Representations of Lie superalgebras. I: Extensions of representations of the even part
type: Item
instance of: scholarly article
title: Representations of Lie superalgebras. I: Extensions of representations of the even part
zbMATH Open document ID: 0674.17010
author: H. Furutsu
author: Takeshi Hirai
published in: Journal of Mathematics of Kyoto University
publication date: 1988-01-01T00:00:00Z
review text: In this paper, representations of Lie superalgebras are studied. Since the classification of all finite-dimensional irreducible representations of basic classical Lie superalgebras by means of their highest weight [\textit{V. G. Kac}, Adv. Math. 26, 8-96 (1977; Zbl 0366.17012)], and the analysis of typical representations [\textit{V. G. Kac}, Lect. Notes Math. 676, 597-626 (1978; Zbl 0388.17002)], most effort has gone in analyzing the remaining atypical highest weight representations. Here, a completely different approach is taken. The authors discuss the following problems, for \({\mathfrak g}={\mathfrak g}_ 0\oplus {\mathfrak g}_ 1\) any Lie superalgebra: Problem 1 (resp. 1bis): Given an irreducible (resp. a reducible) representation \(\rho\) of \({\mathfrak g}_ 0\) on \(V_ 0\), do there exist irreducible representations (\(\pi\),V) of \({\mathfrak g}\) extending \((\rho,V_ 0)?\) Problem 2 (resp. 2bis): Given an irreducible (resp. a reducible) unitary representation \((\rho,V_ 0)\) of \({\mathfrak g}_ 0\), do there exist irreducible unitary extensions (\(\pi\),V)? Having these problems in mind, a theory is developed the main result of which is that such extensions are possible provided a bilinear map B: \({\mathfrak g}\times {\mathfrak g}_ 1\to {\mathfrak gl}(V_ 0)\) exists satisfying certain conditions. Although the mathematics involved in this theory may be interesting, the applicability turns out to be rather disappointing. This is due to the fact that the restriction of an irreducible \({\mathfrak g}\)-module to \({\mathfrak g}_ 0\) is in general reducible (and for the converse problem it turns out to be very difficult to see which reducible \({\mathfrak g}_ 0\)-modules can be extended to an irreducible \({\mathfrak g}\)- module), and that a finite-dimensional irreducible \({\mathfrak g}\)-module is in general not unitary. Finally, some examples are discussed. The finite example is \({\mathfrak osp}(1,2)\) [see also \textit{J. W. B. Hughes}, J. Math. Phys. 22,…
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17A70
zbMATH DE Number: 4103315
zbMATH Keywords: infinite-dimensional representations
zbMATH Keywords: Lie superalgebras
zbMATH Keywords: unitary representation
zbMATH Keywords: unitary extensions
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520352
OpenAlex ID: W1521121803
DOI: 10.1215/KJM/1250520352
description: scientific article; zbMATH DE number 4103315