La déformation de Mackey pour les groupes quantiques semi-simples réels

Mots clés

groupes quantiques, théorie des représentations, géométrie non commutative

Détail de l'offre

Summary
The initial goal of the project is to obtain a quantization of the deformation of a real semisimple Lie group G to its Cartan motion group G0. One should then be able to generalize the works of Higson [1, 2], Afgoustidis [3, 4, 5] and Monk-Voigt [6] to obtain a Mackey-type analogy for the representations of real semisimple quantum groups and a description of the quantum Baum-Connes map.
State of the art
Any semisimple Lie group admits a Cartan decomposition G = K.ep, where K is the maximal compact subgroup and p a particular complementary subspace of the Lie algebra k in g. From this, one can define the Cartan motion group G0 = K n p as the semi-direct product of its maximal compact subgroup K acting on p by the adjoint action.
The Cartan motion group, being a semidirect product of a compact and an abelian group, has a relatively simple representation theory, as described by Mackey. Mackey also observed that the parameters which describe the irreducible representations of G0 bear a strong resemblance to those which describe the far more complicated unitary dual of G. This informal observation has come to be known as the Mackey analogy.
Higson proposed a means to place the Mackey analogy on a solid basis, by describing a 1-parameter family of Lie groups (Gt) indexed by t ∈ [0,1], which deforms G1 = G to its Cartan motion group G0. By studying the representation theory of this deformation family, he was able to give a topological structure to the representations in the Mackey analogy, at least for complex semisimple groups. Afgoustidis generalized this to all real semisimple groups.
Moreover, this deformation family induces a map between K-theory of the group C*-algebras which can be interpreted in terms of the Baum-Connes assembly map.
The complex semisimple Lie groups admit a further noncommutative deformation to a family of quantum groups. These quantum groups are defined as Drinfeld doubles of the quantized compact semisimple groups, and their representation theory is studied in [10]. Again, the representation theory of these quantum groups is very similar to the representation theory of their classical counterparts.
Monk and Voigt [6] showed that these quantum groups also admit a deformation to a quantized Cartan motion group. The group C∗-algebras of all the above-mentioned groups and quantum groups fit into a single continuous field of C∗-algebras over the double parameter space [0, 1]x(0, 1]. Again this can be interpreted as a quantization of the Baum-Connes map.
Immediate goals
The analogues results remain unknown when G is a real semisimple quantum group. In fact, even the definition of the real semisimple quantum groups is unresolved. To point out the subtility of this problem, Woronowicz proved a famous no-go theorem, show- ing that there is an obstruction to defining a q-deformation of the Lie group SL(2, R). Nonetheless, solutions exist. Koelink-Kustermans [9] showed that one can define a q- deformation of a double cover of this group. Their construction, while promising, is extremely technical, and has never been generalized to higher rank Lie groups.
The initial goal of the project will be to obtain a complete description of the Mackey analogy for the quantum group SLq(2,R), as well as the unitary representation theory of all the groups involved. Although there will be some technical difficulties in this, the project is concrete enough that it will be readily achievable.
An incidental goal of the initial project will be to obtain a better un- derstanding of de Commer-Dzokou Talla's quantizations, and their relationship with the previous model by Koelink-Kustermans. If successful, this would be a major step forward for the theory of quantized semisimple Lie groups. Even if only the Mackey analogy could be understood for Commer-Dzokou Talla's q-deformations, this would give significant insight into the structure of these constructions.