Lie bialgebroids and Poisson groupoids

Abstract

Lie bialgebras arise as infinitesimal invariants of Poisson Lie groups. A Lie bialgebra is a Lie algebra g with a Lie algebra structure on the dual g∗ which is compatible with the Lie algebra g in a certain sense. For a Poisson group G, the multiplicative Poisson structure π induces a Lie algebra structure on the Lie algebra dual g∗ which makes (g, g∗) into a Lie bialgebra. In fact, there is a one-one correspondence between Poisson Lie groups and Lie bialgebras if the Lie groups are assumed to be simply connected [7], [16], [19]. The importance of Poisson Lie groups themselves arises in part from their role as classical limits of quantum groups [8] and in part because they provide a class of Poisson structures for which the realization problem is tractable [15]. Poisson groupoids were introduced by Weinstein [24] as a generalization of both Poisson Lie groups and the symplectic groupoids which arise in the integration of arbitrary Poisson manifolds [4], [11]. He noted that the Lie algebroid dual A∗G of a Poisson groupoid G itself has a Lie algebroid structure, but did not develop the infinitesimal structure further. In this paper we introduce and study a natural infinitesimal invariant for Poisson groupoids, the Lie bialgebroids of the title. Our ultimate purpose is to develop a Lie theory for Poisson groupoids parallel to that for Poisson groups. In this paper we are primarily concerned with the first half of this process; that is, with the construction of the Lie bialgebroid of a Poisson groupoid. After the preliminary §2, in which we describe the generalization to arbitrary Lie algebroids of the exterior calculus and Schouten calculus, in §3 we define a Lie bialgebroid to be a Lie algebroid A whose dual A∗ is also equipped with a Lie algebroid structure, such that the coboundary operator d∗ : A −→ ∧(A) associated to A∗ satisfies a cocycle equation with respect to Γ(A), the Lie algebra of sections of A. This is clearly a straightforward extension of the concept of a Lie bialgebra [16] but cannot be formalized in Lie algebroid cohomological terms since there is no satisfactory adjoint representation for a general Lie algebroid. Most of §3 is devoted to proving that this definition is self-dual: if (A,A∗) is a Lie bialgebroid, then (A∗, A) is also. In §4, we briefly consider the special case of Lie bialgebroids satisfying a triangularity condition, which include some important examples such as the usual triangular Lie bialgebras and Lie bialgebroids associated to Poisson manifolds. The techniques used in §§2–4 are similar to those known for Lie bialgebras. It would be possible, by suitably generalizing the proof for Poisson groups, to prove