over B with anchor map a given by the restriction of T[a, j3] to A(T) (here [a, (3](z)
= (a(z),0(z)), zeT) and bracket of sections [X,Y](b) :=
[X1, Yl}r(e(b))
: F Ker{Ta)
is the unique left invariant vector field whose restriction to e(B) is X.
Let V be a vector space and let p : F —• Aut(V) be a smooth groupoid
morphism where Aut(V) is viewed as a groupoid over its unit element Iy.
Definition 2.1.6. A smooth map E : V V is called a groupoid 1-cocycle iff
E(xy) = Y:(x)+p(x)j:(y)
for all (x\y) G T * T. The induced map E* : A(T) » V defined as the restriction of
TYi to A(Y) is called the induced Lie algebroid 1-cocycle.
Finally, we recall the notion of an action of a Lie groupoid V ^ B on a manifold
S with moment map / : S B. (We follow the terminology of [MW].)
r*fS = {(x,s)eTxS | /?(*) = /(*)},
S *f r = {(5, x) G S x F I /(*) = a(x)}.
Definition 2.1.7. (a) A left action ofT on S with moment map f is a smooth
map (pl :T *f S S : (x, s) ^- x s such that
f{x - s) - a(x), y-(x-s) = (yx) s, e(/(t)) t = t,
for all (y, x) G T * T, (x, s) r * / S , t G 5.
f&J A right action ofY on S with moment map f is a smooth map
: S*fT
S : («s, x) H-» 5 x swe/i £/m£
/(s x) - /3(x), (5 x) y = s (xy), t e(/(*)) - t,
for all (x, 2/) T * F, (s, x) G ^ ^ / F t e S .
2.2. Trivial Lie groupoids in C^
Our purpose in this subsection is to provide an explicit class of Poisson brackets
on trivial Lie groupoids which extends the construction of Theorem 2.F4 (a), and
is essential for our subsequent study of duality.
Throughout the paper, we define the bundle map II# associated with the Pois-
son manifold (V, { , }y) using the convention df, U^dg = {/, g}y. Also, we
assume that the Lie subgroup H C G is connected. We begin with a general
Proposition 2.2.1. Let Y be a Poisson groupoid over U with target and source
maps a and (5 and unit map e. If there exists a (base preserving) Poisson groupoid
I :H xU -* y,
(here H x U is the Hamiltonian unit) then Y belongs to CJJ .
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