This article in French, with a large English introduction, is a survey about applications of bi-quantization theory in Lie theory. We focus on a conjecture of M. Duflo. Most of the applications are coming from our article with Alberto Cattaneo [13] and some extensions are relating discussions with my student [9]. The end of the article is completely new. We prove that the conjecture E = 1 impli...

We study the quantum ( $$C^*$$ ) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, -extreme POVMs countable spaces (in particular for finite sets) are always spectral (normalized projection measures). More generally shown atomic spectral. A Krein–Milman type theorem ha...

We extend to the sl(N) case the results that we previously obtained on the construction of Wq,p algebras from the elliptic algebra Aq,p(ŝl(2)c). The elliptic algebra Aq,p(ŝl(N)c) at the critical level c = −N has an extended center containing trace-like operators t(z). Families of Poisson structures indexed by N(N −1)/2 integers, defining q-deformations of the WN algebra, are constructed. The op...

Superimposed D–branes have matrix–valued functions as their transverse coordinates, since the latter take values in the Lie algebra of the gauge group inside the stack of coincident branes. This leads to considering a classical dynamics where the multiplication law for coordinates and/or momenta, being given by matrix multiplication, is nonabelian. Quantisation further introduces noncommutativi...

A simple axiomatic characterization of the general (infinite dimensional , noncommutative) Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Itô B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Itô algebras of quantum Brownian and quantum Lévy motion, and the B*-algebras of vacuum and ...

A simple axiomatic characterization of the general (infinite dimensional , noncommutative) Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Itô B*-algebra, generalizing the C*-algebra is defined to include the Banach infinite dimensional Itô algebras of quantum Brownian and quantum Lévy motion, and the B*-algebras of vacuum and ...

pC++ is an object-parallel extension to the C++ programming language. This paper describes the current language deenition and illustrates the programming style. Examples of parallel linear algebra operations are presented and a fast poisson solver is described in complete detail.

A Poisson algebra is a commutative algebra with a Lie bracket {, } satisfying the Leibniz rule. Such algebras appear in classical mechanics. Namely, functions on the phase space form a Poisson algebra, and Hamilton’s equation of motion is df dt = {f, H}, where H is the Hamiltonian (energy) function. Moreover, the transition from classical to quantum mechanics can be understood in terms of defor...

This paper studies certain relations among vertex algebras, vertex Lie algebras and vertex poisson algebras. In this paper, the notions of vertex Lie algebra (conformal algebra) and vertex poisson algebra are revisited and certain general construction theorems of vertex poisson algebras are given. A notion of filtered vertex algebra is formulated in terms of a notion of good filtration and it i...