Irreducible representations for toroidal Lie algebras
type: Item
instance of: scholarly article
title: Irreducible representations for toroidal Lie algebras
zbMATH Open document ID: 0942.17016
author: Stephen Berman
author: Yuly Billig
published in: Journal of Algebra
publication date: 2000-08-22T00:00:00Z
review text: A toroidal Lie algebra is a Lie algebra \({\mathfrak g}\) which has the structure \[ {\mathfrak g} = \dot{\mathfrak g} \otimes \mathbb C[t_0^{\pm 1}, \ldots, t_n^{\pm 1}] \oplus {\mathcal K} \oplus {\mathcal D}, \] where \(\dot{\mathfrak g}\) is a finite-dimensional simple complex Lie algebra, \({\mathcal K}\) is central and \({\mathcal D}\) is a certain Lie algebra of derivations of \(\mathbb C[t_0^{\pm 1}, \ldots, t_n^{\pm 1}]\). The main purpose of the paper under review is the construction of irreducible representations of toroidal Lie algebras with finite-dimensional weight spaces with respect to certain natural gradings. The techniques to obtain these results combine the previously used vertex operator techniques with classical Verma module techniques to a new construction process satisfying all requirements. More precisely, the construction starts with an irreducible module \(W\) of \({\mathfrak {gl}}(n,\mathbb C)\). Then \(T(W) := \mathbb C[t_1^{\pm 1}, \ldots, t_n^{\pm 1}] \otimes W\) is a module of \(\text{Der}(\mathbb C[t_1^{\pm 1}, \ldots, t_n^{\pm 1}])\) and, in addition, of the zero part \({\mathfrak v}_0\) of a certain \(\mathbb Z\)-graded Lie algebra \({\mathfrak v}\) with decomposition \({\mathfrak v} = {\mathfrak v}_+ \oplus {\mathfrak v}_0 \oplus {\mathfrak v}_-\). Then we pass to the induced module \(M := \text{Ind}_{{\mathfrak v}_0 + {\mathfrak v}_+}^{\mathfrak v}(T(W))\) and further to an \({\mathfrak v}\)-irreducible quotient module \(L\) of \(M\). In the final step an irreducible highest weight module of an affine Lie algebra \(\dot{\mathfrak g} \otimes \mathbb C[t_0, t_0^{-1}] \oplus \mathbb C k_0 \oplus \mathbb C d_0\), \(k_0 \in {\mathcal K}\), \(d_0 \in {\mathcal D}\) is tensored with \(L\), which eventually leads to a module of \({\mathfrak g}\). It is remarkable that the results leading to finite-dimensionality of certain weight spaces work in amazing generality for the class of \(\mathbb Z\)-graded Lie algebras with polynomial multipli…
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B10
zbMATH DE Number: 1384444
zbMATH Keywords: toroidal Lie algebra
zbMATH Keywords: vertex operator
zbMATH Keywords: graded Lie algebras
zbMATH Keywords: finite dimensional weight spaces
reviewed by: Karl-Hermann Neeb
MaRDI profile type: Publication
full work available at URL: jabr.1999.7961
OpenAlex ID: W2066123174
Wikidata QID: Q115395802
cites work: Q3755625
cites work: Enveloping algebras and representations of toroidal Lie algebras
cites work: Quantum tori and the structure of elliptic quasi-simple Lie algebras
cites work: Principal vertex operator representations for toroidal Lie algebras
cites work: An extension of the Korteweg-de Vries hierarchy arising from a representation of a toroidal Lie algebra
cites work: Borel subalgebras and categories of highest weight modules for toroidal Lie algebras
cites work: Virasoro type Lie algebras and deformations
cites work: Vertex representations for \(n\)-toroidal Lie algebras and a generalization of the Virasoro algebra
cites work: Q3223012
cites work: Q3983629
cites work: Higher-Dimensional WZW Model on Kähler Manifold and Toroidal Lie Algebra
cites work: Hirota bilinear forms with \(2\)-toroidal symmetry.
cites work: Q4040877
cites work: Kähler differentials and coverings of complex simple Lie algebras extended over a commutative algebra
cites work: Q3999650
cites work: CONFORMAL FIELDS: A CLASS OF REPRESENTATIONS OF Vect(N)
cites work: Lowest-energy representations of non-centrally extended diffeomorphism algebras
cites work: Toroidal Lie algebras and vertex representations
cites work: Irreducible representations of the Lie-algebra of the diffeomorphisms of a \(d\)-dimensional torus
cites work: IRREDUCIBLE REPRESENTATIONS OF INFINITE-DIMENSIONAL LIE ALGEBRAS OF CARTAN TYPE
cites work: Toroidal actions on level 1 modules of \(U_q(\widehat{sl}_n)\)
cites work: Principal construction of the toroidal Lie algebra of type \(A_1\)
cites work: Double-loop algebras and the Fock space
DOI: 10.1006/JABR.1999.7961
description: scientific article; zbMATH DE number 1384444