CHAPTER 1

Introduction

The classical dynamical Yang-Baxter equation (CDYBE) is a functional differ-

ential equation which first appeared in the context of Wess-Zumino-Witten confor-

mal field theory [BDF], [F]. Subsequently, its geometric meaning was unraveled by

Etingof and Varchenko in the fundamental paper [EV]. While the solutions of the

classical Yang-Baxter equation are related to Poisson Lie groups [D], the authors

in [EV] showed that an appropriate geometrical setting for the CDYBE is that of

a special class of Poisson groupoids (as defined in [Wl]), the so-called coboundary

dynamical Poisson groupoids. Given a Lie group G, a Lie subgroup H C G, an

Ad*H invariant open set U C f)* (here f)* is the dual of f ) = Lie(H)), and a solution

of the CDYBE , Etingof and Varchenko constructed a Poisson bracket on the trivial

Lie groupoid X = U x G xU compatible with its groupoid structure. This Poisson

bracket intertwines terms which are responsible for left and right inclusions of the

restricted symplectic cotangent H x U into X together with a Sklyanin-like term

on G. In addition, the authors in [EV] identified an appropriate abstract context in

which to view these objects as the category of iiT-bi-equivariant Poisson manifolds

Cv.

It is classical that the study of Poisson Lie groups relies in an essential way on

duality and the construction of doubles [D], [STS], [LW1]. For Poisson groupoids,

the notion of duality was introduced by Weinstein in [Wl], and was developed

by Mackenzie and Xu in [MX1],[ MX2]. In the same paper [Wl], Weinstein also

introduced the notion of symplectic double groupoids (see also [M2]), and described

a program for showing that, at least locally, Poisson groupoids in duality arise as

the base of a symplectic double groupoid.

In order to state our objectives and results, let us begin by recalling that

a symplectic groupoid is a pair (r,II), consisting of a Lie groupoid V together

with a non-degenerate Poisson structure II, in such a way that the graph of the

multiplication map is a Lagrangian submanifold of Y x Y x Y [W2],[K]. It is a

classical fact that Poisson structures can be understood at least locally by the notion

of symplectic groupoids. On the other hand, double groupoids are intrinsically

complicated objects introduced by Ehresmann [E] in the 1960's and have found

usage in category theory [E], homotopy theory [BH], differential geometry [P], and

Poisson groups [M3], [LW2]. By definition, a double Lie groupoid is a quadruple

(S;H,V,i3) where H and V are Lie groupoids over J3, and S is equipped with

two Lie groupoid structures, a horizontal structure with base V, and a vertical

structure with base H, such that the structure maps of each groupoid structure on

S are morphisms with respect to the other. Finally, a symplectic double groupoid

is a double Lie groupoid (S; W, V, B) in which S is equipped with a symplectic

structure such that both S =4 V and S =4 H are symplectic groupoids. Note that

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