Representations of Cartan type Lie algebras in characteristic \(p\).
type: Item
instance of: scholarly article
title: Representations of Cartan type Lie algebras in characteristic \(p\).
zbMATH Open document ID: 1047.17009
DOI: 10.1016/S0021-8693(02)00035-2
author: Bin Shu
published in: Journal of Algebra
publication date: 2003-02-12T00:00:00Z
review text: Let \(K\) be an algebraically closed field of characteristic \(p>2\) and let \(L\) denote the derivation algebra of the truncated polynomial algebra \(\mathfrak{A}=K[x_1,\dots,x_m]/ (x_1^p,\dots,x_m^p)\). The aim of the paper under review is a description of certain representations of \(L\) similar to the representations of Lie algebras of reductive groups with \(p\)-characters in standard Levi form which was developed by \textit{E. M. Friedlander} and \textit{B. J. Parshall} [Am. J. Math. 110, No. 6, 1055--1093 (1988; Zbl 0673.17010)] as well as \textit{J. C. Jantzen} [J. Pure Appl. Algebra 152, No. 1-3, 133--185 (2000; Zbl 0976.17004)]. Let \(I\) be a subset of \(S=\{1,\dots,m\}\) and denote by \(\mathfrak{A}(I)\) the subalgebra of \(\mathfrak{A}\) generated by \(\{x_i\mid i\in I\}\), by \(L(I)\) the derivation algebra of \(\mathfrak{A}(I)\), and by \(\mathfrak{h}(I)\) the standard maximal torus \(\bigoplus_{i=1}^m Kx_i\frac{\partial}{\partial x_i}\) of \(L(I)\). Each subset \(I\) of \(S\) determines a grading of \(\mathfrak{A}\) by considering \(x_i\) for \(i\in I\) as a homogeneous element of degree one and for \(i\not\in I\) as a homogeneous element of degree zero. This grading of \(\mathfrak{A}\) induces a \(\mathbb{Z}^m\)-grading \(L=\bigoplus_{\alpha\in\mathbb{Z}^m} L_{\alpha,I}\) and a \(\mathbb{Z}\)-grading \(L=\bigoplus_{q\geq -1}L_{[q],I}\) on \(L\). Here \(L_I=L_{0,I}\) is generated as a module over \(\mathfrak{A}(S\setminus I)\) by \(\{x_i \frac{\partial}{\partial x_i}\mid i\in I\}\cup\{\frac{\partial}{\partial x_i}\mid i\in S\setminus I\}\), i.e., \(L_I\) is a semidirect product of \(L(S\setminus I)\) and an abelian ideal isomorphic to \(\mathfrak{A}(S\setminus I)\otimes\mathfrak{h}(I)\), and \(L_{[0],I}\) is a semidirect product of \(L(S\setminus I)\) and an ideal isomorphic to \(\mathfrak{A}(S \setminus I)\otimes\mathfrak{gl}(I)\). Let \(\chi\) be a linear form on \(L\) and denote by \(u_\chi(\mathfrak{g})\) the \(\chi\)-reduced universal enveloping…
reviewed by: Jörg Feldvoss
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 1868018
zbMATH Keywords: Jacobson-Witt algebra
zbMATH Keywords: grading
zbMATH Keywords: general linear Lie algebra
zbMATH Keywords: \(p\)-character
zbMATH Keywords: reduced universal enveloping algebra
zbMATH Keywords: simple module
zbMATH Keywords: regular semisimple \(p\)-character
zbMATH Keywords: Morita equivalence
Wikidata QID: Q115340549
MaRDI profile type: Publication
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full work available at URL: s0021-8693(02)00035-2
OpenAlex ID: W2026886940
description: scientific article; zbMATH DE number 1868018