Non-associative algebras associated to Poisson algebras

Poisson algebras are usually defined as structures with two operations, a commutative associative one and an anti-commutative one that satisfies the Jacobi identity. These operations are tied up by a distributive law, the Leibniz rule. We present Poisson algebras as algebras with one operation, which enables us to study them as part of non-associative algebras. We study the algebraic and cohomological properties of Poisson algebras, their deformations as non-associative algebras, and give the classification in low dimensions. 1 Poisson algebras presented as non-associative algebras 1.1 Non-associative algebra associated to a Poisson algebra A Poisson algebra over a field K is a K-vector space P equipped with two bilinear products: 1) a Lie algebra multiplication, denoted by {, }, called the Poisson bracket, 2) an associative commutative multiplication, denoted by •. These two operations satisfy the Leibniz condition: {X • Y, Z} = X • {Y, Z}+ {X,Z} • Y, (1) for all X,Y, Z ∈ g. This means that the Poisson bracket acts as a derivation of the associative product. Proposition 1 Let K be a field of characteristic different from 3 or 2. Let · be a bilinear product on the K-vector space P and define the operations {, } and • by {X,Y } = 1 2 (X · Y − Y ·X), (2) X • Y = 1 2 (X · Y + Y ·X). (3) 1 Then these operations endow P with a Poisson algebra structure if and only if the product · satisfies: 3A·(X,Y, Z) = (X · Z) · Y + (Y · Z) ·X − (Y ·X) · Z − (Z ·X) · Y, (4) where A·(X,Y, Z) = (X · Y ) · Z −X · (Y · Z) is the associator of the product ·. Proof: [13]. We shall say that (P , ·) whereX ·Y = {X,Y }+X•Y, is a Poisson algebra instead of “the non-associative algebra associated to the Poisson algebra (P , {, }, •)”, the algebra (P , {, }, •) being a “classical” Poisson algebra. There is a one to one correspondence between the category of Poisson algebras defined as nonassociative algebras (P , ·) satisfying Equation (4) and the category of ”classical” Poisson algebras (P , {, }, •). The aim of this work is to study the non-associative algebra (P , ·) associated to a Poisson algebra. Given a Poisson algebra (P , ·), the corresponding Lie algebra (P , {, }) will be denoted by gP , while the associative commutative algebra (P , •) is denoted by AP ; Notation. In the following, we will denote (when no confusion is possible) the Poisson product by XY instead of X · Y . Proposition 2 A Poisson algebra (P , .) is flexible. Proof. A non-associative algebra is called flexible if its associator satisfies A·(X,Y,X) = 0. If (P , .) is a Poisson algebra, we have from (4) that 3A·(X,Y,X) = X Y + (Y X)X − (Y X)X −XY = 0 , where X = XX. We deduce easily that the associator of Poisson product satisfies A·(X,Y, Z) +A·(Z, Y,X) = 0 (flexibility), (5) A·(X,Y, Z) +A·(Y, Z,X)−A·(Y,X,Z) = 0. (6) This last relation is obtained written the identity (4) for the triples (X,Y, Z), (Y, Z,X) and (Y,X,Z). Remark. The previous system constituted by Equations (5) and (6) is equivalent to 2A·(X,Y, Z) + 1 2 A·(Y,X,Z) +A·(Z, Y,X) +A·(Y, Z,X) + 3 2 A·(Z,X, Y ) = 0 (7) but a non-associative algebra satisfying (7) is not always a Poisson algebra.