Many interesting C∗-algebras can be viewed as quantizations of Poisson manifolds. I propose that a Poisson manifold may be quantized by a twisted polarized convolution C∗-algebra of a symplectic groupoid. Toward this end, I define polarizations for Lie groupoids and sketch the construction of this algebra. A large number of examples show that this idea unifies previous geometric constructions, ...

A symplectic Lie group is a with left-invariant form. Its algebra structure that of quasi-Frobenius algebra. In this note, we identify the groupoid analogue group. We call aforementioned \textit{$t$-symplectic groupoid}; $t$ motivated by fact each target fiber $t$-symplectic manifold. For $\mathcal{G}\rightrightarrows M$, show there one-to-one correspondence between algebroid structures on $A\m...

is the identity on objects so that the latter holonomy groupoid is a quotient of the mon-odromy groupoid. We note also that these Lie groupoids are not etale groupoids. This is one of the distinctions between the direction of this work and that of Kock and Moerdijk 11, 12]. Remark 4.12 The above results also include the notion of a Lie local equivalence relation, and a strong monodromy principl...

This paper deals with Lie groupoids, in particular symplectic and contact groupoids. We formulate a deenition of contact groupoids; we give examples and a counterexample (groupoid with a contact form which is not a contact groupoid). The theory of diierentiable groupoids (called now Lie groupoids) has been introduced by C. Ehresmann E] in 1950 in his paper on connections (where he deened the gr...

Motivated by the computations done in [9], where I introduced and discussed what I called the groupoid of generalized gauge transformations, viewed as a groupoid over the objects of the category BunG,M of principal G-bundles over a given manifold M , I develop in this paper the same ideas for the more general case of principal G-bundles or principal bundles with structure groupoid G, where now ...

In this paper we define complex equivariantK-theory for actions of Lie groupoids. For a Bredon-compatible Lie groupoid G, this defines a periodic cohomology theory on the category of finite G-CW-complexes. A suitable groupoid allows us to define complex equivariant K-theory for proper actions of non-compact Lie groups, which is a natural extension of the theory defined in [24]. For the particul...

We extend some fundamental definitions and constructions in the established generalisation of Lie theory involving Lie groupoids by reformulating them in terms of groupoids internal to a well-adapted model of synthetic differential geometry. In particular we define internal counterparts of the definitions of source path and source simply connected groupoid and the integration of A-paths. The ma...

We study the relationship between general dynamical Poisson groupoids and Lie quasi-bialgebras. For a class of Lie quasi-bialgebras G naturally compatible with a reductive decomposition , we extend the description of the moduli space of classical dynamical r-matrices of Etingof and Schiffmann. We construct, in each gauge orbit, an explicit analytic representative l can. We translate the notion ...

In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group G with dual G⋆ we obtain a suitably connected integrating symplectic double groupoid S. As a consequence, the cotangent lift of a Poisson action on an integrable Poisson manifold P can be integrated to a Poisson actio...