CHAPTER 2
A Class of Biequivariant Poisson Groupoids
2.1. Preliminaries
In this preliminary subsection, we recall some of the basic concepts and con-
structs which we will use in this paper (other results will be recalled when needed).
Let T be a Lie groupoid over B (see [CdSW],[Ml] for details), with target and
source maps a, (3 : V B, and multplication map m : V * Y B defined on the
set of composable pairs r * T := {(x,y) \ (3(x) = a(y)}. We will denote the unit
section by e : B T, and the inversion map by i : T T.
Definition 2.1.1 [Wl]. (Poisson groupoid.) A Lie groupoid T equipped with
a Poisson structure II is called a Poisson groupoid if and only if the graph of the
multiplication map
Gr(m) C T x T x T
is a coisotropic submanifold, i.e. if and only if
(n e n e -n) (CJ, J) = o, Vw , J e (TiGrim))^ c T*(r x r x r).
r is called a symplectic groupoid if II is non degenerate with Gr(m) a La-
grangian submanifold.
In both cases, the Poisson structure and the groupoid structure are said to be
compatible.
Let G be a connected Lie group, H C G a connected Lie subgroup with re-
spective Lie algebras g and \) and let U C f)* be a connected Ad*H- invariant open
subset. In [EV], Etingof and Varchenko introduced the category Cu of biequivariant
Poisson manifolds over U as follows.
An object in Cu is a Poisson manifold (X,{, }x) equipped with commuting
left Hamiltonian i^-action / ~ and right Hamiltonian i^-action 0 + with ZJ-valued
Ad*H-equivariant momentum maps : X —• U satisfying the polarity condition
0 , 3-^}x = 0, for all cp,^ G C°°(U).
A morphism in Cu between (X, { , }x) and (X', { , }x')
ls a n
equivariant Poisson
map a : X —• X' such that j '
±
o a = j±.
Definition 2.1.2[EV]. (Poisson groupoid in Cu-) A Poisson manifold X G Cu
is a Poisson groupoid in Cu iff it is equipped with a compatible groupoid structure
over U such that a = jf_, j3 j+.
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