Odd Jacobi manifolds and Loday-Poisson brackets

TL;DR

This paper introduces Loday-Poisson brackets on odd Jacobi supermanifolds, extending classical Poisson structures to a non-skewsymmetric setting and exploring their properties and relations with Hamiltonian vector fields.

Contribution

It constructs a non-skewsymmetric Loday-Poisson bracket on odd Jacobi manifolds and analyzes its relation to Hamiltonian vector fields and noncommutative products.

Findings

Loday-Poisson brackets satisfy Leibniz rule over noncommutative product

Relations between Hamiltonian vector fields and Loday-Poisson structure are established

Loday-Poisson brackets generalize classical Poisson brackets to odd Jacobi manifolds

Abstract

In this paper we construct a non-skewsymmetric version of a Poisson bracket on the algebra of smooth functions on an odd Jacobi supermanifold. We refer to such Poisson-like brackets as Loday-Poisson brackets. We examine the relations between the Hamiltonian vector fields with respect to both the odd Jacobi structure and the Loday-Poisson structure. Furthermore, we show that the Loday-Poisson bracket satisfies the Leibniz rule over the noncommutative product derived from the homological vector field.