Poisson homogeneous spaces of Poisson 2-groups

TL;DR

The paper extends Drinfeld’s Dirac-structure framework for Poisson homogeneous spaces from Poisson Lie groups to Poisson 2-groups by developing a 2-category approach using crossed modules and quotient Poisson groupoids. It establishes a bijection between Poisson homogeneous spaces of a Poisson 2-group $\mathbb{G}$ and $H_0$-invariant Dirac structures $L$ in the Lie bialgebra $(\mathfrak g_1,\mathfrak g_1^*)$ with $L\cap \mathfrak g_1=\mathfrak h_1$, equivalently described by invariant pairs $(\mathfrak h_1,r)$ with $r\in \wedge^2 \mathfrak g_1$ and $H_0\triangleright r \equiv r \pmod{\mathfrak h_1}$. The associated Poisson structure on the quotient is given by $\Pi=\mathrm{pr}_*(\Pi_{\mathbb{G}}+\overleftarrow{r}^{gp})$, highlighting compatibility with the quotient Poisson groupoid $\mathbb{G}/H_0$ and revealing affine structures in suitable cases (e.g., triangular 2-bialgebras). These results generalize the classical Drinfeld–Liu–Weinstein–Xu theory to categorified symmetry, providing a systematic description of Poisson homogeneous spaces for Poisson 2-groups and their representations via Dirac data.

Abstract

Drinfeld classified Poisson homogeneous spaces of a Poisson Lie group in terms of Dirac structures of the Lie bialgebra. In this paper, we study homogeneous spaces of a 2-group and develop Drinfeld theorem in the Poisson 2-group context.

Theorems & Definitions (57)

• Definition 2.1
• Proposition 2.2
• proof
• Theorem 2.3
• proof
• Example 2.4
• Example 2.5
• Example 2.6
• Proposition 2.7
• proof