Commuting variety of Witt algebra
type: Item
instance of: scholarly article
title: Commuting variety of Witt algebra
zbMATH Open document ID: 1446.17030
author: Y. Aharonov
published in: Frontiers of Mathematics in China
publication date: 2018-11-27T00:00:00Z
review text: Let \(\mathfrak{g}\) be a Lie algebra over an algebraically closed field. Then the commuting variety \(\mathcal{C}(\mathfrak{g})=\{(x,y)\in\mathfrak{g} \times\mathfrak{g}\mid[x,y]=0\}\) of \(\mathfrak{g}\) is a Zariski closed subset of \(\mathfrak{g}\times\mathfrak{g}\). \textit{P. Levy} [J. Algebra 250, No. 2, 473--484 (2002; Zbl 1034.17012)] proved under mild conditions on the characteristic of the ground field that the commuting variety of the Lie algebra of a reductive algebraic group is an irreducible algebraic variety. In the paper under review the authors investigate the commuting variety of the smallest simple Lie algebra that is not the Lie algebra of a reductive algebraic group, namely, the \(p\)-dimensional Witt algebra \(W_1\) over a field of characteristic \(p >3\). It is shown that the commuting variety of \(W_1\) is the union of \(\frac{p-1}{2}\) irreducible components, one of which is of dimension \(p+1\), and the others are all of dimension \(p\). In particular, this variety is never irreducible, and as the origin belongs to any irreducible component, it follows that this variety is also never normal.
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B70
zbMATH DE Number: 6985520
zbMATH Keywords: commuting variety
zbMATH Keywords: prime characteristic
zbMATH Keywords: Witt algebra
MaRDI profile type: Publication
full work available at URL: s11464-018-0725-9
OpenAlex ID: W2892233832
cites work: Über Wittsche Lie-Ringe
cites work: A basis for the space of functions analytic in a disk
cites work: Q4089001
cites work: Commuting varieties of Lie algebras over fields of prime characteristic
cites work: ORBIT CLOSURES IN THE WITT ALGEBRA AND ITS DUAL SPACE
cites work: On Generalized Witt Algebras
cites work: Q4196567
cites work: Q4039778
cites work: Q4674279
cites work: Automorphisms of graded lie algebras op cartan type
cites work: The nilpotent commuting variety of the Witt algebra
cites work: Nilpotent Orbits in the Witt AlgebraW1
DOI: 10.1007/S11464-018-0725-9
Recommended article: Commuting varieties of Lie algebras over fields of prime characteristic
Recommended article: Witt代数的 r 元组交换簇
Recommended article: Q3804872
Recommended article: On commuting varieties of parabolic subalgebras
Recommended article: Irregular and singular loci of commuting varieties
Recommended article: On the irreducibility of commuting varieties associated with involutions of simple Lie algebras
Recommended article: ON VARIETIES OF LIE ALGEBRAS NOT CONTAINING A THREE-DIMENSIONAL SIMPLE ALGEBRA
Recommended article: The nilpotent commuting variety of the Witt algebra
Recommended article: Symmetric pairs and associated commuting varieties
Recommended article: On the conormal bundle of a \(G\)-stable subvariety
description: scientific article; zbMATH DE number 6985520