The versatility of the Drinfeld double of a finite group

TL;DR

This survey analyzes the Drinfeld double $D(G)$ of a finite group, highlighting its role as a braided Hopf algebra whose representation category illuminates connections across harmonic analysis, geometry, and quantum topology. It surveys algebraic, analytic, and geometric perspectives on $D(G)$, including explicit module classifications, centre structures, and equivalences to Yetter–Drinfeld and Hopf–module theories, as well as geometric realizations via $G$-equivariant bundles and sheaves. The article then surveys broad applications: character theory, link invariants, Lusztig’s non-abelian Fourier transform, mapping class group actions, Mackey quantization, orbifold conformal field theory, and Verlinde-type formulas, tying together Nichols algebras and pointed Hopf algebra classification where relevant. Overall, the work emphasizes how $D(G)$ serves as a unifying algebraic framework for symmetry, topology, and representation theory, with concrete tools like Fourier transforms and the Verlinde formula guiding computations and conceptual understanding.

Abstract

In this survey we review different instances in which the Drinfeld double of a finite group and its representations play a role, touching upon some of Tom Koornwinder's research interests: harmonic analysis, Lie algebras, quantum groups, non-commutative geometry, and Verlinde formula.