (R, ), t) = t4 1 t 4 ; P (P H 3 (R, ), t) = t4 1 t 4. The two previous cases have some common properties : they are Jacobian Poisson structures in dimension three given by a weight homogeneous polynomial with an isolated singularity. In ( [19]), Anne Pichereau computes the Poisson homology of Jacobian Poisson structures in dimension three given by a weight homogeneous polynomial with an isolated singularity. In our article ( [25]), we did the same work as her but in dimension four : we computed the Poisson homology of a Jacobian Poisson structure in dimension four given 6

7 by weight homogeneous polynomials Q 1 and Q 2 which form a complete intersection. We proved that the Poincaré series of these homological groups depend only on the weights and the degrees of Q 1 and Q 2. In the quadratic homogeneous case, we obtained the following result : Theorem 1.3. The Poincaré series of the Poisson homological groups of Jacobian Poisson structures in dimension four given by quadratic homogeneous polynomials Q 1 and Q 2 which form a complete intersection, P H i (R), ), i = 0, 1, 2, 3, 4 have : P (P H 0 (R, ), t) = 2t2 +4t+1 P (P H 1 (R, ), t) = t4 +4t 3 +4t 2 +4t P (P H 2 (R, ), t) = 2t4 +4t 3 P (P H 3 (R, ), t) = t 4 P (P H 4 (R, ), t) = t 4 (1 t 2 ) 2. In Sklyanin's case, we obtained the following explicit result : Theorem 1.4. ( [25]) 1. The homological group P H 0 (R, ) is a rank 7 free K[Q 1, Q 2 ]-module generated by (µ i ) 0 i 6 = (1, x 1, x 2, x 3, x 4, x 2 1, x2 3 ); 2. P H 1 (R, ) is a free K[Q 1, Q 2 ] module given by : 6 5 P H 1 (R, ) ( K[Q 1, Q 2 ]dµ k ) ( K[Q 1, Q 2 ]µ k dq 1 ) K[Q 1, Q 2 ]dq 1 K[Q 1, Q 2 ]dq 2 ; k=1 k=1 3. P H 2 (R, ) is a free K[Q 1, Q 2 ] module given by : ( 5 ) P H 2 (R, ) k=1 K[Q 1, Q 2 ](dµ k dq 1 ) K[Q 1, Q 2 ](dq 1 dq 2 ); 4. P H 3 (R, ) is a rank 1 free K[Q 1, Q 2 ]-module generated by π; 5. P H 4 (R, ) is a rank 1 free K[Q 1, Q 2 ]-module generated by δ, where δ = dx 1 dx 2 dx 3 dx 4, and π = x 1 dx 2 dx 3 dx 4 + x 2 dx 3 dx 1 dx 4 + x 3 dx 1 dx 2 dx 4 + x 4 dx 2 dx 1 dx 3. 7

8 1.3 Generalities on Koszul algebras Let V be a nite -dimensional K-vector space ant let T (V ) be the tensor algebra of V over k. Consider a quadratic K-algebra A = T (V )/(W ), where W W W. Let W be the dual space of W and W W W be the orthogonal of W. The dual algebra of A is dened as A! := T (V )/(W ). n 1 Let (x i ) i=0, xn 1 be a basis of V and (ζ i ) i=0, xn 1 its dual basis. Consider e = x i ζ i. We have e 2 = 0 ( [15]). Let K m (A) = A (A! m) and K (A) = m 0 K m (A). Then the right multiplication by e induces a map d : K m (A) K m 1 (A) and thus a dierential d : K A K A. The complex A = A k = A (A! 0) e (A! 1) e (A! 2) e (8) i=0 is called the Koszul complex of A. The augmented Koszul complex of A is the complex : 0 k ε A = A k = A (A! 0) e (A! 1) e (A! 2) e (9) where ε is the canonical projection. Denition 1.1. A quadratic algebra A is said to be a Koszul algebra if the augmented Koszul complex (9) is exact. Proposition 1.1. ( [26]) Suppose that A is a Koszul algebra. Then HH (A) = H (K(A), b). Hence when A is a Koszul algebra, we have two complexes which enable us to compute the Hochschild homology of A : (K(A), b) and (C(A), b). Let us give an explicit quasiisomorphism between them. Since (A! m) = i+j+2=m V i W V j ( [26]), we dene the map q : A (A! m) A A m as being the restriction of the natural inclusion A V m A A m. Proposition 1.2. ( [26]) q : K(A) C(A) is a quasi-isomorphism. 2 Generalities on non commutative Sklyanin algebras Here, we follow the initial description of Sklyanin algebras by Sklyanin in ( [21]). Let τ C so that Im(τ) > 0. Consider Γ = Z Zτ. Let θ 00, θ 01, θ 10, θ 11 be Jacobi's theta functions associated with Γ, as described in ( [22]). These functions satisfy the following properties : θ ab (z + 1) = ( 1) a θ ab (z); θ ab (z + τ) = e ( πiτ 2πiz πib) θ ab (z); 8

