Quasitriangular Hopf algebra

In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of $$H \otimes H$$ such that


 * $$R \ \Delta(x)R^{-1} = (T \circ \Delta)(x) $$ for all $$x \in H$$, where $$\Delta$$ is the coproduct on H, and the linear map $$T : H \otimes H \to H \otimes H$$ is given by $$T(x \otimes y) = y \otimes x$$,


 * $$(\Delta \otimes 1)(R) = R_{13} \ R_{23}$$,


 * $$(1 \otimes \Delta)(R) = R_{13} \ R_{12}$$,

where $$R_{12} = \phi_{12}(R)$$, $$R_{13} = \phi_{13}(R)$$, and $$R_{23} = \phi_{23}(R)$$, where $$\phi_{12} : H \otimes H \to H \otimes H \otimes H$$, $$\phi_{13} : H \otimes H \to H \otimes H \otimes H$$, and $$\phi_{23} : H \otimes H \to H \otimes H \otimes H$$, are algebra morphisms determined by


 * $$\phi_{12}(a \otimes b) = a \otimes b \otimes 1,$$


 * $$\phi_{13}(a \otimes b) = a \otimes 1 \otimes b,$$


 * $$\phi_{23}(a \otimes b) = 1 \otimes a \otimes b.$$

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, $$(\epsilon \otimes 1) R = (1 \otimes \epsilon) R = 1 \in H$$; moreover $$R^{-1} = (S \otimes 1)(R)$$, $$R = (1 \otimes S)(R^{-1})$$, and $$(S \otimes S)(R) = R$$. 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: $$S^2(x)= u x u^{-1}$$ where $$u := m (S \otimes 1)R^{21}$$ (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
 * $$c_{U,V}(u\otimes v) = T \left( R \cdot (u \otimes v )\right) = T \left( R_1 u \otimes R_2 v\right) $$.

Twisting
The property of being a quasi-triangular Hopf algebra is preserved by twisting via an invertible element $$ F = \sum_i f^i \otimes f_i \in \mathcal{A \otimes A} $$ such that $$ (\varepsilon \otimes id )F = (id \otimes \varepsilon)F = 1 $$ and satisfying the cocycle condition


 * $$ (F \otimes 1) \cdot (\Delta \otimes id)( F) = (1 \otimes F) \cdot (id \otimes \Delta)( F) $$

Furthermore, $$ u = \sum_i f^i S(f_i)$$ is invertible and the twisted antipode is given by $$S'(a) = u S(a)u^{-1}$$, 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.