Summary: 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 \(\mathfrak{g}\). 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 \(\mathfrak{g}\) is reductive (e.g. \(\mathfrak{g}\) is simple, semi-simple, abelian, \(\dots)\) a complete classification of (almost-) graded central extensions is given. In particular, for \(\mathfrak{g}\) 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.
For the entire collection see [Zbl 1117.39001].