Rigid current Lie algebras

A current Lie algebra is contructed 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 such algebras and in the problem of rigidity. In particular we prove that a current Lie algebra is rigid if it is isomorphic to a direct product g× g × ...× g where g is a rigid Lie algebra. 1 Current Lie algebras If g is a Lie algebra over a field K and A a Kassociative commutative algebra, then g⊗A, provided with the bracket [X ⊗ a, Y ⊗ b] = [X,Y ]⊗ ab for everyX,Y ∈ g and a, b ∈ A is a Lie algebra. If dim(A) = 1 such an algebra is isomorphic to g. If dim(A) > 1 we will say that g⊗A with the previous bracket is a current Lie algebra. In [6] we have shown that if P is a quadratic operad , there is an associated quadratic operad, noted P̃ such that the tensor product of a P-algebra by a P̃-algebra is a P-algebra for the natural product. In particular, if the operad P is Lie, then L̃ie = Lie = Com and a Com-algebra is a commutative associative algebra. In this context we find again the notion of current Lie algebra. In this work we study the deformations of a current Lie algebra and we show that a current Lie algebra is rigid if and only if it is isomorphic to g × g × ... × g where g is a rigid Lie algebra. The notion of rigidity is related to the second group of the Chevalley cohomology. For the current Lie algebras, this group is not wellknown. Recently some relation between H(g ⊗ A, g ⊗ A) and H(g, g) and H H(A,A) are given in [7] but often when g is abelian. Let us note also that the scalar cohomology has been studied in [5]. [email protected]. corresponding author: e-mail: [email protected]