#
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.