  In this paper we study the relationship between the extended symmetries of
exact Courant algebroids over a manifold $M$, defined by Bursztyn, Cavalcanti
and Gualtieri, and the Poisson algebras of admissible functions associated to
twisted Dirac structures when acted by Lie groups. We show that the usual
homomorphisms of Lie algebras between the algebras of infinitesimal symmetries
of the action, vector fields on the manifold and the Poisson algebra of
observables, appearing in symplectic geometry, generalize to natural maps of
Leibniz algebras induced both by the extended action and compatible moment maps
associated to it in the context of twisted Dirac structures.
