Convolution of Invariant Distributions: Proof of the Kashiwara–Vergne conjecture

Consider the Kontsevich star product on the symmetric algebra of a finite-dimensional Lie algebra g, regarded as the algebra of distributions with support 0 on g. In this Letter, we extend this star product to distributions satisfying an appropriate support condition. As a consequence, we prove a long-standing conjecture of Kashiwara–Vergne on the convolution of germs of invariant distributions on the Lie group G.     

Algebraic deformation program on minimal nilpotent orbit

We construct an algebraic star product on the minimal nilpotent coadjoint orbit of a simple complex Lie group with a Lie algebra which is not of typeA _n. According to the deformation program, we study the representations of the Lie algebra associated to this orbit.     

Poisson Action and Formality

Using a formality on a Poisson manifold, we construct a star product and for each Poisson vector field a derivation of this star product. Starting with a Poisson action of a Lie group, we are able under a natural cohomological assumption to define a representation of its Lie algebra in the space of derivations of the star product. Finally, we use these results to define some generically tangential star products on duals of Lie algebra as in [1] but in a more realistic context. This work was supported by the CMCU contract 00 F 15 02.     

On tangential star products for the coadjoint Poisson structure

We derive necessary conditions on a Lie algebra from the existence of a star product on a neighbourhood of the origin in the dual of the Lie algebra for the coadjoint Poisson structure which is both differential and tangential to all the coadjoint orbits. In particular we show that when the Lie algebra is semisimple there are no differential and tangential star products on any neighbourhood of the origin in the dual of its Lie algebra.Research partially supported by EC contract CHRX-CT920050     

Kontsevich Star Product on the Dual of a Lie Algebra

We show that on the dual of a Lie algebra g of dimension d, the star product recently introduced by M. Kontsevich is equivalent to the Gutt star product on g*. We give an explicit expression for the operator realizing the equivalence between these star products.     

\mathfrak{g}-Relative Star Products

We 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 be a fixed Lie algebra. We shall say that a Kontsevich star product is -relative if, on ^*, its restriction to invariant polynomial functions is the usual pointwise product. We prove that, if is a semi-simple Lie algebra, the only strict Kontsevich -relative star products are the relative (for every Lie algebras) Kontsevich star products.     

Star Products and Quantum Algebras

I show explicitly that the star product on atriangular Poisson Lie group leads to a quantum algebrastructure (triangular Hopf algebra) on the quantizedenveloping algebra of the Lie algebra of the Lie group, and that equivalent star-productsgenerate isomorphic quantum algebras.     

Star exponentials of the elements of the inhomogeneous symplectic Lie algebra

We compute the star exponential of any element of the inhomogeneous symplectic Lie algebra on a 2l-dimensional phase space and show the existence of classical trajectories for a quantum system whose Hamiltonian belongs to this Lie algebra.     

A convergent star product for linear Poisson structures

An explicit star product ⋆ _α ^Γ on the dual of a general Lie algebra equipped with the linear Poisson bracket is constructed. An equivalence operator between this star product and the Kontsevich star product in [K₁] is given and diverse properties of the star product ⋆ _α ^Γ are studied. It is also proved that the star product ⋆ _α ^Γ provides a convergent deformation quantization in the sense of Rieffel [R₁].