The Classification of Dirac Homogeneous Spaces

TL;DR

This paper extends the classification of Poisson Lie groups to Dirac Lie groups and their homogeneous spaces, using Courant algebroids and coisotropic subalgebras, providing a broader framework for understanding these geometric structures.

Contribution

It introduces a classification of homogeneous spaces for Dirac Lie groups via $K$-invariant coisotropic subalgebras, generalizing Drinfeld's Poisson case.

Findings

Classified Dirac homogeneous spaces using coisotropic subalgebras.

Unified Poisson and Dirac morphism frameworks.

Extended Drinfeld's classification to Dirac structures.

Abstract

A well known result of Drinfeld classifies Poisson Lie groups $(H,\Pi)$ in terms of Lie algebraic data in the form of Manin triples $(\mathfrak{d},\mathfrak{g},\mathfrak{h})$; he also classified compatible Poisson structures on $H$-homogeneous spaces $H/K$ in terms of Lagrangian subalgebras $\mathfrak{l}\subset\mathfrak{d}$ with $\mathfrak{l}\cap\mathfrak{h}=\mathfrak{k}=\mathrm{Lie}(K)$. Using the language of Courant algebroids and groupoids, Li-Bland and Meinrenken formalized the notion of \emph{Dirac Lie groups} and classified them in terms of so-called "$H$-equivariant Dirac Manin triples" $(\mathfrak{d}, \mathfrak{g}, \mathfrak{h})_\beta$; this generalizes the first result of Drinfeld, as each Poisson Lie group gives a unique Dirac Lie group structure. In this thesis, we consider a notion of homogeneous space for Dirac Lie groups, and classify them in terms of $K$-invariant coisotropic subalgebras $\mathfrak{c}\subset\mathfrak{d}$, with $\mathfrak{c}\cap\mathfrak{h} = \mathfrak{k}$. The relation between Poisson and Dirac morphisms makes Drinfeld's second result a special case of this classification.