Primitive ideals in the enveloping algebra of the Lie superalgebra sl(2,1)
type: Item
instance of: scholarly article
title: Primitive ideals in the enveloping algebra of the Lie superalgebra sl(2,1)
zbMATH Open document ID: 0799.17009
author: Ian M. Musson
published in: Journal of Algebra
publication date: 1994-06-23T00:00:00Z
review text: Let \(U({\mathfrak g})\) be the enveloping algebra of a classical simple Lie superalgebra \({\mathfrak g}\). In a previous paper [Adv. Math. 91, 252-268 (1992; see the preceding review)] the author has shown that every primitive ideal in \(U({\mathfrak g})\) is the annihilator of a graded simple quotient of a Verma module. In this paper he focusses on the special case \({\mathfrak g}= {\mathfrak {sl}}(2,1)\), obtaining a complete description of all submodules of Verma modules, an analogue of the Beilinson-Bernstein theorem in the so-called atypical case, the Jacobson topology and clique structure of the primitive spectrum, and a formula for the intersection of a primitive ideal with \(U({\mathfrak g}_ 0)\). In particular, he produces an interesting example of an infinite clique with unbounded Goldie dimension that does not satisfy the second layer condition. The methods are mostly those of [loc. cit.] but also involve fundamental techniques of \textit{K. A. Brown} and \textit{R. B. Warfield} [J. Algebra 116, 294-315 (1988; Zbl 0652.16006)]. \textit{G. Pinczon} has undertaken a similar study of \(U({\mathfrak {osp}} (1,2))\) in [J. Algebra 132, 219-242 (1990; Zbl 0702.17005)].
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17A70
zbMATH DE Number: 571348
zbMATH Keywords: enveloping algebra
zbMATH Keywords: simple Lie superalgebra
zbMATH Keywords: primitive ideal
zbMATH Keywords: Verma module
zbMATH Keywords: Beilinson-Bernstein theorem
zbMATH Keywords: primitive spectrum
reviewed by: William M. McGovern
MaRDI profile type: Publication
full work available at URL: jabr.1993.1158
OpenAlex ID: W2041206737
Wikidata QID: Q115396296
DOI: 10.1006/JABR.1993.1158
description: scientific article; zbMATH DE number 571348