Lie-admissible Algebras and Operads

A Lie-admissible algebra gives a Lie algebra by anticommutativity. In this work we describe remarkable types of Lie-admissible algebras such as Vinberg algebras, pre-Lie algebras or Lie algebras. We compute the corresponding binary quadratic operads and study their duality. Considering Lie algebras as Lie-admissible algebras, we can define for each Lie algebra a cohomology with values in an Lie-admissible module. This permits to study some deformations of Lie algebras, in particular classes of Lie-admissible algebras such as Vinberg algebras or pre-Lie algebras. 1 Lie-admissible algebras 1.1 Definition Let A = (A, μ) be a finite dimensional algebra on a commutative field K of characteristic zero. In this notation, μ is the law of A, that is a linear mapping μ : A⊗ A → A on the vector space A. We denote by aμ : A ⊗3 → A the associator of the law μ : aμ (X1, X2, X3) = μ (μ (X1, X2) , X3)− μ (X1, μ (X2, X3)) . Let ∑ n be the symmetric group of degree n. For every σ ∈ ∑ 3, we put σ (X1, X2, X3) = ( Xσ−1(1), Xσ−1(2), Xσ−1(3) ) . Definition 1.1 The algebra A = (A, μ) is called Lie-admissible if the law μ satisfies ∑ σ∈ ∑ 3 (−1) ε(σ) aμ ◦ σ = 0. (∗) This definition ([A]) is equivalent to say that the mapping [, ]μ : A ⊗ A → A defined by [X,Y ] = μ(X,Y ) − μ(Y,X) is a Lie bracket. We will denote by AL the corresponding Lie algebra.