To a generic holomorphic vector bundle on an algebraic curve and an irreducible finite-dimensional representation of a semisimple Lie algebra, we assign a representation of the corresponding affine Krichever–Novikov algebra in the space of semi-infinite exterior forms. It is shown that equivalent pairs of data give rise to equivalent representations and vice versa.

Abstract. We construct algebraic-geometric families of genus one (i.e. elliptic) current and affine Lie algebras of Krichever-Novikov type. These families deform the classical current, respectively affine Kac-Moody Lie algebras. The construction is induced by the geometric process of degenerating the elliptic curve to singular cubics. If the finitedimensional Lie algebra defining the infinite dimensional current algebra is simple then, even if restricted to local families, the constructed families are non-equivalent to the trivial family. In particular, we show that the current algebra is geometrically not rigid, despite its formal rigidity. This shows that in the infinite-dimensional Lie algebra case the relations between geometric deformations, formal deformations and Lie algebra twocohomology are not that close as in the finite-dimensional case. The constructed families are e.g. of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. 1.

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

Abstract. By considering non-trivial global deformations of the Witt (and the Virasoro) algebra given by geometric constructions it is shown that, despite their infinitesimal and formal rigidity, they are globally not rigid. This shows the need of a clear indication of the type of deformations considered. The families appearing are constructed as families of algebras of Krichever-Novikov type. They show up in a natural way in the global operator approach to the quantization of two-dimensional conformal field theory. In addition, a proof of the infinitesimal and formal rigidity of the Witt algebra is presented. We dedicate this article to the memory of our good friend Peter Slodowy who passed away in 2002 1.

Dedicated to Professor S.P.Novikov in honour of his 65th Birthday Abstract. The survey of the current state of the theory of Krichever-Novikov algebras including new results on local central extensions, invariants, representations and casimir operators. Contents

We present a classical r-matrix for the Hitchin system without marked points on an arbitrary non-degenerate algebraic curve of genus g ≥ 2 using Tyurin parameterization of holomorphic vector bundles. 1

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

Abstract. Classical affine Lie algebras appear e.g. as symmetries of infinite dimensional integrablesystems and are related to certain differentialequations. 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. Talk presented at the International Conference on Difference Equations, Special Functions, and Applications, Munich, July 2005

Abstract. In two earlier articles we constructed algebraic-geometric families of genus one (i.e. elliptic) Lie algebras of Krichever-Novikov type. The considered algebras are vector fields, current and affine Lie algebras. These families deform the Witt algebra, the Virasoro algebra, the classical current, and the affine Kac-Moody Lie algebras respectively. The constructed families are not equivalent (not even locally) to the trivial families, despite the fact that the classical algebras are formally rigid. This effect is due to the fact that the algebras are infinite dimensional. In this article the results are reviewed and developed further. The constructions are induced by the geometric process of degenerating the elliptic curves to singular cubics. The algebras are of relevance in the global operator approach to the Wess-Zumino-Witten-Novikov models appearing in the quantization of Conformal Field Theory. 1.

integrable systems