A symmetric version of Kontsevich graph complex and Leibniz homology

TL;DR

This paper introduces a symmetric variant of the Kontsevich graph complex to compute Leibniz homology of symplectic vector fields, establishing a new algebraic isomorphism involving Zinbiel and associative structures.

Contribution

It develops a symmetric graph complex framework for Leibniz homology, extending Kontsevich's Lie homology results with novel algebraic structures.

Findings

Leibniz homology computed via a symmetric graph complex.

Isomorphism as Zinbiel-associative bialgebras.

Extension of Kontsevich's Lie homology results.

Abstract

Kontsevich has proven that the Lie homology of the Lie algebra of symplectic vector fields can be computed in terms of the homology of a graph complex. We prove that the Leibniz homology of this Lie algebra can be computed in terms of the homology of a variant of the graph complex endowed with an action of the symmetric groups. The resulting isomorphism is shown to be a Zinbiel-associative bialgebra isomorphism.