On second cohomology of duals of compact groups

TL;DR

This paper investigates the second cohomology of duals of compact groups, revealing its structure and implications for tensor categories and group classification, using ergodic actions and Hopf algebra theory.

Contribution

It establishes a canonical isomorphism for the second cohomology group of compact connected groups and characterizes groups by their representation categories.

Findings

Second cohomology group is isomorphic to H^2(rac{Z(G)};T)

Autoequivalences of Rep G are described by H^2(rac{Z(G)};T) and Out(G)

Compact connected groups are uniquely determined by their representation categories

Abstract

We show that for any compact connected group G the second cohomology group defined by unitary invariant 2-cocycles on \hat G is canonically isomorphic to H^2(\hat{Z(G)};T). This implies that the group of autoequivalences of the C*-tensor category Rep G is isomorphic to H^2(\hat{Z(G)};T)\rtimes\Out(G). We also show that a compact connected group G is completely determined by Rep G. More generally, extending a result of Etingof-Gelaki and Izumi-Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classification of triangular semisimple Hopf algebras. In two appendices we give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analogue of Radford's theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups.