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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

Online
ISSN
1435-5345
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Volume 2012, Issue 667

Issues

Zeroth Poisson homology of symmetric powers of isolated quasihomogeneous surface singularities

Pavel Etingof / Travis Schedler
Published Online: 2011-08-08 | DOI: https://doi.org/10.1515/CRELLE.2011.124

Abstract

Let X ⊂ ℂ3 be a surface with an isolated singularity at the origin, given by the equation Q(x, y, z) = 0, where Q is a weighted-homogeneous polynomial. In particular, this includes the Kleinian surfaces X = ℂ2/G for G < SL2(ℂ) finite. Let Y ≔ SnX be the n-th symmetric power of X. We compute the zeroth Poisson homology HP0(𝒪Y), as a graded vector space with respect to the weight grading, where 𝒪Y is the ring of polynomial functions on Y. In the Kleinian case, this confirms a conjecture of Alev, that , where Weyl2n is the Weyl algebra on 2n generators. That is, the Brylinski spectral sequence degenerates in degree zero in this case. In the elliptic case, this yields the zeroth Hochschild homology of symmetric powers of the elliptic algebras with three generators modulo their center, Aγ, for all but countably many parameters γ in the elliptic curve. As a consequence, we deduce a bound on the number of irreducible finite-dimensional representations of all quantizations of Y. This includes the noncommutative spherical symplectic reflection algebras associated to Gn ⋊ Sn.

About the article

Received: 2009-07-17

Revised: 2010-02-09

Published Online: 2011-08-08

Published in Print: 2012-06-01


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2012, Issue 667, Pages 67–88, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2011.124.

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©[2012] by Walter de Gruyter Berlin Boston.Get Permission

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