9 and the zeros of θ ab are the points : 1 2 (1 b) + (1 + a)τ + Γ. Fix η C such that η is not of order 4 in C/Γ. Let (ab, ij, kl) be a cyclic permutation of (00, 01, 10, 11). Suppose : [ ] α ab = ( 1) a+b θ11 (η)θ ab (η) 2. θ ij (η)θ kl(η) Consider V a four dimensional vector space with S 0, S 1, S 2, S 3 as a basis. We dene A = T (V )/(I 2 ) where (I 2 ) is the two-sided ideal generated by the subspace I 2 V V with basis : f 0i = [S 0, S i ] α i (S j S k + S k S j ); f jk = [S j, S k ] (S 0 S i + S i S 0 ), α 1 = α 00, α 2 = α 01, α 1 = α 10, and (i, j, k) is a cyclic permutation of (1, 2, 3). A is called the Sklyanin algebra. We have the following results from the paper of S.P. Smith and J.T. Staord ( [22]) : Proposition 2.1. ( [22]) A is a Koszul algebra. Therefore the Hochschild homology of A is given by the complex (K(A), b), where b is the Hochschid boundary. Proposition 2.2. ( [22]) For m 5, A! m = 0 and if m 4, A! m is spanned by the following elements : A! 0 : 1 A! 1 : ζ 0, ζ 1, ζ 2, ζ 3 A! 2 : ζ 0ζ 1, ζ 0 ζ 2, ζ 0 ζ 3, ζ 1 ζ 0, ζ 2 ζ 0 A! 3 : ζ 0ζ 1 ζ 0, ζ 0 ζ 2 ζ 0, ζ 0 ζ 3 ζ 0, ζ 1 ζ 0 ζ 1 A! 4 : ζ 0ζ 1 ζ 0 ζ 1 where ζ 0, ζ 1, ζ 2, ζ 3 is the dual basis of S 0, S 1, S 2, S 3. These elements form a basis for A! and in particular dima! m = ( 4 m). We also have the following Poincaré duality : (A! m) = A! 4 m. If we consider K m (A) as free A-module, the Koszul complex of the algebra A has the following form : where : 0 A t S 4 N S 6 M S 4 x S ε C 0. (10) x is a right multiplication by x = (S 0, S 1, S 2, S 3 ) T ; M is the right multiplication by a matrix M obtained from the relations f 0i, f jk of A : M = S 1 S 0 α 1 S 3 α 1 S 2 S 1 S 0 S 3 S 2 S 2 α 3 S 3 S 0 α 2 S 1 S 2 S 3 S 0 S 1 S 3 α 3 S 2 α 3 S 1 S 0 S 3 S 2 S 1 S 0 9 ;

10 N is the right multiplication by the matrix N given by : 2S 1 2S 2 2S N = 1 0 (1 α 2 )S 3 (1 + α 3 )S 2 2S 0 (1 + α 2 )S 3 (1 α 3 )S 2 2 (1 + α 1 )S 3 0 (1 α 3 )S 1 (1 α 1 )S 3 2S 0 (1 + α 3 )S 1 ; (1 α 1 )S 2 (1 + α 2 )S 1 0 (1 + α 1 )S 2 (1 α 2 )S 1 2S 0 and t is the right multiplication by t = (S 0, S 1, S 2, S 3 ). We denote by the element 1 K 4 (A) and by Π the element (S 0, S 1, S 2, S 3 ) K 3 (A). We have b( ) = b(π) = 0 and : q(π) = 3(S 2 S 3 f 01 + S 3 S 1 f 02 + S 1 S 2 f 03 + S 0 S 1 f 23 ) A 4 ; q( ) = 1 q(π) A 5. Hence Π and dene respectively elements of HH 3 (A) and HH 4 (A) which we will also denote using the same letters. 3 Hochschild homology of the Sklyanin algebra A We suppose that α 1, α 2, α 3 are algebraically independent over Q. Proposition 3.1. as graded C-algebras, and A = C Q(α1,α 2,α 3 ) (Q(α 1, α 2, α 3 ) S 0, S 1, S 2, S 3 /(I 2 )) HH (A) = C Q(α1,α 2,α 3 ) HH (Q(α 1, α 2, α 3 ) S 0, S 1, S 2, S 3 /(I 2 )), as graded C-vector spaces. Proof. The rst isomorphism is a direct consequence of the fact that the relations f 0i = 0, f jk = 0 of the algebra A have their coecients in the eld Q(α 1, α 2, α 3 ). Then we can easily construct a C-algebras isomorphism : with C Q(α1,α 2,α 3 ) (Q(α 1, α 2, α 3 ) S 0, S 1, S 2, S 3 /I 2 ) A, α a αa A C Q(α1,α 2,α 3 ) (Q(α 1, α 2, α 3 ) S 0, S 1, S 2, S 3 /I 2 ), S i 1 S i as the inverse. The second isomorphism is a consequence of the following commutative diagram : A n b A (n 1) C Q(α1,α 2,α 3) (Q(α 1, α 2, α 3 ) S 0, S 1, S 2, S 3 /I 2 ) n 1 C Q(α1,α 2,α 3 )b C Q(α1,α 2,α 3) (Q(α 1, α 2, α 3 ) S 0, S 1, S 2, S 3 /I 2 ) (n 1) where b is the Hochschild boundary. 10

