Leaves of stacky Lie algebroids

Álvarez, Daniel

Comptes Rendus. Mathématique, Tome 358 (2020) no. 2, pp. 217-226.

Nous montrons que les feuilles d’un groupoïde en algébroïde de Lie qui passent par la variété unité sont elles mêmes des groupoïdes de Lie, sous une condition de connexité. Ce résultat est appliqué aux algébroïdes de Lie cotangents des groupoïdes de Poisson. On obtient ainsi une classe intéressante de groupoïdes symplectiques associés à leurs feuilletages symplectiques. Nous montrons en particulier que pour un $2$-groupe de Lie strict, les orbites coadjointes des unités dans le dual de sa $2$-algèbre de Lie sont des groupoïdes symplectiques, en ce sens que les formes symplectiques classiques de Kostant-Kirillov-Souriau sur ces orbites coadjointes spéciales sont mutiplicatives.

We show that the leaves of an LA-groupoid which pass through the unit manifold are, modulo a connectedness issue, Lie groupoids. We illustrate this phenomenon by considering the cotangent Lie algebroids of Poisson groupoids thus obtaining an interesting class of symplectic groupoids coming from their symplectic foliations. In particular, we show that for a (strict) Lie 2-group the coadjoint orbits of the units in the dual of its Lie 2-algebra are symplectic groupoids, meaning that the classical Kostant–Kirillov–Souriau symplectic forms on these special coadjoint orbits are multiplicative.

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