Mathematics – Quantum Algebra
Scientific paper
2005-10-20
Mathematics
Quantum Algebra
9 pages, Talk presented at the International Conference on Difference Equations, Special Functions, and Applications, Munich,
Scientific paper
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.
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