On the differential geometry of Lie groups of Fröbenius type

Pierre Bieliavsky (Université Catholique de Louvain, Belgium)

11-Apr-2022, 14:00-15:00 (24 months ago)

Abstract: The talk will be based on the papers.

(1) In the first one, joint with V. Gayral (Memoirs AMS 2015), we construct universal deformation formulae for actions on topological algebras (C* or Fréchet) of the Lie groups which carries a negatively curved left-invariant Kähler structure.

(2) A second one, joint with V.Gayral, S. Neshveyev and L. Tuset, where we construct locally compact quantum groups from star products on a class of Lie groups.

The Lie groups on which these deformations are performed (in both (1) and (2)) are of ``Frobenius type''. This means that their Lie algebras carry an exact non-degenerate two-cocycle or, equivalently, that they admit an open co-adjoint orbit. In both cases, the star products, say at the formal level, are of Fedosov type i.e. associated with a left-invariant symplectic torsion free affine connection on the group manifold at hand. In particular, they are obtained from differential theoretical considerations.

However, there is a dichotomy: the orderings of the star products considered in (1) and (2) are different. In (1), we deal with Weyl ordered star products, while in (2) with normal (or anti-normal) ones. This has, apparently, a strong effect on the regularity of the categories those constructions live in: smooth versus measurable or topological. More precisely: In (1), we definitely deal with a ``smooth object'', e.g. the universal deformation formula (i.e. the twist) allows to deform smooth vectors of the group action, e.g. they are relevant in differential noncommutative geometry in the sense of A. Connes. But, no locally compact quantum group is present there. And until now, I haven't be able to define a reasonable notion of ``smooth quantum group'' attached to the construction. In (2), the quantum group is present, but the deformation procedure apparently breaks smoothness: smooth vectors of strongly continuous actions (i.e. smooth module-algebras) of the group are not stable under twisting.

In the talk, I will discuss differential geometrical aspects of Frobenius Lie groups within this deformation quantization context. I will end with a suggestion based on the possible use of a Lie group theoretical version of a microlocal analytical tool : Hörmander's smooth wave front set.

operator algebrasquantum algebra

Audience: researchers in the topic


Quantum Groups Seminar [QGS]

Series comments: This seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

The zoom link will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

Organizers: Rubén Martos, Frank Taipe*, Makoto Yamashita
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