Pierre Bieliavsky

Deformation quantization for actions of Kählerian Lie groups

Introduction Notations and conventions Oscillatory integrals Tempered pairs for Kahlerian Lie groups Non-formal star-products Deformation of Frechet algebras Quantization of polarized symplectic symmetric spaces Quantization of Kahlerian Lie groups Deformation of

Oscillatory integral formulae for left-invariant star products on a class of Lie groups

C^*

Deformation Quantization for Heisenberg Supergroup

-algebras Bibliography

The Deformation Quantizations of the Hyperbolic Plane

We use star representation geometric methods to obtain explicit oscillatory integral formulae for strongly invariant star products on Iwasawa subgroups AN of SU(1,n)

Star products on extended massive non-rotating BTZ black holes

We construct a non-formal deformation machinery for the actions of the Heisenberg supergroup analogue to the one developed by M. Rieffel for the actions of R^d. However, the method used here differs from Rieffel’s one: we obtain a Universal Deformation Formula for the actions of R^{m|n} as a byproduct of Weyl ordered Kirillov’s orbit method adapted to the graded setting. To do so, we have to introduce the notion of C∗-superalgebra, which is compatible with the deformation, and which can be seen as corresponding to noncommutative superspaces. We also use this construction to interpret the renormalizability of a noncom- mutative quantum field theory.

Regular Poisson structures on massive non-rotating BTZ black holes

We describe the space of (all) invariant, both formal and non-formal, deformation quantizations on the hyperbolic plane

Symmetric Symplectic Manifolds and Deformation Quantization

{\mathbb D}

Semisimple Symplectic Symmetric Spaces

as solutions of the evolution of a second order hyperbolic differential operator. The construction is entirely explicit and relies on non-commutative harmonic analytical techniques on symplectic symmetric spaces. The present work presents a unified method producing every quantization of