Poisson Algebras and Graphs

Abstract

A non-commutative Poisson algebra is a Lie algebra endowed with a, not necessarily commutative, associative product in such a way that the Lie and associative products are compatible via the Leibniz identity. If we consider a non-commutative Poisson algebra \(\mathcal {P}\) of arbitrary dimension, over an arbitrary base field \({\mathbb F}\), a basis \(\mathcal {B}=\{v_{i}\}_{i \in I}\) of \(\mathcal {P}\) is called multiplicative if for any i, j ∈ I we have that \([v_i,v_j]\in \mathbb {F}v_{r}\) and that \(v_iv_j\in \mathbb {F}v_{s}\) for some r, s ∈ I. We associate an adequate graph (V, E) to \(\mathcal {P}\) relative to \(\mathcal {B}\). By arguing on this graph we show that \(\mathcal {P}\) decomposes as a direct sum of ideals, each one being associated to one connected component of (V, E). Also the minimality of \(\mathcal {P}\) and the division property of \(\mathcal {B}\) are characterized in terms of the weak symmetry of the graph.