Defining relations and Gröbner–Shirshov bases of Poisson algebras as of conformal modules

Abstract

We study the relation between Poisson algebras and representations of Lie conformal algebras. We establish a Gröbner–Shirshov basis theory framework for modules over associative conformal algebras and apply this technique to Poisson algebras considered as conformal modules over appropriate associative conformal envelopes of current Lie conformal algebras. As a result, we obtain a setting for the calculation of a Gröbner–Shirshov basis in a Poisson algebra.