Invariant theory for matrices over the Grassmann algebra.
type: Item
instance of: scholarly article
title: Invariant theory for matrices over the Grassmann algebra.
zbMATH Open document ID: 1276.16015
author: Allan Berele
published in: Advances in Mathematics
publication date: 2013-04-16T00:00:00Z
review text: The finite dimensional complex \(\mathbb Z_2\)-graded simple algebras form three families: (1) the \(n\times n\) matrix algebra \(M_n(\mathbb C)\) with trivial grading; (2) the algebra \(M_{k,\ell}(\mathbb C)=M_{k+\ell}(\mathbb C)\) with even part \(Ae_{11}+De_{22}\) and odd part \(Be_{12}+Ce_{21}\), where the blocks \(A,B,C\), and \(D\) are, respectively, \(k\times k\), \(k\times\ell\), \(\ell\times k\), and \(\ell\times\ell\) matrices; (3) the algebra \(M_n(\mathbb C[u]/(u^2-1))\) where \(M_n(\mathbb C)\) is of even degree and \(u\) is of odd degree. This algebra is isomorphic to the subalgebra of \(M_{n,n}(\mathbb C)\) spanned by \(A(e_{11}+e_{22})\) and \(B(e_{12}+e_{21})\), \(A,B\in M_n(\mathbb C)\). To each \(\mathbb Z_2\)-graded simple algebra one associates a Lie superalgebra, a Lie supergroup, and a verbally prime PI-algebra. (Verbally prime PI-algebras are the key objects in the structure theory of T-ideals developed by Kemer.) The corresponding Lie superalgebras are constructed from the graded algebras using the super Lie bracket, and are equal, respectively, to: (1) the general linear Lie algebra \(gl_n(\mathbb C)\); (2) the general linear superalgebra \(pl(k,\ell)\); (3) the queer superalgebra \(q(n)\). The corresponding verbally prime PI-algebra in the case (1) is \(M_n(\mathbb C)\). In the cases (2) and (3) one has to replace the even and the odd entries of the matrices by even and odd elements of the infinite dimensional Grassmann algebra \(E\). The corresponding verbally prime PI-algebras are \(M_{k,\ell}(E)\) and \(SM_{n,n}(E)\) which is isomorphic to \(M_n(E)\). The related supergroups are: (1) the general linear group \(GL_n(\mathbb C)\); (2) the general linear supergroup \(PL(k,\ell)\) of the invertible matrices in \(pl(k,\ell)\); (3) the queer supergroup \(Q(n)\) of the invertible matrices in \(SM_n(E)\) which is isomorphic to \(GL_n(E)\). Each of the Lie superalgebras has a basic representation \(U\) and there is a double centralizer theorem d…
reviewed by: Vesselin Drensky
Mathematics Subject Classification ID: 16R30
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 15A75
Mathematics Subject Classification ID: 16W22
Mathematics Subject Classification ID: 16W55
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 17A70
Mathematics Subject Classification ID: 16W50
zbMATH DE Number: 6154616
zbMATH Keywords: Grassmann algebras
zbMATH Keywords: invariant theory
zbMATH Keywords: super Lie algebras
zbMATH Keywords: queer superalgebras
zbMATH Keywords: generic algebras
zbMATH Keywords: graded simple algebras
zbMATH Keywords: Lie superalgebras
zbMATH Keywords: Lie supergroups
zbMATH Keywords: verbally prime PI-algebras
MaRDI profile type: Publication
full work available at URL: j.aim.2012.12.021
OpenAlex ID: W2014195297
DOI: 10.1016/J.AIM.2012.12.021
description: scientific article; zbMATH DE number 6154616