Given a bialgebra , a Drinfel’d twist is an invertible element satisfying
satisfying (in fact it is enough to require one out of these two counitality conditions).
In Majid‘s formalism of bialgebra cocycles, this is the same as a counital 2-cocycle in .
Application to twisting
Vladimir Drinfel’d introduced in order to suggest a procedure of twisting Hopf algebras. Given a bialgebra, Hopf algebra or quasitriangular Hopf algebra, one gets another such structure twisting it with a Drinfel’d twist. Let be a quasi-triangular Hopf algebra with comultiplication , antipode , universal R-element and let be a Drinfel’d twist. Then the same algebra becomes another quasitriangular Hopf algebra with formulas for comultiplication , universal R-element and antipode . The counit is not changed. Here where is the flip of tensor factors, and is a notation similar to Sweedler’s convention.
Generalizations
In the original Drinfel’d’s work 2-cocycles for twisting quasi-Hopf algebras were also considered.
There is also a subtle generalization for bialgebroids over nonassociative base due Ping Xu.
Literature
The original references are
V. G. Drinfeld, Quasi-Hopf algebras and Knizhnik-Zanolodchikov equations, Acad. Sci. Ukrainian SSR, Institute for Theoretical Physics, Preprint ITP-89-43B (Kiev 1989) pdf
V. G. Drinfel’d, Quasi-Hopf algebras, Algebra i Analiz, 1:6 (1989) 114-148; Leningrad Math. J., 1:6 (1990) 1419-1457 (pdf in Russian)
V. G. Drinfel’d, Almost commutative Hopf algebras, Algebra i Analiz, 1:2 (1989) 30-46; Leningrad Math. J., 1:2 (1990) 321-342 mathnet.ru
Drinfeld twist and Drinfeld associator have been reproduced as special cases of higher bialgebra cocycles in works of Majid,
Shahn Majid, Cross product quantisation, nonabelian cohomology and twisting of Hopf algebras, in H.-D. Doebner, V.K. Dobrev, A.G. Ushveridze, eds., Generalized symmetries in Physics. World Sci. (1994) 13-41; (arXiv:hep.th/9311184)
Shahn Majid, Foundations of quantum group theory, Cambridge UP