11 Proposition 3.2. There exists an extension k 0 ((h)) of the eld Q(α 1, α 2, α 3 ) such that in this extension α i = β i h 2, where coecients β 1, β 2, β 3 belong to the eld k 0 and are also algebraically independent over Q. Proof. Let us set k 0 = Q(X, Y, Z) where X, Y, Z are variables. The morphism Q(α 1, α 2, α 3 ) k 0 ((h)), α 1 X, α 2 Y, α 3 Z dene an injection. Then we conclude by setting β 1 = X, β 2 = Y, β 3 = Z. Proposition 3.3. The algebra k 0 ((h)) S 0, S 1, S 2, S 3 /(I) is isomorphic to k 0 ((h)) Q(α1,α 2,α 3 ) (Q(α 1, α 2, α 3 ) S 0, S 1, S 2, S 3 /(I 2 )) as graded k 0 ((h))-algebras, while the vector space is isomorphic to HH (k 0 ((h)) S 0, S 1, S 2, S 3 /(I)) k 0 ((h)) Q(α1,α 2,α 3 ) HH (Q(α 1, α 2, α 3 ) S 0, S 1, S 2, S 3 /(I 2 )) as graded k 0 ((h))-vector spaces. Here (I) is the two-sided ideal generated by the subspace I V V with basis : F 0i = [S 0, S i ] β i h 2 (S j S k + S k S j ); F jk = [S j, S k ] (S 0 S i + S i S 0 ), and (i, j, k) is a cyclic permutation of (1, 2, 3). In particular we have HH (Q(α 1, α 2, α 3 ) S 0, S 1, S 2, S 3 /(I 2 )) = Q(α 1, α 2, α 3 ) k0((h)) HH (k 0 ((h)) S 0, S 1, S 2, S 3 /(I)). Therefore the computation of Hochschild's homology of algebra A is equivalent to nding the homology of algebra k 0 ((h)) S 0, S 1, S 2, S 3 /(I)). Let us introduce new variables by setting x 0 = h 1 S 0 ; and x i = S i for i = 1, 2, 3. We denote by A h the k 0 ((h))-algebra generated by x 0, x 1, x 2, x 3 with the relations g 0i = [x 0, x i ] β i h(x j x k + x k x j ); g jk = [x j, x k ] h(x 0 x i + x i x 0 ), where (i, j, k) is a cyclic permutation of (1, 2, 3). We also denote by (I) the two-sided ideal generated by g 0i, g jk. 3.1 Filtration on A h Let us denote by P the vector-subspace of k 0 x 0, x 1, x 2, x 3 generated by the ordered monomials x i 0 0 x i 1 1 x i 2 2 x i 3 3. Using the same method as Marconnet in ( [16]), we obtain the following results : 11

