The structure of standard modules. I: Universal algebras and the Rogers- Ramanujan identities
type: Item
instance of: scholarly article
title: The structure of standard modules. I: Universal algebras and the Rogers- Ramanujan identities
zbMATH Open document ID: 0577.17009
DOI: 10.1007/BF01388447
author: J. Lepowsky
author: Robert Wilson
published in: Inventiones Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143146
review text: A new approach to the construction and study of algebras of endomorphisms of graded vector spaces \(V=\coprod V_ i\), where i ranges over the integers and \(V_ i=0\) for i sufficiently large, is introduced. To each family \(Z_ i\) of endomorphisms of V, where deg \(Z_ i=i\), the formal Laurent series \(Z(\zeta)=\sum Z_ i \zeta^ i\) is associated. In various settings, identities of the form \(a(\zeta_ 1/\zeta_ 2)Z(\zeta_ 1)Z(\zeta_ 2)+b(\zeta_ 2/\zeta_ 1)Z(\zeta_ 2)Z(\zeta_ 1)=c(\zeta_ 1/\zeta\) \({}_ 2)Z(\zeta_ 2)+d(\zeta_ 1/\zeta_ 2)\), where a(\(\zeta)\), b(\(\zeta)\), c(\(\zeta)\), d(\(\zeta)\) are formal Laurent series with complex coefficients, are proved. This improves our Lie-theoretic understanding of the Rogers-Ramanujan identities. Also this approach gives explicit constructions of very general modules for affine Lie algebras - modules whose construction had been inaccessible before.
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 3922917
zbMATH Keywords: algebras of endomorphisms of graded vector spaces
zbMATH Keywords: formal Laurent series
zbMATH Keywords: identities
zbMATH Keywords: explicit constructions
zbMATH Keywords: general modules for affine Lie algebras
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
cites work: Transformation groups for soliton equations. Euclidean Lie algebras and reduction of the KP hierarchy
cites work: Vertex operators and \(\tau\)-functions. Transformation groups for soliton equations. II
cites work: The Weyl-Kac character formula and power series identities
cites work: Q4740715
cites work: Basic representations of affine Lie algebras and dual resonance models
cites work: Lie algebra homology and the Macdonald-Kac formulas
cites work: Q3897665
cites work: Fixed points of diffeomorphisms of two-dimensional spheres preserving an oriented plane
cites work: Infinite-dimensional algebras, Dedekind's \(\eta\)-function, classical Möbius function and the very strange formula
cites work: Realization of the basic representations of the Euclidean Lie algebras
cites work: The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group
cites work: Macdonald-type identities
cites work: Generalized Verma modules, loop space cohomology and MacDonald-type identities
cites work: Application of the numerator formula to k-rowed plane partitions
cites work: Q4740716
cites work: Lie algebras and classical partition identities
cites work: Lie algebraic approaches to classical partition identities
cites work: Construction of the affine Lie algebra \(A^{(1)}_1\)
cites work: The Rogers-Ramanujan identities: Lie theoretic interpretation and proof
cites work: A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities
cites work: A new family of algebras underlying the Rogers-Ramanujan identities and generalizations
cites work: Affine root systems and Dedekind's \(\eta\)-function
cites work: A new class of Lie algebras
cites work: Euclidean Lie Algebras
cites work: Unitary representations of some infinite dimensional groups
cites work: Structure of certain standard modules for \(A_ n^{(1)}\) and the Rogers-Ramanujan identities
cites work: Structure of some standard modules for \(C_ n^{(1)}\)
cites work: The structure of standard modules. II: The case \(A_ 1^{(1)}\), principal gradation
Recommended article: Structure of certain standard modules for \(A_ n^{(1)}\) and the Rogers-Ramanujan identities
Recommended article: Standard modules, induction and the structure of the Temperley-Lieb algebra
Recommended article: Q3220719
Recommended article: Structure of some standard modules for \(C_ n^{(1)}\)
Recommended article: Q3140283
Recommended article: A new family of algebras underlying the Rogers-Ramanujan identities and generalizations
Recommended article: On irreducibility of standard modules for generic representations
Recommended article: On the standard modules conjecture
Recommended article: ON SOME STANDARD GRADED ALGEBRAS IN MODULAR INVARIANT THEORY
Recommended article: A theorem of Mumford and Ramanujam for universal algebras
description: scientific article; zbMATH DE number 3922917