Didier Arnal

Covariance and geometrical invariance in *quantization

Several notions of invariance and covariance for * products with respect to Lie algebras and Lie groups are investigated. Some examples, including the Poincare group, are given. The passage from the Lie‐algebra invariance to the Lie‐group covariance is performed. The compact and nilpotent cases are treated.

Construction of canonical coordinates for exponential Lie groups

Given an exponential Lie group G, we show that the constructions of B. Currey, 1992, go through for a less restrictive choice of the Jordan-Holder basis. Thus we obtain a stratification of g * into G-invariant algebraic subsets, and for each such subset Ω, an explicit cross-section Σ C Ω for coadjoint orbits in Ω, so that each pair (Ω, Σ) behaves predictably under the associated restriction maps on g * . The cross-section mapping σ: Ω → Σ is explicitly shown to be real analytic. The associated Vergne polarizations are not necessarily real even in the nilpotent case, and vary rationally with ∈ Ω. For each Ω, algebras e 0 (Ω) and e 1 (Ω) of polarized and quantizable functions, respectively, are defined in a natural and intrinsic way. Now let 2d > 0 be the dimension of coadjoint orbits in Ω. An explicit algorithm is given for the construction of complex-valued real analytic functions {q 1 , q 2 , ..., q d } and {p 1 ,p 2 ,...,p d } such that on each coadjoint orbit O in Ω, the canonical 2-form is given by Σdp k ^ dq k . The functions {q 1 ,q 2 , ..., q d } belong to e 0 (Ω), and the functions {p 1 ,p 2 , ... ,p d } belong to e 1 (Ω). The associated geometric polarization on each orbit O coincides with the complex Vergne polarization, and a global Darboux chart on O is obtained in a simple way from the coordinate functions (p 1 , ... , p d , q 1 , ... , q d ) (restricted to O). Finally, the linear evaluation functions l ↦ l(X) are shown to be quantizable as well.

Geometrical theory of contractions of groups and representations

The contractions of Lie groups and Lie algebras and their representations are studied geometrically. We prove they can be defined by deformations in Poisson algebras of symplectic manifolds on which the groups act. These deformations are given by Dirac constraints which induce on C∞ functions on the deformed manifold an associative twisted product, characterizing the contracted group or its representations. We treat the contractions of SO(n) to E(n) and apply this theory to thermodynamical limits in spin systems.

The spaces Hn(osp(1∣2),M) for some modules M

We first generalize a result by Bavula on the sl(2) cohomology to the osp(1∣2) cohomology and then we entirely compute the cohomology for a natural class of osp(1∣2) modules M. We study the restriction to the sl(2) cohomology of M and apply our results to the module M=Dλ,μ of differential operators on the superline acting on densities.

The Poincare-Dulac theorem for nonlinear representations of nilpotent lie algebras

The aim of this Letter is to show that the Poincare-Dulac theorem for holomorphic finite-dimensional representation, is valid for any nilpotent Lie algebrag. We reduce the classification problem of representations with a semisimple linear part satisfying the Poincaré condition to an algebraic problem. We develop a complete computation in a particular case.

\(\mathfrak{g}\)-Relative Star Products

AbstractWe consider Kontsevich star products on the duals of Lie algebras. Such a star product is relative if, for any Lie algebra, its restriction to invariant polynomial functions is the usual pointwise product. Let

Hom-Lie quadratic and Pinczon Algebras

\mathfrak{g}

The structure of sl(2,1)-supersymmetry: irreducible representations and primitive ideals

be a fixed Lie algebra. We shall say that a Kontsevich star product is