Leibniz algebras on symplectic plane and cohomological vector fields

TL;DR

This paper explores the connection between Leibniz algebras, symplectic geometry, and cohomological vector fields, introducing new structures and properties within this mathematical framework.

Contribution

It introduces cohomological vector fields on symplectic planes and demonstrates their role in generating Leibniz algebras via derived brackets, extending cohomological field theory to Leibniz structures.

Findings

Cohomological vector fields induce Leibniz algebras.

Properties and factorization of cohomological fields are analyzed.

Introduction of double-algebra concept in Leibniz category.

Abstract

By using help of algebraic operad theory, Leibniz algebra theory and symplectic-Poisson geometry are connected. We introduce the notion of cohomological vector field defined on nongraded symplectic plane. It will be proved that the cohomological vector fields induce the finite dimensional Leibniz algebras by the derived bracket construction. This proposition is a Leibniz analogue of the cohomological field theory in the category of Lie algebras. The basic properties of the cohomological fields will be studied, in particular, we discuss a factorization problem with the cohomological fields and introduce the notion of double-algebra in the category of Leibniz algebras.