nLab Drinfel'd twist

Definition

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

$$(1\otimes\chi)(id\otimes\Delta)\chi = (\chi\otimes 1)(\Delta\otimes id)\chi,$$

satisfying $(\epsilon\otimes id)\chi = (id\otimes\epsilon)\chi = 1$ (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 $H$.

Application to twisting

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 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 $S_\chi b = \sum \chi^{(1)} S(\chi^{(2)}) (S b)(\sum \chi^{(1)}S(\chi^{(2)}))^{-1}$. 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.

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

The bialgebroid generalization is described in

• Ping Xu, Quantum groupoids, Commun. Math. Phys. 216 (2001) 539–581 arXiv:q-alg/9905192

category: algebra