Quasitriangular Hopf algebra

{{Refimprove|date=December 2009}}

In mathematics, a Hopf algebra, H, is quasitriangularMontgomery & Schneider (2002), [{{Google books|plainurl=y|id=I3IK9U5Co_0C|page=72|text=Quasitriangular}} p. 72]. if there exists an invertible element, R, of $H\; \backslash otimes\; H$ such that

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

:*$(\backslash Delta\; \backslash otimes\; 1)(R)\; =\; R\_\{13\}\; \backslash \; R\_\{23\}$,

:*$(1\; \backslash otimes\; \backslash Delta)(R)\; =\; R\_\{13\}\; \backslash \; R\_\{12\}$,

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

:$\backslash phi\_\{12\}(a\; \backslash otimes\; b)\; =\; a\; \backslash otimes\; b\; \backslash otimes\; 1,$

:$\backslash phi\_\{13\}(a\; \backslash otimes\; b)\; =\; a\; \backslash otimes\; 1\; \backslash otimes\; b,$

:$\backslash phi\_\{23\}(a\; \backslash otimes\; b)\; =\; 1\; \backslash otimes\; a\; \backslash otimes\; b.$

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, $(\backslash epsilon\; \backslash otimes\; 1)\; R\; =\; (1\; \backslash otimes\; \backslash epsilon)\; R\; =\; 1\; \backslash in\; H$; moreover

$R^\{-1\}\; =\; (S\; \backslash otimes\; 1)(R)$, $R\; =\; (1\; \backslash otimes\; S)(R^\{-1\})$, and $(S\; \backslash otimes\; S)(R)\; =\; R$. 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 $u\; :=\; m\; ((S\; \backslash otimes\; 1)\; \backslash circ\; T)R$ (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

:$c\_\{U,V\}(u\backslash otimes\; v)\; =\; T\; \backslash left(\; R\; \backslash cdot\; (u\; \backslash otimes\; v\; )\backslash right)\; =\; T\; \backslash left(\; R\_1\; u\; \backslash otimes\; R\_2\; v\backslash right)$.