Irreducible representations of the generalized Jacobson-Witt algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1242.17012
DOI: 10.1142/S1005386712000041
author: Bin Shu
author: Yu-Feng Yao
published in: Algebra Colloquium
publication date: 2012-04-12T00:00:00Z
full work available at URL: S1005386712000041.html
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B50
Mathematics Subject Classification ID: 17B70
Mathematics Subject Classification ID: 17B66
zbMATH DE Number: 6023014
zbMATH Keywords: Cartan type Lie algebra
zbMATH Keywords: generalized Jacobson-Witt algebra
zbMATH Keywords: generalized restricted Lie algebra
zbMATH Keywords: irreducible module
zbMATH Keywords: generalized \(p\)-character
zbMATH Keywords: height of a generalized \(p\)-character
zbMATH Keywords: generalized reduced universal enveloping algebra
zbMATH Keywords: modified induced module
zbMATH Keywords: \((R,L)\)-module
zbMATH Keywords: exceptional module
zbMATH Keywords: de Rham complex
zbMATH Keywords: number of isomorphism classes of irreducible modules
zbMATH Keywords: dimension formula
MaRDI profile type: Publication
cites work: Über Wittsche Lie-Ringe
cites work: Extension functors of modular Lie algebras
cites work: Simple modules with character height at most one for the restricted Witt algebras
cites work: The generalized restricted representations of graded Lie algebras of Cartan type
cites work: Generalized restricted Lie algebras and representations of the Zassenhaus algebra
cites work: Conjugation in representations of the Zassenhaus algebra
cites work: Representations of the Witt algebra
cites work: Q4039778
cites work: A structural characterization of the simple Lie algebras of generalized Cartan type over fields of prime characteristic
cites work: IRREDUCIBLE REPRESENTATIONS OF THE HAMILTONIAN ALGEBRAH(2r;n)
cites work: Representations of the restricted Lie algebras of Cartan type
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title: Irreducible representations of the generalized Jacobson-Witt algebras
review text: Let \(\mathbf{n}=(n_1,\dots,n_m)\) be an arbitrary \(m\)-tuple of positive integers and let \(L=W(m;\mathbf{n})\) denote a graded generalized Jacobson-Witt algebra over an algebraically closed field of characteristic \(p>3\). The goal of the paper under review is to use the setup of generalized restricted Lie algebras to prove that, up to finitely many exceptions, irreducible representations of \(L\) with generalized \(p\)-characters of height at most \(\min\{p^{n_i}-p^{n_i-1} \mid 1\leq i\leq m\}-1\) can be obtained as modules induced from an irreducible representation of the distinguished maximal subalgebra \(L_0\) of \(L\). (Here the \(L\)-action on the induced modules is twisted in order to ensure that the exceptions are exactly the modules induced from the irreducible \(\mathfrak{gl}_m\)-modules having as highest weight some fundamental weight.) In particular, every irreducible module of \(L\) with generalized \(p\)-character of height between \(1\) and \(\min\{ p^{n_i}-p^{n_i-1}\mid 1\leq i\leq m\}-1\) is induced from some irreducible \(L_0\)-module, and the number of isomorphism classes of irreducible \(L\)-modules with such a generalized \(p\)-character \(\chi\) is the same as the number of isomorphism classes of irreducible \(L_0\)-modules with \(p\)-character \(\chi_{\mid L_0}\). NEWLINENEWLINENEWLINENEWLINEThe main technical tool is the concept of a \(\mathfrak{C}\)-category (in the paper under review called a category of \((R,L)\)-modules). The definition of a \(\mathfrak{C}\)-category involves an \(L\)-module structure, its restriction to \(L_0\), a module structure coming from the defining divided power algebra of \(L\), and several compatibility conditions, and was introduced by \textit{S. Skryabin} [``Independent systems of derivations and Lie algebra representations'', Algebra and Analysis. Proceedings of the International Centennial Chebotarev Conference, Kazan, Russia, June 5-11, 1994. Arslanov, M.\ M.\ (ed.) et al.\ Berlin: Walter de Gruyter.\ 11…
reviewed by: Jörg Feldvoss
description: scientific article; zbMATH DE number 6023014