# Quasitriangular Hopf algebra

In mathematics, a Hopf algebra, H, is quasitriangular[1] if there exists an invertible element, R, of ${\displaystyle H\otimes H}$ such that

• ${\displaystyle R\ \Delta (x)R^{-1}=(T\circ \Delta )(x)}$ for all ${\displaystyle x\in H}$, where ${\displaystyle \Delta }$ is the coproduct on H, and the linear map ${\displaystyle T:H\otimes H\to H\otimes H}$ is given by ${\displaystyle T(x\otimes y)=y\otimes x}$,
• ${\displaystyle (\Delta \otimes 1)(R)=R_{13}\ R_{23}}$,
• ${\displaystyle (1\otimes \Delta )(R)=R_{13}\ R_{12}}$,

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

${\displaystyle \phi _{12}(a\otimes b)=a\otimes b\otimes 1,}$
${\displaystyle \phi _{13}(a\otimes b)=a\otimes 1\otimes b,}$
${\displaystyle \phi _{23}(a\otimes b)=1\otimes a\otimes b.}$

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, ${\displaystyle (\epsilon \otimes 1)R=(1\otimes \epsilon )R=1\in H}$; moreover ${\displaystyle R^{-1}=(S\otimes 1)(R)}$, ${\displaystyle R=(1\otimes S)(R^{-1})}$, and ${\displaystyle (S\otimes S)(R)=R}$. 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: ${\displaystyle S^{2}(x)=uxu^{-1}}$ where ${\displaystyle u:=m(S\otimes 1)R^{21}}$ (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.

## Twisting

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

${\displaystyle (F\otimes 1)\circ (\Delta \otimes id)F=(1\otimes F)\circ (id\otimes \Delta )F}$

Furthermore, ${\displaystyle u=\sum _{i}f^{i}S(f_{i})}$ is invertible and the twisted antipode is given by ${\displaystyle S'(a)=uS(a)u^{-1}}$, 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.