# Quasitriangular Hopf algebra

In mathematics, a Hopf algebra, *H*, is **quasitriangular**[1] if there exists an invertible element, *R*, of such that

- for all , where is the coproduct on
*H*, and the linear map is given by ,

- for all , where is the coproduct on

- ,

- ,

where , , and , where , , and , are algebra morphisms determined by

*R* is called the R-matrix.

As a consequence of the properties of quasitriangularity, the R-matrix, *R*, is a solution of the Yang–Baxter equation (and so a module *V* of *H* can be used to determine quasi-invariants of braids, knots and links). Also as a consequence of the properties of quasitriangularity, ; moreover
, , and . One may further show that the
antipode *S* must be a linear isomorphism, and thus *S ^{2}* is an automorphism. In fact,

*S*is given by conjugating by an invertible element: where (cf. Ribbon Hopf algebras).

^{2}It is possible to construct a quasitriangular Hopf algebra from a Hopf algebra and its dual, using the Drinfeld quantum double construction.

If the Hopf algebra *H* is quasitriangular, then the category of modules over *H* is braided with braiding

- .

## Twisting

The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element such that and satisfying the cocycle condition

Furthermore, is invertible and the twisted antipode is given by , with the twisted comultiplication, R-matrix and co-unit change according to those defined for the quasi-triangular quasi-Hopf algebra. Such a twist is known as an admissible (or Drinfeld) twist.

## Notes

- Montgomery & Schneider (2002), p. 72.

## References

- Montgomery, Susan (1993).
*Hopf algebras and their actions on rings*. Regional Conference Series in Mathematics.**82**. Providence, RI: American Mathematical Society. ISBN 0-8218-0738-2. Zbl 0793.16029. - Montgomery, Susan; Schneider, Hans-Jürgen (2002).
*New directions in Hopf algebras*. Mathematical Sciences Research Institute Publications.**43**. Cambridge University Press. ISBN 978-0-521-81512-3. Zbl 0990.00022.