Annihilating ideals of standard modules of \({\mathfrak sl}(2,{\mathbb{C}})^\sim\) and combinatorial identities
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)].
scientific article; zbMATH DE number 4033960
scientific article