On group theoretical Hopf algebras and exact factorizations of finite groups

By Sonia Natale. 2002

69 citations recorded by the source corpus.

Abstract We show that a semisimple Hopf algebra A is group theoretical if and only if its Drinfeld double is a twisting of the Dijkgraaf–Pasquier–Roche quasi-Hopf algebra D ω ( Σ ), for some finite group Σ and some ω ∈ Z 3 ( Σ , k × ). We show that semisimple Hopf algebras obtained as bicrossed products from an exact factorization of a finite group Σ are group theoretical. We also describe their Drinfeld double as a twisting of D ω ( Σ ), for an appropriate 3-cocycle ω coming from the Kac exact sequence.

Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/eedcc93e70eaa6d8a26ded219dc47b7ba6392b4f

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Citations in the corpus

Braid group representations from twisted quantum doubles of finite groups (2007, background): “Since C is group theoretical, Z(C) is equivalent to Mod -D(G) for some G, ω; see [Natale 2003].”
WEAKLY GROUP-THEORETICAL AND SOLVABLE FUSION CATEGORIES (2008, background): “2.13. Kac algebras (abelian extensions). (see [ N1 ])”
Fusion subcategories of representation categories of twisted quantum doubles of finite groups (2008, background): “for some finite group G and 3-cocycle !. (This follows from [ENO, Sect. 8.8]. See also the proof of [ Na , Thm.”
On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups (2006, result): “There are connections here to some results of Natale [ N1 ].”
Non-group-theoretical semisimple Hopf algebras from group actions on fusion categories (2007, methodology): “Note that the Hopf algebra (kG)J is necessarily non cocommutative, since otherwise Hp would be group-theoretical by the result of S. Natale [ Nt ], see Remark 5.1.”