Quasifinite representations of a family of Lie algebras of Block type
type: Item
instance of: scholarly article
title: Quasifinite representations of a family of Lie algebras of Block type
zbMATH Open document ID: 1051.17006
author: Yucai Su
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: The author investigates representations of Lie algebras \(B(s,G)\) over an algebraically closed field \(\mathbb F\) of characteristic zero, where \(s\) is 0 or 1 and \(G\) is an additive subgroup of \(\mathbb F\). \(B(s,G)\) has basis \(\{ x^{a,i} \mid a \in G, i \in \mathbb Z_+\}\), and \([x^{a,i},x^{b,j}] = s(b-a)x^{a+b,i+j} + ((a-1+s)j -(b-1+s)i)x^{a+b,i+j-1}\). It is shown that \(B(s,G)\) has a nontrivial quasifinite module (i.e. \(G\)-graded with all components finite dimensional) if and only if \(s = 1\) and \(G\) is isomorphic to \(\mathbb Z\). These modules are highest or lowest weight modules. A necessary and sufficient condition for an irreducible highest weight \(B(1,\mathbb Z)\)-module to be quasifinite is determined, and it is shown that the unitary ones are trivial.
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B60
zbMATH DE Number: 2093435
zbMATH Keywords: quasifinite module
zbMATH Keywords: highest weight modules
zbMATH Keywords: lowest weight modules
MaRDI profile type: Publication
full work available at URL: j.jpaa.2004.02.001
OpenAlex ID: W2050963884
cites work: On Torsion-Free Abelian Groups and Lie Algebras
cites work: Quasifinite highest weight modules over the Lie algebra of matrix differential operators on the circle
cites work: Integrable representations of affine Lie-algebras
cites work: Q4890829
cites work: \({\mathcal W}_{1+\infty}\) and \({\mathcal W}(gl_ N)\) with central charge \(N\)
cites work: Unitary quasi-finite representations of \(W_\infty\)
cites work: Quasifinite highest weight modules over the Lie algebra of differential operators on the circle
cites work: Quasifinite representations of classical Lie subalgebras of \({\mathcal W}_{1+\infty}\)
cites work: Classification of Harish-Chandra modules over the Virasoro Lie algebra
cites work: Q4934900
cites work: Infinite-dimensional Lie algebras of generalized Block type
cites work: On indecomposable modules over the Virasoro algebra
cites work: Derivations and structure of the Lie algebras of Xu type
cites work: Classification of quasifinite modules over the Lie algebras of Weyl type.
cites work: Classification of Harish-Chandra modules over the higher rank Virasoro algebras
cites work: Structure of Lie Superalgebras of Block Type Related to Locally Finite Derivations
cites work: Quasifinite representations of a Lie algebra of Block type
cites work: STRUCTURE OF THE LIE ALGEBRAS RELATED TO THOSE OF BLOCK
cites work: Generalizations of the Block algebras
cites work: Representations of the virasoro-like algebra and itsq-Analog
cites work: A Class of Infinite Dimensional Simple Lie Algebras
cites work: Structure of Degenerate Block Algebras
DOI: 10.1016/J.JPAA.2004.02.001
description: scientific article; zbMATH DE number 2093435