Generalized restricted Lie algebras and representations of the Zassenhaus algebra
type: Item
instance of: scholarly article
title: Generalized restricted Lie algebras and representations of the Zassenhaus algebra
zbMATH Open document ID: 0913.17010
author: Bin Shu
published in: Journal of Algebra
publication date: 1999-01-22T00:00:00Z
review text: The remarkable result of this paper is a description of all irreducible representations of the Zassenhaus algebra \(W(1:n)\) over an algebraically closed field \(F\) of characteristic \(p>3.\) In the standard basis \(\{ e_i, i=-1,\ldots ,p^n-2\}\) of \(L=W(1: n)\) \[ [e_i, e_j]=c_{ij}e_{i+j}, c_{ij}=\binom {i+j+1}j -\binom {i+j+1}i, \] \(\{ L_i=\langle{} e_j, j\geq i \rangle{}, i\geq -1\}\) is the standard filtration of \(L\). It is well known that \(L_0\) is a \(p\)-subalgebra of \(L\) with respect to the natural embedding \(L\) into \(\mathcal L=\text{Der }L=L+\langle{} e_{-1}^{p^k}, 0<k<n \rangle{}.\) Let \(\rho \) be an irreducible representation of \(L\) on a space \(V\). There exists a linear function \(\varTheta \in L^*\) such that \[ \rho (e_{-1})^{p^n}=\varTheta (e_{-1})^{p^n}I, \] \[ \rho (e_i)^p-\rho (e_i^{[p]})=\varTheta (e_i)^p, i\geq 0. \] The author shows that the representation \(\rho \) (and the function \(\varTheta ,\) as well) may be continued to \(\mathcal L.\) Thus, \(\rho \) may be considered as an irreducible representation of the \(p\)-algebra \(\mathcal L\) with \(p\)-character \(\varTheta\). Let \(r\) be the minimal integer \(i\) such that \(\varTheta (L_i)=0.\) All irreducible modules for \(r=-1\) (i.e. \(\varTheta =0\)) are well known. If \(0\leq r<p^n-1\) then \(V\cong \text{Ind }^{\mathcal L}_Q(Fu, \varTheta).\) Here \(Q\) is a special subalgebra in \(L_0\), \(Fu\) is a one-dimensional \(Q\)-module and induction is understood in terms of the \(\varTheta\)-reduced enveloping algebra \(U(\mathcal L)/J\) where \(J\) is an ideal generated by \(e^p-e^{[p]}-\varTheta (e)^p1, e\in \mathcal L.\) Thus, \(\dim V=p^{n+\dim L_0/Q}.\) If \(r=p^n-1\) then as \(L_0\)-module \(V\cong \text{Ind }^{L_0}_{L_s}(Fu, \varTheta), s=\frac{p^n-1}2 \) where \(Fu\) is a one-dimensional \(L_{\frac{p^n-1}2}\)-module. Thus, \(\dim V=p^{\frac{p^n-1}2}.\)
Mathematics Subject Classification ID: 17B50
zbMATH DE Number: 1218674
zbMATH Keywords: generalized restricted Lie algebra
zbMATH Keywords: Zassenhaus algebra
zbMATH Keywords: representation
reviewed by: Michael Kuznetsov
MaRDI profile type: Publication
full work available at URL: jabr.1997.7344
OpenAlex ID: W2059329999
Wikidata QID: Q115395897
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DOI: 10.1006/JABR.1997.7344
description: scientific article; zbMATH DE number 1218674