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 j± : 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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