Related papers: Applications de la bi-quantification \`a la th\'eo…

We apply the star product quantization to the Lie algebra. The quantization in terms of the star product is well known and the commutation relation in this case is called the $\theta$-deformation where the constant $\theta$ appears as a…

We show that the Kashiwara-Vergne (KV) problem for quadratic Lie algebras (that is, Lie algebras admitting an invariant scalar product) reduces to the problem of representing the Campbell-Hausdorff series in the form…

Quantum and q-deformed algebras find their application not only in mathematical physics and field theoretical context, but also in phenomenology of particle properties. We describe (i) the use of quantum algebras U_q(su_n) corresponding to…

Let g be a Lie bialgebra and let V be a finite-dimensional g-module. We study deformations of the symmetric algebra of V which are equivariant with respect to an action of the quantized enveloping algebra of g. In particular we investigate…

This dissertation is an exposition of Kontsevich's proof of the formality theorem and the classification of deformation quantisation on a Poisson manifold. We begin with an account of the physical background and introduce the Weyl-Moyal…

In these lecture notes I give an introduction to deformation quantization. The quantization problem is discussed in some detail thereby motivating the notion of star products. Starting from a deformed observable algebra, i.e. the star…

Cohomology and deformation theories are developed for Poisson algebras starting with the more general concept of a Leibniz pair, namely of an associative algebra $A$ together with a Lie algebra $L$ mapped into the derivations of $A$. A…

Alternative titles of this paper would have been `Index theory without index' or `The Baum-Connes conjecture without Baum.' In 1989, Rieffel introduced an analytic version of deformation quantization based on the use of continuous fields of…

We compare the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones by a very elementary approach. The comparison gives some conditions, which are easy to verify for a given Lie algebra, for deciding…

We prove that the biquantization character of Cattaneo-Torossian for the reduction algebra is the character of the Penney eigendistribution from harmonic analysis on Lie groups. Part of the author's PhD thesis at University Paris 7, 2009.

We develop a new approach to deformation quantizations of Lie bialgebras and Poisson structures which goes in two steps. In the first step one associates to any Poisson (resp. Lie bialgebra) structure a so called quantizable Poisson (resp.…

A new application of polytope theory to Lie theory is presented. Exponential sums of convex lattice polytopes are applied to the characters of irreducible representations of simple Lie algebras. The Brion formula is used to write a polytope…

In this article, we give Maurer-Cartan characterizations of equivariant Lie superalgebra structures. We introduce equivariant cohomology and equivariant formal deformation theory of Lie superalgebras. As an application of equivariant…

The "quantum duality principle" states that the quantization of a Lie bialgebra - via a quantum universal enveloping algebra (QUEA) - provides also a quantization of the dual Lie bialgebra (through its associated formal Poisson group) - via…

Recently M. Kontsevich found a combinatorial formula defining a star-product of deformation quantization for any Poisson manifold. Kontsevich's formula has been reinterpreted physically as quantum correlation functions of a topological…

An explicit realization of the W(2,2) Lie algebra is presented using the famous bosonic and fermionic oscillators in physics, which is then used to construct the q-deformation of this Lie algebra. Furthermore, the quantum group structures…

We give new applications of graded Lie algebras to: identities of standard polynomials, deformation theory of quadratic Lie algebras, cyclic cohomology of quadratic Lie algebras, $2k$-Lie algebras, generalized Poisson brackets and so on.

We consider a vector space V over K=R or C, equipped with a skew symmetric bracket [.,.]: V x V --> V and a 2-form omega:V x V --> K. A simple change of the Jacobi identity to the form…

We consider gauge theories on Poisson manifolds emerging as semiclassical approximations of noncommutative spacetime with Lie algebra type noncommutativity. We prove an important identity, which allows to obtain simple and manifestly…

The limiting transitions between different types of quantizations are studied by the deformation theory methods. We prove that for the first order coboundary deformation (g,g*_1 + x g*_2) of a Lie bialgebra (g,g*) one can always get the…