2003/11/02 by Pavel Etingof, Etingof, Pavel, Alexei Oblomkov +1
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #math.AG #math.QA
paper · pdf · doi:10.48550/arxiv.math/0311005
11 pages, no figures; proof and statement of Cor. 3.3, Cor 3.7 as well as proof of Theorem 4.1. are corrected; two references are added in the new version; minor corrections
openalex publication_date 2003/11/02 · arxiv created 2006/05/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We compute the Hochschild homology of the crossed product \Bbb C[Sn]\ltimes A⊗ n in terms of the Hochschild homology of the associative algebra A (over \Bbb C). It allows us to compute the Hochschild (co)homology of \Bbb C[W]\ltimes A⊗ n where A is the q-Weyl algebra or any its degeneration and W is the Weyl group of type An-1 or Bn. For a deformation quantization A+ of an affine symplectic variety X we show that the Hochschild homology of Sn A, A=A+[ℏ-1] is additively isomorphic to the Chen-Ruan orbifold cohomology of SnX with coefficients in \Bbb C((ℏ)). We prove that for X satisfying H1(X,\Bbb C)=0 (or A∈ VB(d)) the deformation of SnX (\Bbb C[Sn]\ltimes A⊗ n) which does not come from deformations of X (A) exists if and only if dim X=2 (d=2). In particular if A is q-Weyl algebra (its trigonometric or rational degeneration) then the corresponding nontrivial deformations yield the double affine Hecke algebras of type An-1 (its trigonometric or rational versions) introduced by Cherednik.