Higher Genus Affine Lie Algebras of Krichever -- Novikov Type

Abstract

Classical affine Lie algebras appear e.g. as symmetries of infinite dimensional integrable systems and are related to certain differential equations. They are central extensions of current algebras associated to finite-dimensional Lie algebras g. In geometric terms these current algebras might be described as Lie algebra valued meromorphic functions on the Riemann sphere with two possible poles. They carry a natural grading. In this talk the generalization to higher genus compact Riemann surfaces and more poles is reviewed. In case that the Lie algebra g is reductive (e.g. g is simple, semi-simple, abelian, ...) a complete classification of (almost-) graded central extensions is given. In particular, for g simple there exists a unique non-trivial (almost-)graded extension class. The considered algebras are related to difference equations, special functions and play a role in Conformal Field Theory.

9 pages, Talk presented at the International Conference on Difference Equations, Special Functions, and Applications, Munich, July 2005

References in corpus (5)

• The Wess-Zumino-Witten-Novikov theory, Knizhnik-Zamolodchikov equations, and Krichever-Novikov algebras, I
• Degenerations of Generalized Krichever Novikov algebras on Tori
• 2D Toda Chain, Commuting Difference Operators and Holomorphic Bundles
• Differential Operator Algebras on compact Riemann Surfaces
• Extensions of diffeomorphism and current algebras