12 Consider the following commutative diagram where arrows are injective morphisms of k 0 [[h]]-algebras : k 0 ((h)) x 0, x 1, x 2, x 3 j 1 k 0 [[h]] x 0, x 1, x 2, x 3 j 2 j 3 k 0 x 0, x 1, x 2, x 3 ((h)) j 4 k 0 x 0, x 1, x 2, x 3 [[h]] We denote by k 0 x 0, x 1, x 2, x 3 i the homogeneous component of degree i of the graded k 0 -algebra k 0 x 0, x 1, x 2, x 3. We can observe that k 0 [[h]] x 0, x 1, x 2, x 3 is a subalgebra of k 0 x 0, x 1, x 2, x 3 [[h]] whose elements are all formal series n N Q n h n which satisfy : there exists N N such that for all n N, Q n N i=0k 0 x 0, x 1, x 2, x 3 i. In the same way k 0 ((h)) x 0, x 1, x 2, x 3 is a subalgebra of k 0 x 0, x 1, x 2, x 3 ((h)) whose elements are all Laurent series n Z Q n h n which satisfy : there exists N Z such that for all n Z, Q n N i=0k 0 x 0, x 1, x 2, x 3 i. On the other hand, we have the following commutative diagram : A h = k 0 ((h)) x 0, x 1, x 2, x 3 /(I) i 1 A h = k 0 [[h]] x 0, x 1, x 2, x 3 /(I) i 2 i 3 k 0 x 0, x 1, x 2, x 3 ((h))/(i) = Âh i 4 k 0 x 0, x 1, x 2, x 3 [[h]]/(i) = Âh Proposition 3.4. The morphisms i 1, i 2, i 3, i 4 are injective. Proposition 3.5. For all a Âh, there exists a unique family {P i P, i N} such that a = i N P i h i. Proposition 3.6. For all a Âh, there exists a unique family {P i P, i Z} bounded from below such that a = i ZP i h i. Corollary 3.1. For all a A h, there exists a unique family {P i P, i Z} bounded from below such that a = i ZP i h i. For all p Z, we dene a k 0 -subspace F p (Âh) of Âh by F p (Âh) = {a Âh : a = P i h i, P i P} Then F p (Âh) F p+1 (Âh); 1 F 0 (Âh) and for all p, q Z, it is easy to see that the product of A h induces an application F p (Âh) F q (Âh) F p+q (Âh). Thus F is a ltration on  h. Since Âh = p Z F p (Âh) and p Z F p (Âh) = 0 (Proposition 3.6) the ltration F on i p 12

13 Â h is exhaustive and separated. By (3.4) the restriction of F to the subalgebra A h is still an exhaustive separated ltration of the algebra A h. This ltration is compatible with the weight graduation of A h : F p (A h ) (A h ) n = {a A h : a = P i h i, P i P n }. Proposition 3.7. The algebra Âh (respectively Âh) is a completion of A h (respectively A h ) for the topology induced by the ltration F. The ltred algebra A h is not complete. However each homogeneous component (A h ) n, n N, is complete. Therefore the graded ring gr F (A h ) associated with the ltration F is gr p (A h ) = F p (A h )/F p 1 (A h ) = h p P. On the ordered basis of gr p (A h ) the product gr p (A h ) gr q (A h ) gr p+q (A h ) can be written as follows : i p (h p x i 0 0 x i 1 1 x i 2 2 x i 3 3, h q x j 0 0 x j 1 1 x j 2 2 x j 3 3 ) h p q x i 0+j 0 0 x i 1+j 1 1 x i 2+j 2 2 x i 3+j 3 3. The graded ring gr F (A h ) can be identied with k 0 [x 0, x 1, x 2, x 3 ][h, h 1 ]. 3.2 The algebra A h seen as a deformation Let p be the projection p : Â h R = k 0 k0 [[h]] A h, a 1 a. Kerp = ha h = F 1 A h. Consider the commutator bracket [, ] : A h A h A h. The terms [x i, x j ] can be written as formal series in h with no constant coecients in A h. Therefore the map h 1 [, ] : A h A h A h is a biderivation which satises the Jacobian identity. Let us consider the Sklyanin Poisson bracket : dened by {, } : R R R {x 0, x i } = 2β jk x j x k where (i, j, k) is a cyclic permutation of (1, 2, 3). We have the following commutative diagram : {x j, x k } = 2x 0 x i (11) A h A h 1 h [, ] p p A h R R {, } R Using the ordering decomposition of elements of Âh, we dene the following isomorphism of k 0 ((h)) graded vector spaces : Θ : Â h R((h)) i h i pp i P i h i i p 13 p

