Ping Xu
Dirac Submanifolds and Poisson Involutions
Preprint series: ESI preprints

MSC:

58F05 Hamiltonian and Lagrangian systems; symplectic geometry, See also {70Hxx, 81S10}

22E65 Infinite-dimensional Lie groups and their Lie algebras, See also {17B65, 58B25, 58H05}

22A22 Topological groupoids (including differentiable and Lie groupoids)

58B25 Group structures and generalizations on infinite-dimensional manifolds, See also {22E65, 58D05}

58H05 Pseudogroups and differentiable groupoids, See also {22A22,

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 $B_{+}*B_{-}$.