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 .
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 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.
Revised on March 22, 2011 12:43:39
by Tim Porter