14 R((h)) is endowed with the natural increased ltration and the associated topology. We have Θ(F p (Âh) = F p (R((h))). Thus Θ is a homeomorphism with respect to the associated topologies. The product on Âh is carried to a product on R((h)) such that the following diagram commutes :  h Âh  h Θ Θ R((h)) R((h)) Θ R((h)) Therefore the algebra Âh is isomorphic to R((h)) endowed with the product. For a, b R, we can write a b = i 0h i µ i (a, b), where µ i : R R R are k 0 -bilinear maps. On the other hand, from the relations of Âh, if a, b P = R then ā b = ab + h( ), where ab is the product of R. Thus Âh is a deformation of the polynomial algebra R. The bracket 1 h [, ] of Âh is transported to a bracket 1 h [, ] on R((h)). The bracket associated with the deformation, R R R, (a, b) µ 1 (a, b) µ 1 (b, a) coincides with the Sklyanin Poisson bracket {, }. Notice that the subalgebras of Âh which appear in the following commutative diagram i 2 A h  h i 1 i 4 A h i 3  h are identied via the isomorphism Θ with the subalgebras of (R((h)), ) which appear in the following commutative diagram : k 0 ((h))[x 0, x 1, x 2, x 3 ] k 0 [x 0, x 1, x 2, x 3 ]((h)) k 0 [[h]][x 0, x 1, x 2, x 3 ] k 0 [x 0, x 1, x 2, x 3 ][[h]] Thus the algebra A h is identied with the algebra (k 0 ((h))[x 0, x 1, x 2, x 3 ], ). In other words, A h is a subalgebra of a deformation of R. 3.3 Hochschild homology of algebra A h The ltration F on A h can be extended to a ltration on C(A h ), also denote by F in the following way : for all n N and for all p Z, we set F p (A n h ) = p 1 + +p n=pf p1 (A h ) F pn (A h ). The ltration thus obtained is exhaustive and separated. The Hochschild boundary respects this ltration. Thus we can dene the graded complex (gr F (C(A h ), b). One has the natural isomorphism gr F (A n h ) = gr F (A h ) n for all n 1. Then we deduce an isomorphism in homology : H (C(A h ), b) = HH (gr F (A h )). 14

15 Let us just recall general notation on spectral sequence. Let (C, d) be a chain complex and F be an increased ltration on C, compatible with d (d(f p C) F p C). Let p, q Z. We set E 0 p,q = F p C p+q /F p 1 C p+q. Consider the canonical projection : We dene for each r N : η p,q : F p C p+q F p C p+q /F p 1 C p+q = E 0 p,q. A r p,q = {c F p C p+q : d(c) F p r C p+q 1 }, Z r p,q = η p,q (A r p,q), B r p,q = η p,q (da r 1 p+r 1,q r+2 ). We set Ep,q r = Zp,q/B r p,q, r in particular Ep,q r = H p+q F p C p+q /F p 1 C p+q. Let Zp,q = Zp,q, r Bp,q = Bp,q r and Ep,q = Zp,q/B p,q. r=1 r=1 We have the following increasing inclusions : 0 = B 0 p,q B r p,q B p,q z p,q Z r p,q Z 0 p,q = E 0 p,q For all r N the map d induces the dierential complex d r : Ep,q r Ep r,q+r 1 r such that Ep,q r+1 = Kerd r p,q/imd r p+r,q r+1. The spectral sequence E r is said to be regular if there exists r 0 N such that E r = E r 0 for r r 0. And we say that the spectral sequence E r converges weakly to H (C) if Ep,q = F p H p+q /F p 1 H p+q. We have the following result on weak convergence : Proposition 3.8. ( [28]) Let (C, d) be a chain complex endowed with an increasing complete and exhaustive Z-ltration, compatible with the dierential d. If the associated spectral sequence is regular, then this spectral sequence converges weakly to H (C, d). Let A p,q = Ker{d : F p C p+q F p C p+q 1 } and consider the composite Φ r p,q : A p,q A r p,q Z r p,q E r p,q. We are using the following standard criterion for degeneration of spectral sequence : Lemma 3.1. ( [26]) If there exists r such that Φ r p,q is surjective for all p, q, then the spectral sequence degenerates at E r. Let us now apply these results to our situation. Consider the spectral sequence E r associated with the ltration on C(A h ). Theorem 3.1. The spectral sequence E r associated with the ltration F weakly converges to the Hochschild homology HH (A h ) of A h. 15

