Remarks on quiver varieties
type: Item
instance of: scholarly article
title: Remarks on quiver varieties
zbMATH Open document ID: 1017.20040
DOI: 10.1215/S0012-7094-00-10523-6
author: G. Lusztig
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: The quiver varieties introduced by \textit{H. Nakajima} [Duke Math. J. 91, No. 3, 515-560 (1998; Zbl 0970.17017)] play a role in the representation theory of quantized affine algebras similar to that played by varieties of Borel subalgebras of a simple Lie algebra for the representation theory of affine Hecke algebras. In previous work [Represent. Theory 2, No. 9, 298-369 (1998; Zbl 0901.20034), ibid. 3, No. 11, 281-353 (1999; Zbl 0999.20036)], the author conjectured a canonical basis for the equivariant K-theory of the variety of Borel subalgebras containing a given nilpotent element. In this paper, the author's goal is to use that framework to also construct a canonical basis for the equivariant K-theory of the quiver variety. To this end, the author constructs a key missing ingredient -- a certain involution. This is based on re-interpreting the quiver variety as a kind of Grassmannian variety of modules over a Gelfand-Ponomarev algebra as done in the author's work [Adv. Math. 136, No. 1, 141-182 (1998; Zbl 0915.17008)]. He then proceeds to propose how the canonical basis might be defined. Recently, \textit{M. Varagnolo} and \textit{E. Vasserot} [``Canonical bases and quiver varieties'', Represent. Theory 7, 227--258 (2003; Zbl 1055.17007) \texttt{http://arXiv.org/abs/math.RT0107177}] have proven the existence of this conjectured basis. In the final section of the paper, again by analogy with the variety of Borel subalgebras, the author finds a partition of the quiver variety into locally closed smooth subvarieties.
reviewed by: Christopher P. Bendel
Mathematics Subject Classification ID: 20G42
Mathematics Subject Classification ID: 16G20
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 14M15
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 19L47
zbMATH DE Number: 1820762
zbMATH Keywords: quiver varieties
zbMATH Keywords: Borel subalgebras
zbMATH Keywords: Gelfand-Ponomarev algebras
zbMATH Keywords: equivariant K-theory
zbMATH Keywords: canonical bases
MaRDI profile type: Publication
cites work: Homology of the Zero-Set of a Nilpotent Vector Field on a Flag Manifold
cites work: Bases in equivariant 𝐾-theory
cites work: On quiver varieties
cites work: Bases in equivariant 𝐾-theory. II
cites work: Quiver varieties and Kac-Moody algebras
description: scientific article; zbMATH DE number 1820762