Hopf Bimodules, Coquasibialgebras, and an Exact Sequence of Kac

By Peter Schauenburg. 2002

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Abstract Based on the ideas of Tannaka–Krein reconstruction, we present a categorical construction that assigns to any cleft Hopf algebra inclusion K ⊂ H a coquasibialgebra having K * as a Hopf subalgebra. As a special case, the construction gives an intrinsic connection between the bismash product K # Q and the double cross- product Q ⋈ K * constructed from the same combinatorial data. A cocommutative coquasibialgebra is the same as a cocommutative bialgebra equipped with a Sweedler three-cocycle. Thus our construction assigns to every bicrossproduct (or Hopf algebra extension) of a commutative and a cocommutative factor a corresponding cocommutative double crossproduct equipped with a Sweedler three-cocycle. Based on this observation we use the construction to prove generalizations of Kac's exact sequence for the group of Hopf algebra extensions of a group algebra by a dual group algebra.

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

View Paper: https://www.semanticscholar.org/paper/d3ab81e38b9b211da960cb5edbc1f8c212c6452d

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Citations in the corpus

Lazy cohomology: an analogue of the Schur multiplier for arbitrary Hopf algebras (2004, methodology): “ohomology, since the category of comodules over a coquasibialgebra is a monoidal category, we recover a natural way to deﬁne the lazy cohomology groups for coquasibialgebras as done by Schauenburg in [29]. For a cocommutative (k-ﬂat) Hopf algebra A, there is a group isomorphism AutHopf(A)⋉Gal(A) ∼=B”
On the Gauge Equivalence of Twisted Quantum Doubles of Elementary Abelian and Extra-Special 2-Groups (2006, background): “[Ma1]), we have DζG(G)-Mod∼=Z(BM) as braided tensor categories. We proceed to show that B∼=Cω′Eusing [S2, 3.4.2 and 3.4.5], where ω′ = στω·ζ E and τ: A→A, τ: a→a−1. We will use the notation deﬁned in [S2] for the remaining discussion. By [S2, Lemma 3.4.2], B= CG⊗K∗ as vector space, and we write x⋊a ”