On twisting of finite-dimensional Hopf algebras

TY - JOUR

T1 - On twisting of finite-dimensional Hopf algebras

AU - Aljadeff, Eli

AU - Etingof, Pavel

AU - Gelaki, Shlomo

AU - Nikshych, Dmitri

N1 - Funding Information: P.E. and D.N. were partially supported by the NSF grant DMS-9988796. P.E. and D.N. partially conducted their research for the Clay Mathematics Institute as a Clay Mathematics Institute Prize Fellow and Liftoff Mathematician, respectively. D.N. thanks the Mathematics Department of MIT for the warm hospitality during his visit. S.G. research was supported by the VPR-Fund at the Technion and by the fund for the promotion of research at the Technion.

PY - 2002/10/15

Y1 - 2002/10/15

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0037110405&partnerID=8YFLogxK

U2 - 10.1016/S0021-8693(02)00092-3

DO - 10.1016/S0021-8693(02)00092-3

M3 - 文章

AN - SCOPUS:0037110405

SN - 0021-8693

VL - 256

SP - 484

EP - 501

JO - Journal of Algebra

JF - Journal of Algebra

IS - 2

ER -