Semicanonical bases arising from enveloping algebras
type: Item
instance of: scholarly article
title: Semicanonical bases arising from enveloping algebras
zbMATH Open document ID: 0983.17009
published in: Advances in Mathematics
publication date: 2002-01-19T00:00:00Z
review text: Let \(U^+\) be the plus part of the enveloping algebra of a Kac-Moody Lie algebra \({\mathfrak g}\) with a symmetric Cartan datum. In [\textit{G. Lusztig}, J. Am. Math. Soc. 3, 447-498 (1990; Zbl 0703.17008)] a canonical basis of \(U^+\) is given under the assumption that the Cartan datum is of finite type. The basis of \(U^+\) is obtained from a canonical basis of the quantized version of \(U^+\) by specializing the quantum parameter to 1. In [\textit{G. Lusztig}, Publ. Math., Inst. Hautes Étud. Sci. 76, 111-163 (1992; Zbl 0776.17013)] a basis of \(U^+\) is constructed in terms of constructible functions on a Lagrangian variety with the Cartan datum of affine type, which is called the semicanonical basis of \(U^+\). In this paper, the author extends the definition of semicanonical basis to include the case where the Cartan datum is not necesarily of affine or finite type. When the semicanonical basis is not necessarily the same as the canonical basis, it was shown that the semicanonical basis has a number of properties in common with the canonical basis: compatibility with various filtrations of \(U^+\), compatibility with the canonical antiautomorphism of \(U^+\).
reviewed by: Fang Li
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 16S30
zbMATH DE Number: 1471652
zbMATH Keywords: enveloping algebra
zbMATH Keywords: Kac-Moody Lie algebra
zbMATH Keywords: semicanonical basis
zbMATH Keywords: Cartan datum
zbMATH Keywords: canonical basis
author: G. Lusztig
MaRDI profile type: Publication
full work available at URL: aima.1999.1873
OpenAlex ID: W2079778585
cites work: On crystal bases of the \(q\)-analogue of universal enveloping algebras
cites work: Geometric construction of crystal bases
cites work: Canonical Bases Arising from Quantized Enveloping Algebras
cites work: Canonical Bases Arising from Quantized Enveloping Algebras. II
cites work: Quivers, Perverse Sheaves, and Quantized Enveloping Algebras
cites work: Affine quivers and canonical bases
cites work: Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras
DOI: 10.1006/AIMA.1999.1873
description: scientific article; zbMATH DE number 1471652