Integrability of quotients in Poisson and Dirac geometry

Reduction of symplectic groupoids and quotients of quasi-Poisson manifolds

“…For instance, the q-Poisson G-manifold associated to an annulus is G itself and the integrability of the Lie algebroid structure on (T * G) g is automatic, being an action Lie algebroid [6]. The difficulty in dealing with these spaces lies in the fact that the G-action on them is not free, see [4].…”

“…In both of these situations, the Lie algebroid C that appears in Theorem 1.1 controlling the integrability of the quotient can be interpreted as the Lie algebroid of the level set corresponding to the unit of a Lie group valued moment map as in [27] in the former case and in the sense of [2] in the latter. This last observation is related to the integration of Poisson structures on moduli spaces of flat G-bundles that shall be studied in a companion paper, see [4].…”

Integrability of quotients in Poisson and Dirac geometry

“…If L is a split Dirac structure, then B and C constitute a matched pair of Lie algebroids; accordingly, we denote L as B ⋈ C. We shall say that L is a split Dirac structure which is complete if (B, C) is a complete matched pair of Lie algebroids. Such is the case of the Dirac structures induced by a q-Poisson manifold as in (6).…”

“…Poisson groups and the Cartan-Dirac structure provide examples of exact Dirac-Lie groups so this result already covers the situations more commonly studied in the literature. Dirac homogeneous spaces [38] corresponding to Dirac-Lie groups in the sense of [60] are also integrable, see [6]. Poisson groups are always integrable [47] so they allow us to consider two kinds of Poisson homogeneous spaces:…”

“…In both of these situations, the Lie algebroid C that appears in Theorem 5.15 controlling the integrability of the quotient can be interpreted as the Lie algebroid of the level set corresponding to the unit of a Lie group valued moment map as in [47] in the former case and in the sense of [3] in the latter. This last observation is related to the integration of Poisson structures on moduli spaces of flat G-bundles that shall be studied in a companion paper, see [6].…”