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"Quantum double finite group algebras and their representations", Mark Gould, 1993

Reviewed August 25, 2023

Citation: Gould, M. D. "Quantum double finite group algebras and their representations." Bulletin of the Australian Mathematical Society 48.2 (1993): 275-301.

Web: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/D98C7E96811FB1C6E0CE05D250D7F527/S0004972700015707a.pdf/quantum-double-finite-group-algebras-and-their-representations.pdf

Tags: Kitaev-quantum-double, Mathematical

This is a good general/original reference for the quantum double algebras of finite groups. What's good is that this reference puts the most important feature of the theory first, though in different language than we would use today. The way the author puts it, "The irreducible matrix representations are classified and applied to the explicit construction of R-matrices: this affords solutions to the Yang-Baxter equation associated with certain induced representations of a finite group". In more modern, this "R-matrix" is "quasi-triangular structure".

Why is this solution to the Yang-Baxter equation useful? Well, the Yang-Baxter equation exactly corresponds to the braid relation. In particular, categorical solutions to the Yang-Baxter equation yield braided categories. Especially when these solutions are highly-nontrivial one can get highly-nontrivial (in particular, non-symmetric) braidings on fusion categories, which then give modular tensor categories.

On the level of Hopf algebras, ignoring the R-matrix, quantum doubles are sorta lame. For instance, suppose your input group is abelian. The semi-direct product between the Hopf algebra and its dual becomes a direct product, and the function algebra on G is isomorphic to the group algebra on G^*. Hence, on the level of Hopf algebras, we have a canonical isomorphism with C[G \times G^*]. The distinguishing feature between C[G \times G^*] and C[H] where H happens to be isomorphic to G \times G^* is that C[G\times G^*] has a natural quasi-triangular structure.

This gives quantum double algebras, even more abelian groups, a special character which generic group algebras won't have.