nLab bialgebroid 2-cocycle

Idea

Invertible counital bialgebra 2-cocycles are the basic datum in Drinfel'd twist theory for twisting bialgebras and Hopf algebras.

A generalization to associative bialgebroids was introduced in Xu 1999:

using an invertible bialgebroid 2-cocycle $F$ and a, say, left $R$-algebroid $H$, one obtains a new base algebra $R_F$ and a left $R_F$-algebroid $H_F$ whose underlying associative algebra structure is unchanged but whose coproduct is appropriately twisted. The corresponding cocycles and the twisting procedure are now also referred to as Drinfeld–Xu twists.

There is also a dual theory of cotwists.

Literature

The notion is properly introduced in

• Ping Xu: Quantum groupoids, Commun. Math. Phys. 216 (2001) 539–581 [arXiv:q-alg/9905192, doi:10.1007/s002200000334]