2020/07/02 by Alexander Vitanov, Vitanov, Alexander
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2007.01385
openalex publication_date 2020/07/02 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28
We show how a novel construction of the sheaf of Cherednik algebras on a quotient orbifold Y=X/G by virtue of formal geometry in author's prior work leads to results for the sheaf of Cherednik algebra which until recently were viewed as intractable. First, for every orbit type stratum in X, we define a trace density map for the Hochschild chain complex of the sheaf of Cherednik algebras, which generalises the standard Engeli-Felder's trace density construction for the sheaf of differential operators. Second, by means of the newly obtained trace density maps, we prove an isomorphism in the derived category of complexes of \mathbb CY[[ℏ]]-modules which computes the hypercohomology of the Hochschild chain complex C\bullet of the sheaf of formal Cherednik algebras. We show that this hypercohomology is isomorphic to the Chen-Ruan cohomology of the orbifold X/G with values in the ring of formal power series \mathbb C[[ℏ]]. We infer that the Hochschild chain complex of the sheaf of skew group algebras DX\rtimes G has a well-defined Euler characteristic which is proportional to the topological Euler characteristic of X/G. Finally, we prove an algebraic index theorem.