Poisson color algebras of arbitrary degree

TL;DR

This paper introduces and studies Poisson color algebras of arbitrary degree, a broad class encompassing many known algebraic structures, without restrictions on dimension or base field.

Contribution

It extends the concept of Poisson algebras to color and arbitrary degree, unifying various algebraic structures under a common framework.

Findings

Characterization of Poisson color algebras of arbitrary degree

Inclusion of Lie, Poisson, and Gerstenhaber algebras as special cases

Structural properties without restrictions on dimension or base field

Abstract

A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a graded-commutative associative product and a graded-version Leibniz rule we get a so-called Poisson color algebra (of degree zero). This concept can be extended to any degree so as to obtain the class of Poisson color algebras of arbitrary degree. This class turns out to be a wide class of algebras containing the ones of Lie color algebras (and so Lie superalgebras and Lie algebras), Poisson algebras, graded Poisson algebras, $z$-Poisson algebras, Gerstenhaber algebras and Schouten algebras among others classes of algebras. The present paper is devoted to the study of the structure of Poisson color algebras of arbitrary degree, with restrictions neither on the dimension nor the base field.