BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Pierre Bieliavsky (Université Catholique de Louvain\, Belgium) DTSTART:20220411T140000Z DTEND:20220411T150000Z DTSTAMP:20260422T181110Z UID:QGS/57 DESCRIPTION:Title: On the differential geometry of Lie groups of Fröbenius type\nby Pierre Bieliavsky (Université Catholique de Louvain\, Belgium) as part of Quantu m Groups Seminar [QGS]\n\n\nAbstract\nThe talk will be based on the papers .\n\n(1) In the first one\, joint with V. Gayral (Memoirs AMS 2015)\, we c onstruct universal deformation formulae\nfor actions on topological algebr as (C* or Fréchet) of the Lie groups which carries a negatively curved le ft-invariant Kähler structure.\n\n(2) A second one\, joint with V.Gayral\ , S. Neshveyev and L. Tuset\, where we construct locally compact quantum g roups from star products on a class of Lie groups.\n\nThe Lie groups on wh ich these deformations are performed (in both (1) and (2)) are of ``Froben ius type''. This means that their Lie algebras carry an exact non-degenera te two-cocycle or\, equivalently\, that they admit an open co-adjoint orbi t. In both cases\, the star products\, say at the formal level\, are of Fe dosov type i.e. associated with a left-invariant symplectic torsion free a ffine connection on the group manifold at hand. In particular\, they are o btained from differential theoretical considerations.\n\nHowever\, there i s 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: s mooth versus measurable or topological.\nMore precisely:\nIn (1)\, we defi nitely deal with a ``smooth object''\, e.g. the universal deformation form ula (i.e. the twist) allows to deform smooth vectors of the group action\, e.g. they are relevant in differential noncommutative geometry in the sen se 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 ``smoot h quantum group'' attached to the construction.\nIn (2)\, the quantum grou p is present\, but the deformation procedure apparently breaks smoothness: smooth vectors of strongly continuous actions (i.e. smooth module-algebra s) of the group are not stable under twisting.\n\nIn the talk\, I will dis cuss differential geometrical aspects of Frobenius Lie groups within this deformation quantization context. I will end with a suggestion based on t he possible use of a Lie group theoretical version of a\nmicrolocal analyt ical tool : Hörmander's smooth wave front set.\n LOCATION:https://researchseminars.org/talk/QGS/57/ END:VEVENT END:VCALENDAR