Varieties of nilpotent elements for simple Lie algebras. I: Good primes.
type: Item
instance of: scholarly article
title: Varieties of nilpotent elements for simple Lie algebras. I: Good primes.
zbMATH Open document ID: 1063.17007
author: University of Georgia VIGRE Algebra Group
published in: Journal of Algebra
publication date: 2005-02-22T00:00:00Z
review text: Let \(G\) be a simple algebraic group over \(k=\mathbb C\) or \(\overline {\mathbb F}_p\) where \(p\) is good, \(\mathfrak g = \operatorname{Lie} G\), \(\rho \) an irreducible representation of minimal dimension or the adjoint representation. In the paper the description of a variety of nilpotent elements \(N_{r,\rho }=\{ x\in \mathfrak g: \rho (x)^r=0 \}\) is given in terms of orbit closures (with respect to \(G\)). The approach to the problem is based on ideas of \textit{D. K. Nakano} and \textit{T. Tanisaki} [On the realization of orbit closures as support varieties, preprint (2003), J. Pure Appl. Algebra 206, No. 1--2, 66--82 (2006; Zbl 1156.14326)] who have shown how to use orbit closure in \(\mathfrak g\mathfrak l(V)\) after intersecting with \(\mathfrak g\). The case of classical groups is considered in detail. For exceptional groups the authors provide tables listing in Bala-Carter notations the orbit closures which describe nilpotent varieties \(N_{r, \rho }(\mathfrak g)\). All non-Richardson orbits are indicated. In particular, when \(r=p=\text{char}\,k\) the results of the paper recover the calculations of the restricted nullcone \(N_1(\mathfrak g)=N_{p, \rho }(\mathfrak g)\) performed by \textit{J. F. Carlson, Z. Lin, D. K. Nakano} and \textit{B. J. Parshall} [Contemp. Math. 325, 51--75 (2003; Zbl 1059.17013)]. Note that \(N_1(\mathfrak g)\) plays a crucial role in the cohomology theory of restricted Lie algebras. The authors present an alternate method to find \(N_{r, \rho }\) when \(k=\mathbb C, ~\mathfrak g\) is classical, and \(\rho \) is the adjoint representation using Jacobson-Morozov theorem.
reviewed by: Michael Kuznetsov
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 20G10
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 2136744
zbMATH Keywords: simple algebraic groups
zbMATH Keywords: varieties of nilpotent elements
zbMATH Keywords: support varieties
zbMATH Keywords: orbit closure
Wikidata QID: Q115351729
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.05.023
OpenAlex ID: W2900038731
cites work: Q4422481
cites work: Q3682663
cites work: Q4863383
cites work: Cohomology of Lie Algebras and Algebraic Groups
cites work: Support varieties for restricted Lie algebras
cites work: On Richardson classes of unipotent elements in semisimple algebraic groups
cites work: Q4842690
cites work: Kohomologie von p-Lie-Algebren und nilpotente Elemente. (Cohomology of p- Lie-algebras and nilpotent elements)
cites work: Q3797389
cites work: Support Varieties of Weyl Modules
cites work: Jordan block sizes of unipotent elements in exceptional algebraic groups
cites work: On the finiteness of the number of unipotent classes
cites work: Support varieties for algebraic groups
cites work: On the realization of orbit closures as support varieties
cites work: Infinitesimal 1-parameter subgroups and cohomology
cites work: Support varieties for infinitesimal group schemes
DOI: 10.1016/J.JALGEBRA.2004.05.023
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description: scientific article; zbMATH DE number 2136744