Level three standard modules \(A^{(2)}_ 2\) and combinatorial identities
type: Item
instance of: scholarly article
title: Level three standard modules \(A^{(2)}_ 2\) and combinatorial identities
zbMATH Open document ID: 0854.17029
DOI: 10.1016/0022-4049(94)00127-8
author: Manvendra Tamba
author: Chuanfu Xie
published in: Journal of Pure and Applied Algebra
publication date: 1996-06-12T00:00:00Z
review text: There is a close connection between the theory of representations of affine Lie algebras and combinatorial identities of the Rogers-Ramanujan type. Work of J. Lepowsky and R. L. Wilson gave a Lie-theoretic explanation and proof of the classic Rogers-Ramanujan identities, via the level 3 standard representations of the rank one affine algebra \(A_1^{(1)}\). They also laid the foundations for much of the work that was later done on this line of research by several researchers. Typically these works led to, or made use of, well-known partition identities. The first time that this method gave rise to partition identities that appeared to be unknown to specialists was in the paper by the reviewer [J. Algebra 154, 335-355 (1993; Zbl 0774.17029)]. This involved the study of the level 3 modules of the affine algebra \(A_2^{(2)}\). A proof of these identities was not included in that paper. The authors of the paper under review provide such a proof along precisely the lines of the \(\mathbb{Z}\)-algebra theory of Lepowsky and Wilson. They prove generalized anticommutation relations which, together with the Lepowsky-Wilson theory, allow them to prove the spanning of a suitable set of vectors in the level 3 representation space of \(A_2^{(2)}\). The proof of independence is carried out through a delicate analysis of relations among generating functions of operators acting on the vacuum space. As a consequence they obtain a proof of the conjectured identities. It must be noticed that a proof of one of the identities had been obtained by \textit{G. E. Andrews} [Contemp. Math. 166, 141-154 (1994; Zbl 0811.05001)], using combinatorial methods. A combinatorial proof of the other identity was presented in the reviewer's article [Commun. Algebra 23, 2959-2969 (1995; Zbl 0830.17012)]. A proof by the reviewer using Lie theory but a different approach has also appeared in Trans. Am. Math. Soc. 348, 481-501 (1996).
reviewed by: Stefano Capparelli
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 856979
zbMATH Keywords: representations of affine Lie algebras
zbMATH Keywords: combinatorial identities of the Rogers-Ramanujan type
zbMATH Keywords: partition identities
zbMATH Keywords: Lepowsky-Wilson theory
zbMATH Keywords: standard modules
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)00127-8
OpenAlex ID: W1994046140
cites work: Q4310745
cites work: On some representations of twisted affine Lie algebras and combinatorial identities
cites work: Q3983629
cites work: Q3693664
cites work: Lie algebraic approaches to classical partition identities
cites work: A new family of algebras underlying the Rogers-Ramanujan identities and generalizations
cites work: The structure of standard modules. I: Universal algebras and the Rogers- Ramanujan identities
cites work: The structure of standard modules. II: The case \(A_ 1^{(1)}\), principal gradation
cites work: Q3752559
cites work: Annihilating ideals of standard modules of \({\mathfrak sl}(2,{\mathbb{C}})^\sim\) and combinatorial identities
cites work: Structure of the level two standard modules for the affine lie algebra D(3)4
cites work: Level two standard \(D_{I+1}^{(2)}\)-modules
cites work: Structure of the level two Standard Modules for the affine Lie algebra
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description: scientific article; zbMATH DE number 856979