Noncommutative deformations of type-\(A\) Kleinian singularities
type: Item
instance of: scholarly article
title: Noncommutative deformations of type-\(A\) Kleinian singularities
zbMATH Open document ID: 0807.16029
author: Timothy J. Hodges
published in: Journal of Algebra
publication date: 1995-02-28T00:00:00Z
review text: Let \(k\) be an algebraically closed field of characteristic zero. The author studies noncommutative \(k\)-algebras generated by the elements \(a,b\) and \(h\) satisfying the relations \(ha - ah = a\), \(hg - bh = -b\), \(ba = v(h)\), \(ab = v(h - 1)\), where \(v(x)\) is a polynomial with coefficients in \(k\). Such an algebra denoted by \(T(v)\) is a Noetherian ring of Krull dimension one. For example, if \(v(x)\) is quadratic then \(T(v)\) is isomorphic to an infinite dimensional primitive factor of the enveloping algebra of the Lie algebra \(sl(2,k)\). The author proves that the associated graded ring of \(T(v)\) is isomorphic to the coordinate ring of type-A Kleinian singularities. In the case of the complex ground field such a singularity may be presented as the quotient of an affine complex plane by a finite cyclic subgroup of \(SL_ 2(\mathbb{C})\). It is proved that \(T(v)\) is an Auslander-Gorenstein ring and its global dimension is infinite if and only if the polynomial \(v\) has repeated roots. In case the roots of \(v\) are distinct there are only the following two possibilities. When any two roots of \(v\) differ by a nonzero integer then the global dimension of \(T(v)\) is equal to 2, otherwise it is equal to 1. Furthermore, in this case it turns out that the Grothendieck group \(K_ 0(T(v))\) is a free abelian group of rank equal to the degree of \(v\). In conclusion the author formulates some open problems related to the subject.
Mathematics Subject Classification ID: 16S80
Mathematics Subject Classification ID: 16S15
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 16S30
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 16E20
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 13H10
zbMATH DE Number: 554242
zbMATH Keywords: relations
zbMATH Keywords: Noetherian rings
zbMATH Keywords: Krull dimension
zbMATH Keywords: infinite dimensional primitive factor
zbMATH Keywords: enveloping algebra
zbMATH Keywords: associated graded ring
zbMATH Keywords: coordinate ring of type-\(A\) Kleinian singularities
zbMATH Keywords: Auslander-Gorenstein ring
zbMATH Keywords: global dimension
zbMATH Keywords: Grothendieck group
reviewed by: Aleksandr G. Aleksandrov
MaRDI profile type: Publication
full work available at URL: jabr.1993.1219
OpenAlex ID: W2018021371
DOI: 10.1006/JABR.1993.1219
description: scientific article; zbMATH DE number 554242