Dirac Lie groups, Dirac homogeneous spaces and the Theorem of Drinfeld

TL;DR

This paper extends the concepts of Poisson Lie groups and homogeneous spaces to the Dirac setting, proving that Drinfeld's correspondence theorem remains valid in this broader context.

Contribution

It generalizes the classical theory to Dirac structures and establishes the validity of Drinfeld's theorem for Dirac Lie groups and homogeneous spaces.

Findings

Drinfeld's correspondence theorem holds in the Dirac setting.

Poisson Lie groups and homogeneous spaces are extended to Dirac structures.

The paper provides a framework connecting Dirac geometry with Lie bialgebras.

Abstract

The notions of \emph{Poisson Lie group} and \emph{Poisson homogeneous space} are extended to the Dirac category. The theorem of Drinfel$'$d (\cite{Drinfeld93}) on the one-to-one correspondence between Poisson homogeneous spaces of a Poisson Lie group and a special class of Lagrangian subalgebras of the Lie bialgebra associated to the Poisson Lie group is proved to hold in this more general setting.