On the number of simple modules of a supersolvable restricted Lie algebra
type: Item
instance of: scholarly article
title: On the number of simple modules of a supersolvable restricted Lie algebra
zbMATH Open document ID: 0978.17010
author: Jörg Feldvoss
published in: Journal of Algebra
publication date: 2001-08-22T00:00:00Z
review text: A Lie algebra \(L\) is called supersolvable if there is a chain of ideals \(L_i\) \[ 0=L_n\subset\cdots\subset L_1\subset L_0=L \] such that \(\dim L_i/L_{i+1}=1\). Let \(L\) be a supersolvable Lie \(p\)-algebra over an algebraically closed field \(F\) of characteristic \(p\), \(u(L,\chi)\) be a reduced enveloping algebra of \(L\) corresponding to a \(p\)-character \(\chi\in L^*\). The aim of the paper is to obtain the number of isomorphism classes of simple \(u(L,\chi)\)-modules. For this purpose the simple \(u(L,\chi)\)-modules with a fixed eigenvalue function are considered. Let \(T_p(L)\) be the maximal torus in the center of \(L\). Given a simple \(u(L,\chi)\)-module \(S\), there exists \(\sigma\in T_p(L)^*\) such that \((t)_S=\sigma(t)\cdot 1_S\) for any \(t\in T_p(L)\). Obviously, \[ \sigma(t)^p-\sigma(t^{[p]})=\chi(t)^p. \] \(\sigma\) is called the eigenvalue function of \(S\). The set of all eigenvalue functions of simple \(u(L,\chi)\)-modules is denoted by \(\Pi(L,\chi)\) and the set of isomorphism classes of simple \(u(L,\chi)\)-modules with a fixed eigenvalue function \(\tau\) is denoted by \(\text{Irr}_{\tau}(L,\chi)\). Let \(G_0^L\) be the set of all \(\gamma\in L^*\) such that \(\gamma([L,L])=\gamma(T_p(L))=0\), \(\gamma(x^{[p]})=\gamma(x)^p\) for any \(x\in L\). It is shown that if \(M\) and \(N\) are simple \(u(L,\chi)\)-modules with the same eigenvalue function, then \(N=F_\gamma\otimes M\) for some \(\gamma\in G_0^L\) (Theorem 1). Here \(F_\gamma\) is one-dimensional \(L\)-module corresponding to \(\gamma\), i.e. \(x\cdot 1_\gamma=\gamma(x)1_\gamma\), \(x\in L\). It means that \(G_0^L\) acts transitively on \(\text{Irr}_\tau(L,\chi)\) for any \(\tau\in\Pi(L,\chi)\). Therefore, \(|\text{Irr}_\tau(L,\chi)|\) is a \(p\)-power (Proposition 1). For \(\tau\in\Pi(L,\chi)\) let \[ u_\tau(L,\chi)=u(L,\chi)/u(L,\chi)\{t-\tau(t)\cdot 1\mid t\in T_p(L)\}. \] For an arbitrary Lie \(p\)-algebra \(L\) it is shown that \(u(L,\chi)=\oplus_{\tau\in\Pi(L,\chi)}u_\ta…
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B35
zbMATH DE Number: 1512971
zbMATH Keywords: restricted Lie algebra
zbMATH Keywords: representations of Lie algebras
zbMATH Keywords: supersolvable Lie algebra
zbMATH Keywords: nilpotent Lie algebra
reviewed by: Michael Kuznetsov
MaRDI profile type: Publication
full work available at URL: jabr.1999.8281
OpenAlex ID: W2041091198
Wikidata QID: Q115395774
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DOI: 10.1006/JABR.1999.8281
description: scientific article; zbMATH DE number 1512971