Annihilating ideals of standard modules of \({\mathfrak sl}(2,{\mathbb{C}})^\sim\) and combinatorial identities
type: Item
instance of: scholarly article
title: Annihilating ideals of standard modules of \({\mathfrak sl}(2,{\mathbb{C}})^\sim\) and combinatorial identities
zbMATH Open document ID: 0635.17006
DOI: 10.1016/0001-8708(87)90008-9
published in: Advances in Mathematics
publication date: 1987-01-01T00:00:00Z
review text: The paper is devoted to the study of modules over the twisted affine Lie algebra \({\mathfrak g}={\mathfrak sl}(2,{\mathbb{C}})^{\sim}\). Let \({\mathcal F}\) be a family of modules which contains a copy of each standard (i.e. highest weight) \({\mathfrak g}\)-module. The universal enveloping algebra \(U=U({\mathfrak g})\) acts on a space \(W_{{\mathcal F}}=\prod_{V\in {\mathcal F}}V\) which can be regarded as a discrete topological space. Using the topology of pointwise convergence on \(U\subset End(W_{{\mathcal F}})\) (which the authors prove to be Hausdorff) one can define the completion \(\bar U\) of U. Each \(V\in {\mathcal F}\) is also a \(\bar U\)-module. The authors use certain formal Laurent series with coefficients in U and \(\bar U\) as generating functions for the elements which annihilate standard \({\mathfrak g}\)-modules. These Laurent series generalize the so called vertex operator construction for fundamental modules which was introduced by \textit{J. Lepowsky} and \textit{R. L. Wilson} [Commun. Math. Phys. 62, 43-53 (1978; Zbl 0388.17006)]. Further, the authors describe the bases of standard \({\mathfrak g}\)-modules and the bases of maximal \({\mathfrak g}\)-submodules of Verma modules. This description combined with the Weyl-Kac character formula provides a general relation which comprises in itself the combinatorial identities of Rogers and Ramanujan as well as their generalizations due to Gordon, Andrews and Bressoud. As the authors point out, their approach to the subject follow in many ways the basic ideas of Lepowsky and Wilson [Invent. Math. 77, 199-290 (1984; Zbl 0577.17009); ibid. 79, 417-442 (1985; Zbl 0577.17010)].
Mathematics Subject Classification ID: 17B65
Mathematics Subject Classification ID: 17B10
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 4033960
zbMATH Keywords: highest weight modules
zbMATH Keywords: standard module
zbMATH Keywords: basis of standard module
zbMATH Keywords: twisted affine Lie algebra
zbMATH Keywords: formal Laurent series
zbMATH Keywords: combinatorial identities
author: Arne Meurman
author: Mirko Primc
MaRDI profile type: Publication
full work available at URL: 0001-8708(87)90008-9
OpenAlex ID: W2080947546
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description: scientific article; zbMATH DE number 4033960