Quasitriangular Hopf algebra

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

⊗

ϕ

ϕ

ϕ

ϕ

ϕ

ϕ

, 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,

( ϵ ⊗ 1 )

= ( 1 ⊗ ϵ )

One may further show that the antipode S must be a linear isomorphism, and thus S2 is an automorphism.

In fact, S2 is given by conjugating by an invertible element:

(cf.

Ribbon Hopf algebras).

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 The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element

( ε ⊗ i d )

= ( i d ⊗ ε )

{\displaystyle (\varepsilon \otimes id)F=(id\otimes \varepsilon )F=1}

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.