nLab
Drinfel'd twist

Given a bialgebra HH, a Drinfel’d twist is an invertible element χH 2\chi\in H^{\otimes 2} 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.

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 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.

Revised on March 22, 2011 12:43:39 by Tim Porter (95.147.236.255)