The representations of solvable Lie algebras with a triangular decomposition
type: Item
instance of: scholarly article
zbMATH Open document ID: 0988.17007
author: Zhihong Jiang
publication date: 2002-07-08T00:00:00Z
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B50
zbMATH DE Number: 1671599
zbMATH Keywords: solvable Lie algebra
zbMATH Keywords: generalized restricted Lie algebra
zbMATH Keywords: representation
zbMATH Keywords: triangular decomposition
MaRDI profile type: Publication
title: The representations of solvable Lie algebras with a triangular decomposition
published in: Chinese Annals of Mathematics. Series A
review text: Let \(L\) be a Lie algebra over an algebraically closed field \(F\) of characteristic \(p>0\). If \(L=T\oplus N\) where \(T\) is an \(ad\)-torus and \(N\) a nilpotent subalgebra of \(L\) with \([T,N]\subset N\), then \(L\) is called a solvable Lie algebra with a triangular decomposition (SLATD) (this terminology seems to the reviewer somewhat misleading). The author gives a sufficient condition for a SLATD to be generalized restricted (with respect to a basis \(B\) of \(L\)) in the sense of [\textit{B. Shu}, J. Algebra 194, 157-177 (1997; Zbl 0885.17014)]. For such a generalized restricted SLATD \(L\) and a ``generalized character'' \(S\in \text{Map}(B,F)\) (cf. [B. Shu, ibid.]) such that \(S(B\cap N)=0\), let \(u(S)\) be the \(S\)-reduced enveloping algebra of \(L\). The author determines the irreducible and the indecomposable modules of \(u(S)\). The blocks of \(u(S)\) are described, in particular, the number and the dimensions of the blocks are computed.
reviewed by: Guangyu Shen
description: scientific article; zbMATH DE number 1671599