Defining relations and Gr\"obner--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 setting for the calculation of a Gr\"obner--Shirshov basis in a module over an associative conformal algebra and apply this technique to Poisson algebras considered as conformal modules over appropriate associative envelopes of current Lie conformal algebras. As a result, we obtain a setting for the calculation of a Gr\"obner--Shirshov basis in a Poisson algebra.