The Classification of Dirac Homogeneous Spaces

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.

arXiv comment110 pages, PhD Thesis

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