Quasitriangular Hopf algebra

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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.

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

${\displaystyle c_{U,V}(u\otimes v)=T\left(R\cdot (u\otimes v)\right)=T\left(R_{1}u\otimes R_{2}v\right)}$.