We analyze various generalized two-dimensional lattice sums, one of which arose from the solution to a certain Poisson equation. We evaluate certain lattice sums in closed form using results from Ramanujan's theory of theta functions, continued fractions and class invariants. Many explicit examples are given.
The main purpose of this paper is to suggest a method of computing Poisson cohomology of a Poisson manifold by means of symplectic groupoids. The key idea is to convert the problem of computing Poisson cohomology to that of computing de Rham cohomology of certain manifolds. In particular, we shall derive an explicit formula for the Poisson cohomology of a regular Poisson manifold where the symplectic foliation is a trivial fibration.
The purpose of this paper is to establish a connection between various objects such as dynamical -matrices, Lie bialgebroids, and Lagrangian subalgebras. Our method relies on the theory of Dirac structures and Courant algebroids. In particular, we give a new method of classifying dynamical -matrices of simple Lie algebras , and prove that dynamical -matrices are in one-one correspondence with certain Lagrangian subalgebras of .
We prove the universal lifting theorem: for an -simply connected and -connected Lie groupoid with Lie algebroid , the graded Lie algebra of multi-differentials on is isomorphic to that of multiplicative multi-vector fields on . As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases. The second goal of the paper is the study of basic properties of quasi-Poisson groupoids. In particular,...
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