Classification of pointed weak torsion free representations for classical Lie algebras
type: Item
instance of: scholarly article
title: Classification of pointed weak torsion free representations for classical Lie algebras
zbMATH Open document ID: 0814.17018
author: Zhiyong Shi
published in: Journal of Algebra
publication date: 1995-01-01T00:00:00Z
review text: Let \({\mathfrak g}\) be a classical Lie algebra over an algebraically closed field of characteristic \(p>3\) defined by a Cartan matrix, Serre's relations, and generators \(E_ i\), \(H_ i\), \(F_ i\). A representation \(V\) of \({\mathfrak g}\) is called pointed weak torsion free if \(V\) is irreducible with a one-dimensional weight space and the generators \(E_ i\), \(F_ i\) act injectively on \(V\). Assuming further that \({\mathfrak g}\) is not of type \(A_{ip -1}\), the author classifies the pointed weak torsion free representations of \({\mathfrak g}\) up to isomorphism. Such representations exist if and only if \({\mathfrak g}\) is of type \(A_ \ell\) or \(C_ \ell\). Explicit constructions of these representations are given by specifying the actions of \(E_ i\), \(F_ i\), \(H_ i\) on a certain basis. It is also shown that all of these representations have modular Weyl algebra realizations.
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 735322
zbMATH Keywords: modular Lie algebras
zbMATH Keywords: explicit constructions
zbMATH Keywords: pointed weak torsion free representations
zbMATH Keywords: modular Weyl algebra realizations
Wikidata QID: Q115396216
MaRDI profile type: Publication
full work available at URL: jabr.1995.1033
OpenAlex ID: W2078547515
DOI: 10.1006/JABR.1995.1033
description: scientific article; zbMATH DE number 735322