CENTERS AS UNIVERSAL GRADING GROUPS OF REPRESENTATION CATEGORIES

It is a result of Müger that, for a compact group G, associating to every irreducible G-representation its central character identifies the Pontryagin dual of the center Z(G) with the universal group that, in a certain sense, labels the grading of the category of G-representations. It turns out the phenomenon is pervasive, and I will discuss several facets of it. For instance, versions of the universal grading group mentioned above make sense for locally compact groups as well as Hopf algebras. Analogous reconstruction results then go through in a broad range of cases: large classes of locally compact or Lie groups, finite-dimensional Lie algebras (cast as Hopf algebras via their enveloping algebras) that are either solvable or semisimple, etc.