Extended Symmetries and Poisson Algebras Associated to Twisted Dirac Structures

TL;DR

This paper explores the connection between extended symmetries of Courant algebroids and Poisson algebras in twisted Dirac structures, revealing generalized algebraic homomorphisms in geometric contexts.

Contribution

It introduces a framework linking extended symmetries and Poisson algebras via Leibniz algebra maps in twisted Dirac structures, generalizing classical symplectic geometry results.

Findings

Generalized homomorphisms between symmetry algebras and Poisson algebras

Extension of Lie algebra homomorphisms to Leibniz algebra maps

Compatibility of extended actions with moment maps in twisted Dirac structures

Abstract

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.