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Drinfel'd twist

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

$$(1\otimes \chi )(\mathrm{id}\otimes \Delta )\chi =(\chi \otimes 1)(\Delta \otimes \mathrm{id})\chi ,$$(1\otimes\chi)(id\otimes\Delta)\chi = (\chi\otimes 1)(\Delta\otimes id)\chi,

satisfying $(ϵ\otimes \mathrm{id})\chi =(\mathrm{id}\otimes ϵ)\chi =1$ (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 $H$.

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