16 Proof. For all n N, we denote by C(A h ) n the sub-complex of C(A h ) which is formed by the homogeneous components of degree n for the initial graduation (weight graduation) A h. Since the dierential b preserves the graduation of C(A h ), we have C(A h ) = n N C(A h ) n. The complex (C(A h ) n, b) is a complex of nite dimensional vector spaces on which the ltration F is complete and exhaustive. Let p Z and q N. gr p (C q (A h ) n ) = (k 0 [x 0, x 1, x 2, x 3 ] (q+1) ) n h p. Hence we have an isomorphism of k 0 [h, h 1 ]-modules : gr(c(a h ) n ) = C(k 0 [x 0, x 1, x 2, x 3 ]) n [h, h 1 ]. We now consider the spectral sequence associated with the ltration F on C(A h ) n. E 0 p,q = F p (C p+q (A h ) n /F p 1 (C p+q (A h ) n = h p C p+q (k 0 [x 0, x 1, x 2, x 3 ]) n. Therefore Ep,q 0 = Zp,q 0 is a k 0 -vector space of nite dimension. Since we have an innite sequence of inclusions Zp,q r Zp,q 1 Zp,q, 0 the k 0 -vector subspaces Zp,q r of Zp,q 0 are nite dimensional. Thus there exists r 0 N such that Zp,q r = Z r 0 p,q for all r r 0. Using the Proposition (3.8), we conclude that the spectral sequence associated with the ltration F on the complex C(A h ) n converges weakly to H (C(A h ) n ) : Since C(A h ) = n NC(A h ) n, E p,q(c(a h ) n ) = F p (C p+q (A h ) n )/F p 1 (C p+q (A h ) n, b). E p,q(c(a h )) = ni nnf p (C p+q (A h ) n )/F p 1 (C p+q (A h ) n ) = F p (C p+q (A h ))/F p 1 (C p+q (A h )) = F p HH p+q (A h )/F p 1 HH p+q (A h ). As a consequence, the spectral sequence associated with the ltration F on C(A h ) converges weakly to HH (A h ). Since the graded ring gr F (A h ) associated with the ltration F on A h is a polynomial algebra with coecients in the ring k 0 [h, h 1 ], by the Hochschild-Kostant-Rosenberg theorem, we have for all n N, a quasi-isomorphism of k 0 [h, h 1 ]-modules : C n (gr F (A h )) Ω n gr F (A h ) k0 [h,h 1 ] r 0 r 1 r n 1 n! r 0dr 1 dr n. Here Ω n gr F (A h ) is the k 0 [h, h 1 ]-module of dierential forms of degree n of gr F (A h ) k0 [h,h 1 ] on k 0 [h, h 1 ], with zero dierential. Hence HH n gr F (A h ))Ω n gr F (A h ) Under this quasi- k0 [h,h ]. 1 isomorphism, the Connes's coboundary corresponds to the de Rham dierential. On the other hand, we have a canonical isomorphism of k 0 [h, h 1 ]-modules Ω gr F (A h ) = k0 [h,h 1 ] Ω R k0 k0 k 0 [h, h 1 ]. We will denote Ω R k0 by Ω (R). We borrow the following commutative diagram from brylinski's paper ( [3]) : E 1 n = HH n (gr F (A h )) d 1 E 1 n+1 = HH n 1(gr F (A h )) = Ω n (R) k0 k 0 [h, h 1 ] h = Ω n 1 (R) k0 k 0 [h, h 1 ] 16

17 where d 1 is the dierential which computes the second term E 2 of the spectral sequence, h is the multiplication by h and is the boundary Poisson operator associated with the Sklyanin Poisson bracket : n (F 0 df 1... df n ) = ( 1) i+1 {F 0, F i }df 1... df i... df n + ( 1) i+j F 0 d{f i, F j } df 1... df i... df j... df n where F 0,..., F n R. Using the isomorphism E 1 p,q = Ω p+q (R) k0 h p, the rst term of this spectral sequence can be explained as follow : Ω 1 (R) h 3 R h q = 2 Ω 2 (R) h 3 Ω 1 (R) h 2 R h q = 1 Ω 3 (R) h 3 Ω 2 (R) h 2 Ω 1 (R) h 1 R q = 0 Ω 4 (R) h 3 Ω 3 (R) h 2 Ω 2 (R) h 1 Ω 1 (R) 1 R h 0 q = 1 0 Ω 4 (R) h 2 Ω 3 (R) h 1 Ω 2 (R) 1 Ω 1 (R) h R h 2 q = 2 p = 3 p = 2 p = 1 p = 0 p = 1 p = 2 The second term E 2 of the spectral sequence is given by the homology of the lines with respect to the dierential h. Since the multiplication by h is a k 0 [h, h 1 ]-isomorphism, to have this second term, we only have to nd the Poisson homology : 0 Ω 4 (R) 4 Ω 3 (R) 3 Ω 2 (R) 2 Ω 1 (R) 1 R (12) Theorem 1.4 gives this Poisson homology. Proposition 3.9. The spectral sequence associated with the ltration F degenerates at E 2. Proof. This spectral sequence degenerates at E 2 if the map Φ 2 p,q is surjective for all p, q Z. But the columns of E 2 are the same up to a multiplication by h. Thus we only have to give a proof for Φ 2 0,q, q Z. E0,0 2 = P H 0 (R, ) is the quotient of R. Let v E0,0 2. Then v can be lifted to an element u of R = F 0 A h /F 1 A h. On other hand, u can be lifted to an element U of F 0 A h and Φ 2 0,0 (U) = v. Then let P i F 0 (A h ), i = 1, 2 be an element which lifts P i, where P 1, P 2 are Casimirs which give the Sklyanin Poisson bracket. Since k 0 [P 1, P 2 ] is the center of the Sklyanin Poisson algebra (R, {, }), k 0 [ P 1, P 2 ] is the center of algebra A h. We endow A h with a natural structure of k 0 [ P 1, P 2 ]-module. Then using the result (1.4), as a k 0 [P 1, P 2 ]-module, E0,1 2 = P H 1 (R, ) is generated by the class an element f(p 1, P 2 )dψ Ω 1 (R), where d is the de Rham dierential, ψ R. Let Ψ F 0 (A h ) be an element which lifts ψ. B(Ψ) F 0 (A 2 h ) and b(b(ψ)) = B(b(Ψ)) = 0. We have Φ 2 0,1 (B(Ψ)) = dψ. 17

