Rigid Current Lie Algebras

Abstract

A current Lie algebra is constructed from a tensor product of a Lie algebra and a commutative associative algebra of dimension greater than 2. In this work we are interested in deformations of finite dimensional current Lie algebras and in the problem of rigidity. In particular we prove that a complex finite dimensional current Lie algebra with trivial center is rigid if it is isomorphic to a direct product $$\mathfrak {g}\times \mathfrak {g}\times \cdots \times \mathfrak {g}$$ where $$\mathfrak {g}$$ is a rigid Lie algebra.