Dirac submanifolds and Poisson involutions

Abstract

Dirac submanifolds are a natural generalization in the Poisson category for symplectic submanifolds of a symplectic manifold. In a certain sense they correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable locus of a Poisson involution. In this paper, we provide a general study for these submanifolds including both local and global aspects. In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable locuses. We discuss the Poisson involutions on a special class of Poisson groups, and more generally Poisson groupoids, called symmetric Poisson groups (and symmetric Poisson groupoids). Many well-known examples, including the standard Poisson group structures on semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie groups, and Poisson groupoids connecting with dynamical $r$-matrices of semi-simple Lie algebras are symmetric, so they admit a Poisson involution. For symmetric Poisson groups, the relation between the stable locus Poisson structure and Poisson symmetric spaces is discussed. As a consequence, we show that the Dubrovin-Ugaglia-Boalch-Bondal Poisson structure on the space of Stokes matrices $U_{+}$ appearing in Dubrovin's theory of Frobenius manifolds is indeed a Poisson symmetric space for the Poisson group $B_{+}*B_{-}$.