Abstract:
We provide a general study for triangular dynamical r-matrices using Poisson geometry. We show that a triangular dynamical r-matrix always gives rise to a regular Poisson manifold. Using the Fedosov method, we prove that non-degenerate (i.e., the corresponding Poisson manifolds are symplectic) triangular dynamical r-matrices (over $ \frakh^* $ and valued in $\wedge^{2}\frakg$) are quantizable, and the quantization is classified by the relative Lie algebra cohomology $H^{2}(\frakg, \frakh)[[\hbar ]]$. We also generalize this quantization method to splittable triangular dynamical r-matrices, which include all the known examples of triangular dynamical r-matrices. Finally, we arrive a conjecture that the quantization for an arbitrary triangular dynamical r-matrix is classified by the formal neighbourhood of this r-matrix in the modular space of triangular dynamical r-matrices. The dynamical r-matrix cohomology is introduced as a tool to understand such a modular space.

Abstract:
An anologue of the Calabi invariant for Poisson manifolds is considered. For any Poisson manifold $P$, the Poisson bracket on $C^{\infty}(P)$ extends to a Lie bracket on the space $\Omega^{1}(P)$ of all differential one-forms, under which the space $Z^{1}(P)$ of closed one-forms and the space $B^{1}(P)$ of exact one-forms are Lie subalgebras. These Lie algebras are related by the exact sequence: $$0\lon \reals \lon C^{\infty}(P)\stackrel{d}{\lon} Z^{1}(P)\stackrel{f}{\lon} H^{1}(P, \reals)\lon 0, $$ where $H^{1}(P,\reals)$ is considered as a trivial Lie algebra, and $f$ is the map sending each closed one-form to its cohomology class. The goal of the present paper is to lift this exact sequence to the group level for compact Poisson manifolds under certain integrability condition. In particular, we will give a geometric description of a Lie group integrating the underlying Poisson algebra $C^{\infty}(P) $.

Abstract:
The main purpose of the paper is to study hyperkahler structures from the viewpoint of symplectic geometry. We introduce a notion of hypersymplectic structures which encompasses that of hyperkahler structures. Motivated by the work of Kronheimer on (co)adjoint orbits of semi-simple Lie algebras, we define hyper-Lie Poisson structures associated with a compact semi-simple Lie algebra and give criterion which implies their existence. We study an explicit example of a hyper-Lie Poisson structure, in which the moduli spaces of solutions to Nahm's equations assocaited to Lie algebra $\frak{su}(2)$ are realized as hypersymplectic leaves and are related to the (co)adjoint orbits of $\frak{sl}(2, \complex)$.

Abstract:
In this paper we consider dynamical r-matrices over a nonabelian base. There are two main results. First, corresponding to a fat reductive decomposition of a Lie algebra $\frakg =\frakh \oplus \frakm$, we construct geometrically a non-degenerate triangular dynamical r-matrix using symplectic fibrations. Second, we prove that a triangular dynamical r-matrix $r: \frakh^* \lon \wedge^2 \frakg$ corresponds to a Poisson manifold $\frakh^* \times G$. A special type of quantizations of this Poisson manifold, called compatible star products in this paper, yields a generalized version of the quantum dynamical Yang-Baxter equation (or Gervais-Neveu-Felder equation). As a result, the quantization problem of a general dynamical r-matrix is proposed.

Abstract:
Dirac submanifolds are a natural generalization in the Poisson category for symplectic submanifolds of a symplectic manifold. In a certain sense they correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable locus of a Poisson involution. In this paper, we provide a general study for these submanifolds including both local and global aspects. In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable locuses. We discuss the Poisson involutions on a special class of Poisson groups, and more generally Poisson groupoids, called symmetric Poisson groups (and symmetric Poisson groupoids). Many well-known examples, including the standard Poisson group structures on semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie groups, and Poisson groupoids connecting with dynamical $r$-matrices of semi-simple Lie algebras are symmetric, so they admit a Poisson involution. For symmetric Poisson groups, the relation between the stable locus Poisson structure and Poisson symmetric spaces is discussed. As a consequence, we show that the Dubrovin-Ugaglia-Boalch-Bondal Poisson structure on the space of Stokes matrices $U_{+}$ appearing in Dubrovin's theory of Frobenius manifolds is indeed a Poisson symmetric space for the Poisson group $B_{+}*B_{-}$.

Abstract:
We introduce quasi-symplectic groupoids and explain their relation with momentum map theories. This approach enables us to unify into a single framework various momentum map theories, including the ordinary Hamiltonian $G$-spaces, Lu's momentum maps of Poisson group actions, and group valued momentum maps of Alekseev--Malkin--Meinrenken. More precisely, we carry out the following program: (1) Define and study properties of quasi-symplectic groupoids; (2) Study the momentum map theory defined by a quasi-symplectic groupoid. In particular, we study the reduction theory and prove that the reduced space is always a symplectic manifold. More generally, we prove that the classical intertwiner space between two Hamiltonian $\Gamma$-spaces is always a symplectic manifold whenever it is a smooth manifold; (3) Study the Morita equivalence of quasi-symplectic groupoids. In particular, we prove that Morita equivalent quasi-symplectic groupoids give rise to equivalent momentum map theories and that the intertwiner space depends only on the Morita equivalence class. As a result, we recover various well-known results concerning equivalence of momentum maps including Alekseev-- Ginzburg--Weinstein linearization theorem and Alekseev--Malkin--Meinrenken equivalence theorem between quasi-Hamiltonian spaces and Hamiltonian loop group sapces.

Abstract:
We study various aspects of Fedosov star-products on symplectic manifolds. By introducing the notion of "quantum exponential maps", we give a criterion characterizing Fedosov connections. As a consequence, a geometric realization is obtained for the equivalence between an arbitrary *-product and a Fedosov one. Every Fedosov *-product is shown to be a Vey *-product. Consequently, one obtains that every *-product is equivalent to a Vey * -product, a classical result of Lichnerowicz. Quantization of a hamiltonian G-space, and in particular, quantum momentum maps are studied. Lagrangian submanifolds are also studied under a deformation quantization.

Abstract:
The purpose of this Note is to unify quantum groups and star-products under a general umbrella: quantum groupoids. It is shown that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. The converse question, i.e., the quantization problem, is posed. In particular, any regular triangular Lie bialgebroid is shown quantizable. For the Lie bialgebroid of a Poisson manifold, its quantization is equivalent to a star-product.

Abstract:
We introduce a general notion of quantum universal enveloping algebroids (QUE algebroids), or quantum groupoids, as a unification of quantum groups and star-products. Some basic properties are studied including the twist construction and the classical limits. In particular, we show that a quantum groupoid naturally gives rise to a Lie bialgebroid as a classical limit. Conversely, we formulate a conjecture on the existence of a quantization for any Lie bialgebroid, and prove this conjecture for the special case of regular triangular Lie bialgebroids. As an application of this theory, we study the dynamical quantum groupoid ${\cal D}\otimes_{\hbar} U_{\hbar}(\frakg)$, which gives an interpretation of the quantum dynamical Yang-Baxter equation in terms of Hopf algebroids.

Abstract:
The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber algebras and BV-algebras. Some applications are discussed. In particular, we found an explicit connection between the Koszul-Brylinski operator of a Poisson manifold and its modular class. As a consequence, we prove that Poisson homology is isomorphic to Poisson cohomology for unimodular Poisson structures.