The cohomology of the \(\mathfrak {spl}(2,1)\) superalgebra and reducibility of its indecomposable constituents
type: Item
instance of: scholarly article
title: The cohomology of the \(\mathfrak {spl}(2,1)\) superalgebra and reducibility of its indecomposable constituents
zbMATH Open document ID: 0597.17004
author: Bani Mitra
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1985-01-01T00:00:00Z
review text: In the paper one of the classical simple Lie superalgebras is studied. Its underlying Lie algebra is \(\mathfrak{sl}(2)\times\mathfrak{gl}(1)\) which has many physical applications. The irreducible representations of \(\mathfrak{spl}(2,1)\) were studied by \textit{M. Scheunert}, \textit{W. Nahm} and \textit{V. Rittenberg} [J. Math. Phys. 18, 155--162 (1977; Zbl 0354.17005)]. In this paper the cohomology of the irreducible representations of the \(\mathfrak{spl}(2,1)\) superalgebra is computed. It turns out that \(H^ n_{\varphi}(\mathfrak{spl}(2,1), W)=0\) if \(n>0\) for all the typical irreducible representations \(W\), and for the trivial representation. For the nontypical irreducible representations \(H^ n(\mathfrak{spl}(2,1), W)=0\) for \(q>n/2\) where \(\varphi: \mathfrak{spl}(2,1)\to \mathrm{End}\;W\). This information is applied in studying the reducibility of the indecomposable representations.
reviewed by: Alice Fialowski
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B56
zbMATH DE Number: 3961905
zbMATH Keywords: simple Lie superalgebras
zbMATH Keywords: cohomology
zbMATH Keywords: irreducible representations
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3961905