Orbits and representation of Krichever-Novikov affine-type algebras
type: Item
instance of: scholarly article
title: Orbits and representation of Krichever-Novikov affine-type algebras
zbMATH Open document ID: 0890.17026
DOI: 10.1007/BF02362645
published in: Journal of Mathematical Sciences (New York)
publication date: 1997-12-08T00:00:00Z
review text: In the global approach to Conformal Field Theory on higher genus Riemann surfaces initiated by Krichever and Novikov, one replaces the Virasoro algebra and the related algebras by the corresponding algebras which are given by meromorphic objects with poles only at two points (see the review Zbl 0820.17036 for more background information). Starting from any simple Lie algebra \(\mathfrak g\) (with associated group \(G\)) the generalized affine Lie algebra \(\hat{\mathfrak g}\) was introduced by the author in [\textit{O. K. Sheinman}, Affine Lie algebras on Riemann surfaces, Funct. Anal. Appl. 27, 266-272 (1993; Zbl 0820.17036)]. In this article the author describes representations of these affine-type Lie algebras. A suitable Weil group is defined and the coadjoint orbit method is used to classify representations. For the analogues of highest weight representations the concept of the weight has to be generalized. He introduces the notion of Bloch weights. The weight has \(2g+1\) components (\(g\) the genus of the Riemann surface). It can be related to the global meromorphic differentials which are holomorphic outside the two punctures and have at the punctures at most a pole of order 1. For these modules formal characters are introduced and an analogue of the Weyl-Kac character formula is given. The proof of the character formula can be found in [\textit{O. K. Sheinman}, Modules with highest weight for affine Lie algebras on Riemann surfaces, Funct. Anal. Appl. 29, 44-55 (1995; Zbl 0848.17023)].
Mathematics Subject Classification ID: 17B67
Mathematics Subject Classification ID: 14H55
Mathematics Subject Classification ID: 81R10
Mathematics Subject Classification ID: 22E67
Mathematics Subject Classification ID: 30F30
Mathematics Subject Classification ID: 81T40
zbMATH DE Number: 1017611
zbMATH Keywords: Kac-Moody algebras
zbMATH Keywords: Krichever-Novikov algebras
zbMATH Keywords: affine algebras
zbMATH Keywords: almost graded algebras
zbMATH Keywords: central extensions
zbMATH Keywords: Weyl-Kac character formula
zbMATH Keywords: highest weight modules
zbMATH Keywords: coadjoint orbit method
author: O. K. Sheinman
reviewed by: M. Schlichenmaier
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1017611