18 Let v E0,2 2 = P H 2 (R, ). From (1.4), v is the class of an element f(p 1, P 2 )dp 1 dψ, f(p 1, P 2 ) k 0 [P 1, P 2 ]. Since P 1 dψ P H 1 (R, ), there exists Ψ A 0,1 such that Φ 2 0,1 (Ψ) = P 1dψ. We have b(ψ) = 0 and therefore b(b(ψ)) = 0. Φ 2 0,2 (B(Ψ)) = dp 1 dψ. Then f( P 1, P 2 )B(Ψ) lifts v. Since the image of the Hochschild's cycle Π A 4 h in gr F (A 4 h ) is the generator π of P H 3 (R, ) = E0,3 2, as a free k 0[P 1, P 2 ]-module, f( P 1, P 2 )Π lifts the element f(p 1, P 2 )π E0,3 2. Similarly, since the image of the Hochschild's cycle A 5 h in gr F (A 5 h ) is the generator δ of P H 4 (R, ) = E0,4 2, as a free k 0[P 1, P 2 ]-module, f( P 1, P 2 )Π lifts the element f(p 1, P 2 )δ E0,4 2. Since the spectral sequence E r weakly converges to HH (A h ), and identifying E 2 and P H (R, ) k0 k 0 [h, h 1 ], we have : gr F (HH i (A h )) = P H i (R, ) k0 k 0 [h, h 1 ]. We have the following classical result : Let R be an k-algebra endowed with a N-graduation and a Z-increasing ltration F. By setting F p (R n ) = F p (R) R n, we naturally have a ltration on R n. We assume that the ltration F and the graduation of R are compatible ie., F p (R) = n N F p (R n ). In this case F p (R)/F p 1 (R) = ( n N F p (R n )/F p 1 (R n )) and gr F (R) = n N gr F (R n ). Proposition [16] Let M be an R-module endowed with a ltration F and a graduation which are compatible and such that F is exhaustive and separated. Assume that the ltration on R n is complete for N N. If gr F (M) is a free graded gr F (R)-module of rank m with a basis { x 1,, x m }, where for all i, x i is the class of a homogeneous element x i F 0 (M) in the graded associated, then M is a free graded R-module with {x 1,, x m } as a basis. In our case R = k 0 ((h))[ P 1, P 2 ] endowed with the ltration of A h and M = HH i (A h ) endowed with the ltration of C(A h ). The ltration F on HH i (A h ) (respectively on R) is exhaustible, separated and compatible with the graduation. On the other hand the ltration F on R n is complete for all n N. Theorem 3.2. HH 4 (A h ) is a free k 0 ((h))[ P 1, P 2 ]-module of rank 1 generated by the homogeneous element of degree 4. HH 3 (A h ) is a free k 0 ((h))[ P 1, P 2 ]-module of rank 1 generated by the homogeneous element Π of degree 4. 18

