On twisting of finite-dimensional Hopf algebras
By E. Aljadeff, Pavel Etingof, Shlomo Gelaki, D. Nikshych. 2001
55 citations recorded by the source corpus.
In this paper we study the properties of Drinfeld’s twisting for finite-dimensional Hopf algebras. We determine how the integral of the dual to a unimodular Hopf algebra H changes under twisting of H . We show that the classes of cosemisimple unimodular, cosemisimple involutive, cosemisimple quasitriangular finite-dimensional Hopf algebras are stable under twisting. We also prove the cosemisimplicity of a coalgebra obtained by twisting of a cosemisimple unimodular Hopf algebra by two different twists on two sides (such twists are closely related to bi-Galois extensions), and describe the representation theory of its dual. Next, we define the notion of a non-degenerate twist for a Hopf algebra H , and set up a bijection between such twists for H and H ∗ . This bijection is based on Miyashita–Ulbrich actions of Hopf algebras on simple algebras. It generalizes to the non-commutative case the procedure of inverting a non-degenerate skew-symmetric bilinear form on a vector space. Finally, we apply these results to classification of twists in group algebras and of cosemisimple triangular finite-dimensional Hopf algebras in positive characteristic, generalizing the previously known classification in characteristic zero.
View Paper: https://www.semanticscholar.org/paper/83514b44be8fc88c204110bc4c4a72fbbb17e896
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- E. Aljadeff
- V. Ostrik
- A. Kanel’-Belov
- D. Nikshych
- D. Venku Naidu
- Alexei Davydov
- Michael Mueger
- Pavel Etingof
Citations in the corpus
- REPRESENTABILITY AND SPECHT PROBLEM FOR G-GRADED ALGEBRAS (2009, result): “Interestingly, this is in contrast to the case where the “grading” is given by other type of Hopf algebras (see [1], [4]).”
- Module categories over the Drinfeld double of a finite group (2002, background): “Then the twisted group algebraA(H,ψ) = Cψ[H] is an associative algebra in the category VecGω; for any 1-cochain χ ∈ C1(H,C∗), the algebras A(H,ψ) and A(H,ψ + dχ) are isomorphic.”