Noncommutative Algebraic Geometry Shanghai September 12-16, 211 Calabi-Yau algebras linked to Poisson algebras Roland Berger (Saint-Étienne, France (jointly Anne Pichereau Calabi-Yau algebras viewed as deformations of Poisson algebras, arxiv:117.4472 Aim: define a family of 3-CY algebras B and compute Hochschild homology of some ones with the help of Poisson homology. PART I - The family of 3-CY algebras B Based on the family of the following algebras A: A = A(f = k x 1,...,x n /(f = f ij x i x j 1 i,j n Proposition (RB, JLMS 29. 1 The quadratic algebra A is Koszul. 2 The global dimension of A is equal to 2, except if f is symmetric of rank 1 (in this case, the global dimension is infinite. 3 A is AS-Gorenstein if and only if f is nondegenerate. 1
4 A is 2-Calabi-Yau if and only if f is non-degenerate and skew-symmetric (i.e. f is symplectic. 5 If the global dimension of A is equal to 2, the Hilbert series of the graded algebra A is given by Otherwise, one has h A (t = (1 nt + t 2 1. h A (t = (1 nt + t 2 t 3 + t 4 1. Set F = k x 1,...,x n, z where z is an extra generator of degree 1. Then B is the algebra defined by the potential w = fz = f ij x i x j z F cyc = F/[F, F] : 1 i,j n B = B(f = F/( x1 (w,..., xn (w, z (w, xi (w = (f ij x j z + f ji zx j, Clearly A = B/(z. 1 j n z (w = f. 2
Reducing the linear system xi (w = 1 i n to a triangular form shows that, if f is non-degenerate, - the xi (w and f form a basis of relations of B, - z is normal in B: Bz = zb, - the left (or right sub-a module Az = za of B is free, - B = A[z; σ] the skew polynomial algebra defined by the automorphism σ of A such that za = σ(az. Consequence. For any non-degenerate f, the quadratic algebra B is Koszul. The global dimension of B is equal to 3 if n 2, to if n = 1, and h B (t = (1 (n+1t+(n+1t 2 t 3 1 if n 2 h B (t = (1 2t + 2t 2 2t 3 + 1 if n = 1. Moreover, z is central in B if and only if f is symplectic. 3
Theorem. For any non-degenerate f and for any n 2, B is 3-Calabi-Yau. Sketch of proof. The Koszul resolution K w of B is self-dual: B(kc(wB d 3 BR B B d 2 BV B B d 1 BkB f 3 f 2 f 1 f BkB d 1 BVB B d 2 BRBB d 3 B(kc(w B f (1 = c(w, f 1 (x i = r i, f 2 (r i = x i, f 3 (c(w = 1. PART II - Classification of the algebras B Set M = (f ij 1 i,j n. It is immediate that A(M = A(N if and only if there exists P GL n (k such that N = t PMP. We prove that it is same as for the algebras B, but the proof is more involved. When n = 2 and k = C, this leads to a classification in three types: First type (classical type: B = C[x, y, z] Second type (Jordan type: The relations of B are 4
the following: zy = yz + 2xz, zx = xz, yx = xy + x 2. Third type (quantum type: The relations of B are the following: xy = qyx, yz = qzy, zx = qxz where {q C \ {, 1}}/(q q 1. Hochschild homology is known in the first and third types. PART III - Hochschild homology of B for the second type B = C x, y, z /(zy = yz+2xz, zx = xz, yx = xy+x 2. Consider the filtration F of B given by the degree of y. It is clear from the relations of B that gr F (B C[x, y, z]. Thus the filtered algebra B is almost commutative and gr F (B is equipped with the Poisson bracket defined by: {z, y} = 2xz, {z, x} =, {y, x} = x 2. So we can consider the non-commutative algebra B as a deformation of the Poisson algebra T = (C[x, y, z], {, }. 5
The Poisson bracket of T can be written {x, y} = φ z, φ {y, z} = x, φ {z, x} = y, where φ = x 2 z C[x, y, z] is the symmetrization of the potential w = fz = (yx xy x 2 z of B. Then {, } is called the Poisson bracket derived from the Poisson potential φ. The first step is to compute the Poisson homology of T. Two difficulties: 1 {, } is not diagonalizable (see Monnier in the diagonalizable case. 2 the origin is a singularity of φ which is nonisolated (see Van den Bergh, Marconnet, Pichereau, Pelap in the isolated case. So there are more technical problems to overcome. We get 6
HP (T = xc[y] C[y, z] HP 1 (T = C[φ] ( xz x 2 n N k n ( y n n N C n N 1 k n+1 C[z] ( C n xy n 1 ( z 2 xz ky k 1 z n k (n ky k z n 1 k ( C (2n+3 yz 3k xz y k 1 z n+1 ( 2n+3(k 1 xy HP 2 (T = C[φ] ( xy z n N k n C 1 (xc[φ] zc[z]( ( (k+1x (2(n k+1 y 2(k+1 z y k z n k HP 3 (T = C[φ] 7
The second step is to prove the following. Theorem. The Hochschild homology HH (B of B is isomorphic to the Poisson homology HP (T. Sketch of proof. 1 We use the Brylinski spectral sequence of the almost commutative algebra B, in which we replace the Hochschild complex by the Koszul complex B B e K w (since B is Koszul. 2 The graded complex associated to B B e K w is isomorphic to the Poisson homology complex δ 3 δ 2 T T 3 T 3 T with the differentials given, for F T, and F := (F 1, F 2, F 3 T 3, by: δ 1 ( F = φ ( F = Div( F φ, δ 2 ( F = ( F φ + Div( F φ, δ 3 (F = F φ, 3 So the (converging Brylinski spectral sequence δ 1 8
becomes E 1 = HP (T = HH (B. We prove that it degenerates at E 1 by showing that any Poisson cycle lifts in a Koszul cycle. Remark. Since B is 3-Calabi-Yau, we deduce Hochschild cohomology of B from the previous theorem: HH (B = HH 3 (B. In particular the center of the algebra B is the polynomial algebra generated by the element Φ = x 2 z which is a lifting of the Poisson potential φ = x 2 z. 9