Martin Schlichenmaier

Geometric Deformations of Witt and Virasoro algebra by algebras of Krichever-Novikov type

Finite-dimensional Lie algebras have been the subject of extensive studies for quite a long time. Some basic results true in finite dimensions, will become false in infinite dimensions.

In joint work with Alice Fialowski I constructed geometric families of infinite dimensional Lie algebras of vector fields and currents over the moduli space of complex one-dimensional tori with marked points. These algebras are algebras of Krichever-Novikov type and they consist of meromorphic objects over the tori.

The families are non-trivial deformations of the (infinite dimensional) Witt algebra resp. of the classical current algebras. The result should be compared to the fact, that these algebras are formally rigid, i.e. on the formal level every deformation will be trivial. In finitely many dimensions formal rigidity implies rigidity. The construction shows that this is not true anymore in infinite dimensions.