Rigid Current Lie Algebras
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
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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 g × g × · · · × g where g is a rigid Lie algebra.
1 Current Lie Algebras If g is a Lie algebra over a algebraically closed field K and A a K-associative commutative algebra, then g ⊗ A , provided with the bracket [X ⊗ a, Y ⊗ b] = [X, Y ] ⊗ ab for every X, 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 [16] we have shown that if P is a quadratic operad, there is an associated ˜ quadratic operad, noted P˜ such that the tensor product of a P-algebra by a Palgebra is a P-algebra for the natural product. In particular, if the operad P is L ie, then L˜ie = L ie! = C om and a C om-algebra is a commutative associative algebra. In this context we find again the notion of current Lie algebra. E. Remm (B) · M. Goze Université de Haute Alsace, 68093 Mulhouse, France e-mail: [email protected] M. Goze e-mail: [email protected] A. Makhlouf et al. (eds.), Algebra, Geometry and Mathematical Physics, 247 Springer Proceedings in Mathematics & Statistics 85, DOI: 10.1007/978-3-642-55361-5_14, © Springer-Verlag Berlin Heidelberg 2014
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E. Remm and M. Goze
Remark In [3], the notion of duplication of algebras constructed by tensor product is presented. If g is a Lie algebra, we define on g ⊗ g the product μ(X ⊗ Y , X ⊗ Y ) = [X, Y ] ⊗ [X , Y ]. But, in this case, g ⊗ g is not a Lie algebra, but is related with the notion of n-Lie algebras. In this work we study the deformations of finite dimensional current Lie algebras and we study the rigidity. The notion of rigidity is related to the second group of the Chevalley-Eilenberg cohomology. For the current Lie algebras, this group is not well known. Recently some relations between H 2 (g ⊗ A , g ⊗ A ), H 2 (g, g) and HH2 (A , A ) have been given in [18] but often when g is abelian. Let us note also that the scalar cohomology has been studied in [15].
2 Determination of Rigid Current Lie Algebras In all this work, Lie algebras or associative algebras are of finite dimension over the algebraically closed field K.
2.1 On the Rigidity of Lie Algebras Let us remind briefly some properties of the variety of Lie algebras (for more details, see [1]). Let g be a n-dimensional K-Lie algebra. Since the underlying vector space is isomorphic to Kn , there exists a one-to-one correspondance between the set of Lie brackets of n-dimensional Lie algebras and the skew-symmetric bilinear maps μ : Kn × Kn → Kn satisfying the Jacobi identity. We denote by μg this bilinear map corresponding to g. In this framework, we can identify g with the pair
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