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Deformations of the Virasoro algebra of Krichever-Novikov type. (English) Zbl 1196.17023
Summary: It is the general impression that deformation problems are always governed by cohomology spaces. In this contribution we consider the deformation of Lie algebras. There this close connection is true for finite-dimensional algebras, but fails for infinite dimensional ones. We construct geometric families of infinite dimensional Lie algebras over the moduli space of complex one-dimensional tori with marked points. These algebras are algebras of Krichever-Novikov type which consist of meromorphic vector fields of certain type over the tori. The families are non-trivial deformations of the (infinite dimensional) Witt algebra, and the Virasoro algebra, respectively, despite the fact that the cohomology space associated to the deformation problem of the Witt algebra vanishes, and hence the algebra is formally rigid. A similar construction works for current algebras.
The presented results are jointly obtained with Alice Fialowski.
##### MSC:
 17B68 Virasoro and related algebras 17B55 Homological methods in Lie (super)algebras 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, $$W$$-algebras and other current algebras and their representations