Orbits and representation of Krichever-Novikov affine-type algebras
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)].
scientific article; zbMATH DE number 1017611