nLab Drinfel'd twist


Given a bialgebra HH, a Drinfel’d twist is an invertible element χHH\chi\in H\otimes H satisfying

(1χ)(idΔ)χ=(χ1)(Δid)χ, (1\otimes\chi)(id\otimes\Delta)\chi = (\chi\otimes 1)(\Delta\otimes id)\chi,

satisfying (ϵid)χ=(idϵ)χ=1(\epsilon\otimes id)\chi = (id\otimes\epsilon)\chi = 1 (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 HH.

Application to twisting

Vladimir Drinfel’d introduced χ\chi 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 HH be a quasi-triangular Hopf algebra with comultiplication Δ\Delta, antipode SS, universal R-element RR and let χ\chi be a Drinfel’d twist. Then the same algebra becomes another quasitriangular Hopf algebra with formulas for comultiplication Δ χb=χ(Δb)χ 1\Delta_\chi b = \chi (\Delta b)\chi^{-1}, universal R-element R χ=χ 21RχR_\chi = \chi_{21} R\chi and antipode S χb=χ (1)S(χ (2))(Sb)(χ (1)S(χ (2))) 1S_\chi b = \sum \chi^{(1)} S(\chi^{(2)}) (S b)(\sum \chi^{(1)}S(\chi^{(2)}))^{-1}. The counit is not changed. Here χ 21=τ(χ)\chi_{21}=\tau(\chi) where τ\tau is the flip of tensor factors, and χ=χ (1)χ (2)\chi = \sum \chi^{(1)}\otimes\chi^{(2)} is a notation similar to Sweedler’s convention.


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.


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

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; (
  • Shahn Majid, Foundations of quantum group theory, Cambridge UP

The bialgebroid generalization is described in

category: algebra

Last revised on August 13, 2023 at 10:22:21. See the history of this page for a list of all contributions to it.