Contra-variant form on Kac's modules and some series compositions
type: Item
instance of: scholarly article
title: Contra-variant form on Kac's modules and some series compositions
zbMATH Open document ID: 0849.17029
author: Avital Frumkin
published in: Advances in Mathematics
publication date: 1996-11-14T00:00:00Z
review text: The representation theory of classical Lie superalgebras \(G= G^0 \oplus G^1\) begins with what is called the ``Kac module''. For a dominant weight \(\varphi\), the Kac module is the module \(K_\varphi= U(G) \bigotimes_{U(P)} V^0_\varphi\), where \(V^0_\varphi\) is the simple \(G^0\)-module of \(\varphi\), \(U\) is for the universal enveloping algebra, \(P= G^0\oplus (B\cap G^1)\) and \(B\) is the Borel subalgebra of \(G\). The dominant weight \(\varphi\) is called typical if \(K_\varphi\) is irreducible. In the paper under review the author studies the case \(G= sl(n,m)\), the Lie superalgebra of all linear transformations with supertrace equal to zero in the \(n+m\)-dimensional vector space \(V\). In this case \(\varphi\) is called tensorial if \(K_\varphi\) is embeddable into a tensor power of \(V\). The main results of the paper are the following: (i) Let \(\tau^*\) be a dual basis of a basis of the Cartan algebra \(H\) of \(G\). For every dominant weight \(\varphi\) there exists \(r>0\) such that \(\varphi+ (r+i) \tau^*\) is a typical tensorial weight for each natural \(i\). (ii) If \(\varphi\) is a typical tensorial weight, then the Kac module \(K_\varphi\) carries a contravariant symmetric form which is positive and gives a lot of information on \(K_\varphi\). (iii) The series compositions of \(K_{\varphi+ x\tau^*}\) is obtained for any \(x\), where \(\varphi\) is the trivial weight of \(sl (n) \oplus sl (m) \subset G^0\).
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B20
Mathematics Subject Classification ID: 17B45
zbMATH DE Number: 848978
zbMATH Keywords: simple Lie superalgebras
zbMATH Keywords: representations
zbMATH Keywords: dominant weights
zbMATH Keywords: Kac module
zbMATH Keywords: contravariant symmetric form
zbMATH Keywords: series compositions
MaRDI profile type: Publication
full work available at URL: aima.1995.1048
OpenAlex ID: W2078611343
DOI: 10.1006/AIMA.1995.1048
description: scientific article; zbMATH DE number 848978