Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities
type: Item
instance of: scholarly article
title: Representations of symplectic reflection algebras and resolutions of deformations of symplectic quotient singularities
zbMATH Open document ID: 1062.14020
published in: Mathematische Annalen
publication date: 2004-10-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0310187
review text: Let \(\Gamma \) be a finite subgroup of \(SL(2,\mathbb{C}).\) The wreath product \(\Gamma _n=S_n\ltimes \Gamma ^n\) acts on \((\mathbb{C}^2)^n\) preserving the product symplectic form. The symplectic reflection algebra \({\mathbb{H}_c}\) is the \(\mathbb{C}\)-algebra, defined as the quotient of the skew group ring \(T(\mathbb{C}^2)^n\star \Gamma _n\) by the relations \(x\otimes y-y\otimes x=\sum _{\gamma } c_{\gamma }\omega _{\gamma }(x,y)\gamma \) for all \(x,y \in (\mathbb{C}^2)^n\) and the symplectic reflections \(\gamma \) and their corresponding forms \(\omega _{\gamma }.\) The authors prove that there is a crepant resolution (i.e. preserving the symplectic forms) \(\pi _{\mathbf c}:Y_{\mathbf c}\to X_{\mathbf c}\) of the spectrum of the centre of the algebra \({\mathbb{H}_c}\) such that there is an equivalence between the bounded derived category of finitely generated \({\mathbb{H}_c}\)-modules and the bounded derived category of coherent sheaves on \(Y_{\mathbf c}.\) As the consequence they get that for all \(x\in X_{\mathbf c}\), there is an isomorphism of Grothendieck groups \(K({\mathbb{H}_c}/m_x{\mathbb{H}_c})\cong K(\pi _c^{-1}(x)).\)
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 16S38
Mathematics Subject Classification ID: 18E30
zbMATH DE Number: 2105002
zbMATH Keywords: symplectic structure
zbMATH Keywords: resolution of singularities
zbMATH Keywords: representation theory
author: Iain Gordon
author: S. Paul Smith
reviewed by: Stanisław Janeczko
MaRDI profile type: Publication
OpenAlex ID: W2041276723
arXiv ID: math/0310187
DOI: 10.1007/S00208-004-0545-Y
description: scientific article