  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.
