Poisson structures on sets of Maurer-Cartan elements

TL;DR

This paper introduces Poisson structures on the gauge orbits of Maurer-Cartan elements in differential graded Lie algebras, linking them to Batalin-Vilkovisky algebras and providing new geometric and algebraic invariants.

Contribution

It constructs Maurer-Cartan-Poisson structures for certain dglas, connecting them to Batalin-Vilkovisky algebras and Lie algebroids, with applications to symplectic geometry.

Findings

Existence of MCP structures for dglas related to Frobenius algebras and Poisson deformations

Definition of Hamiltonian flows and Lie algebroids on gauge orbits

Construction of a finite-dimensional Lie algebra for symplectic manifolds

Abstract

Given a differential graded Lie algebra (dgla) L satisfying certain conditions, we construct Poisson structures on the gauge orbits of its set of Maurer-Cartan (MC) elements, termed Maurer-Cartan-Poisson (MCP) structures. They associate a compatible Batalin-Vilkovisky algebra to each MC element of L. An MCP structure is shown to exist for a number of dglas associated to commutative Frobenius algebras, deformations of Poisson and symplectic structures, as well as the Chevally-Eilenberg complex. MCP structures yield a notion of hamiltonian flow of MC elements, and also define Lie algebroids on gauge orbits, whose isotropy algebras give invariants of MC elements. As an example, this gives a finite-dimensional two-step nilpotent graded Lie algebra associated to any closed symplectic manifold.