19 HH 2 (A h ) is a free k 0 ((h))[ P 1, P 2 ]-module of rank 6 generated by homogeneous elements of respective degrees 3, 3, 3, 3, 4, 4. HH 1 (A h ) is a free k 0 ((h))[ P 1, P 2 ]-module of rank 13 generated by homogeneous elements of respective degrees 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4. HH 0 (A h ) is a free k 0 ((h))[ P 1, P 2 ]-module of rank 7 generated by homogeneous elements of respective degrees 0, 1, 1, 1, 1, 2, 2. We can also deduce the following result : Corollary 3.2. As k 0 ((h))-vector spaces, the homological groups HH i (A h ) have the following Poincaré series : References P (P (HH 0 (A h ), t) = 2t2 +4t+1 P (HH 1 (A h ), t) = t4 +4t 3 +4t 2 +4t P (HH 2 (A h ), t) = 2t4 +4t 3 P (HH 3 (A h ), t) = t 4 P (HH 4 (A h ), t) = t 4 (1 t 2 ) 2. [1] Michael Artin and William F. Schelter, Graded algebras of global dimension 3, Adv. in Math. 66 (1987), no. 2, [2] H. Boos, M. Jimbo, T. Miwa, F. Smirnov, and Y. Takeyama, Traces on the Sklyanin algebra and correlation functions of the eight-vertex model, J. Phys. A 38 (2005), no. 35, [3] Jean-Luc Brylinski, A dierential complex for Poisson manifolds, J. Dierential Geom. 28 (1988), no. 1, [4] V. G. Drinfel d, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations, Dokl. Akad. Nauk SSSR 268 (1983), no. 2, [5] David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995, With a view toward algebraic geometry. [6] Benoit Fresse, Théorie des opérades de Koszul et homologie des algèbres de Poisson, Ann. Math. Blaise Pascal 13 (2006), no. 2, [7] Viktor L. Ginzburg, Grothendieck groups of Poisson vector bundles, J. Symplectic Geom. 1 (2001), no. 1,

20 [8] Viktor L. Ginzburg and Jiang-Hua Lu, Poisson cohomology of Morita-equivalent Poisson manifolds, Internat. Math. Res. Notices (1992), no. 10, [9] Viktor L. Ginzburg and Alan Weinstein, Lie-Poisson structure on some Poisson Lie groups, J. Amer. Math. Soc. 5 (1992), no. 2, [10] G. Khimshiashvili, On one class of exact Poisson structures, Proc. A. Razmadze Math. Inst. 119 (1999), [11], On one class of ane Poisson structures, Bull. Georgian Acad. Sci. 161 (2000), no. 3, [12] G. Khimshiashvili and R. Przybysz, On generalized Sklyanin algebras, Georgian Math. J. 7 (2000), no. 4, [13] André Lichnerowicz, Les variétés de Poisson et leurs algèbres de Lie associées, J. Dierential Geometry 12 (1977), no. 2, [14] Yu. I. Manin, Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier (Grenoble) 37 (1987), no. 4, [15], Quantum groups and noncommutative geometry, Université de Montréal Centre de Recherches Mathématiques, Montreal, QC, [16] Nicolas Marconnet, Homologies of cubic Artin-Schelter regular algebras, J. Algebra 278 (2004), no. 2, [17] A. V. Odesski and B. L. Fe gin, Sklyanin's elliptic algebras, Funktsional. Anal. i Prilozhen. 23 (1989), no. 3, 4554, 96. [18] A. V. Odesski and V. N. Rubtsov, Polynomial Poisson algebras with a regular structure of symplectic leaves, Teoret. Mat. Fiz. 133 (2002), no. 1, 323. [19] Anne Pichereau, Poisson (co)homology and isolated singularities, J. Algebra 299 (2006), no. 2, [20] E. K. Sklyanin, Some algebraic structures connected with the Yang-Baxter equation, Funktsional. Anal. i Prilozhen. 16 (1982), no. 4, 2734, 96. [21], Some algebraic structures connected with the Yang-Baxter equation. Representations of a quantum algebra, Funktsional. Anal. i Prilozhen. 17 (1983), no. 4, [22] S. P. Smith and J. T. Staord, Regularity of the four-dimensional Sklyanin algebra, Compositio Math. 83 (1992), no. 3, [23], Regularity of the four-dimensional Sklyanin algebra, Compositio Math. 83 (1992), no. 3, [24] Joanna M. Staniszkis, The 4-dimensional Sklyanin algebra, J. Algebra 167 (1994), no. 1,

21 [25] Serge Roméo Tagne Pelap, Poisson (co)homology of polynomial poisson algebras in dimension four : Sklyanin's case, Preprint LAREMA (2008). [26] Michel Van den Bergh, Noncommutative homology of some three-dimensional quantum spaces, Proceedings of Conference on Algebraic Geometry and Ring Theory in honor of Michael Artin, Part III (Antwerp, 1992), vol. 8, 1994, pp [27] Pol Vanhaecke, Integrable systems in the realm of algebraic geometry, second ed., Lecture Notes in Mathematics, vol. 1638, Springer-Verlag, Berlin, [28] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, [29] Ping Xu, Poisson cohomology of regular Poisson manifolds, Ann. Inst. Fourier (Grenoble) 42 (1992), no. 4, [30], Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), no. 3, Laboratoire Angevin de Recherche en Mathématiques Université D'Angers Département de Mathématiques address : pelap@math.univ-angers.fr 21