Related papers: Pseudodifferential operators on differential group…

A bicovariant calculus of differential operators on a quantum group is constructed in a natural way, using invariant maps from \fun\ to \uqg\ , given by elements of the pure braid group. These operators --- the `reflection matrix' $Y \equiv…

We study Lie-Rinehart algebra structures in the framework provided by a duality pairing of modules over a unital commutative associative algebra. Thus, we construct examples of Lie brackets corresponding to a fixed anchor map whose image is…

Let h \subset g be an inclusion of Lie algebras with quotient h-module n. There is a natural degree filtration on the h-module U(g)/U(g)h whose associated graded h-module is isomorphic to S(n). We give a necessary and sufficient condition…

The ring $\text{Diff}_{\mathbf{h}}(n)$ of $\mathbf{h}$-deformed differential operators appears in the theory of reduction algebras. In this thesis, we construct the rings of generalized differential operators on the $\mathbf{h}$-deformed…

It is well known that the Lie-algebra structure on quantum algebras gives rise to a Poisson-algebra structure on classical algebras as the Planck constant goes to 0. We show that this correspondance still holds in the generalization of…

We develop a geometric framework for Weyl quantization on pseudo-Riemannian manifolds, in which pseudodifferential operators act on sections of vector bundles equipped with arbitrary connections. We construct the associated star product and…

In this paper, we generalize all the results obtained on para-K\"ahler Lie algebras in Journal of Algebra {\bf 436} (2015) 61-101 to para-K\"ahler Lie algebroids. In particular, we study exact para-K\"ahler Lie algebroids as a…

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie…

This paper has been withdrawn by the authors. A class of pseudodifferential operators on the Heisenberg group is defined. As it should be, this class is an algebra containing the class of differential operators. Furthermore, those…

Let X be a smooth compact manifold with boundary. For smooth foliations on the boundary of X admitting a `resolution' in terms of a fibration, we construct a pseudodifferential calculus generalizing the fibred cusp calculus of Mazzeo and…

We propose a q-difference version of the Drinfeld-Sokolov reduction scheme, which gives us q-deformations of the classical W-algebras by reduction from Poisson-Lie loop groups. We consider in detail the case of SL(2). The nontrivial…

Poisson algebras are, just like Lie algebras, particular cases of Lie-Rinehart algebras. The latter were introduced by Rinehart in his seminal 1963 paper, where he also introduces the notion of an enveloping algebra and proves --- under…

We introduce a notion of duality for a Lie-Rinehart algebra giving certain bilinear pairings in its cohomology generalizing the usual notions of Poincar\'e duality in Lie algebra cohomology and de Rham cohomology. We show that the duality…

We investigate the real Lie algebra of first-order differential operators with polynomial coefficients, which is subject to the following requirements. (1) The Lie algebra should admit a basis of differential operators with homogeneous…

We generalize Roe's Index Theorem for operators of Dirac type on open manifolds to elliptic pseudodifferential operators. To this end we introduce a class of pseudodifferential operators on manifolds of bounded geometry which is more…

We extend the notion of Poisson-Lie groups and Lie bialgebras from Poisson to g-quasi-Poisson geometry and provide a quantization to braided Hopf algebras in the corresponding Drinfeld category. The basic examples of these g-quasi-Poisson…

We introduce and study a class of Lie algebroids associated to faithful modules which is motivated by the notion of cotangent Lie algebroids of Poisson manifolds. We also give a classification of transitive Lie algebroids and describe…

For an arbitrary Poisson algebra $\CP$ over an arbitrary field, an (analogue of) the Poincar\'{e}-Birkhof-Witt Theorem is proven and several presentations/constructions for its Poisson enveloping algebra $\CU (\CP )$ are given. As a result,…

We describe how Lie groupoids are used in singular analysis, index theory and non-commutative geometry and give a brief overview of the theory. We also expose groupoid proofs of the Atiyah-Singer index theorem and discuss the Baum-Connes…

Given a pair of smooth transversally intersecting manifolds in some ambient manifold, we construct an operator algebra generated by pseudodifferential operators and the (co)boundary operators associated with the submanifolds. We show that…