nLab
quasitriangular bialgebra

Let $A$ be an algebra in a symmetric monoidal category $C$ with symmetry $τ$ ; fix $m, l$ and $D \in A^{\otimes k}$ and let $1 \leq i_{r} \leq l$ for $1 \leq r \leq m$ be different. Then denote $D_{i_{1}, \dots, i_{m}} \in A^{\otimes n}$ as the image of $R \otimes 1^{\otimes (l - m)}$ under the permutation which is the composition of the $m$ transpositions $(r, i_{r})$ of tensor factors interchanging $r$ and $i_{r}$ . In the following $C$ is the monoidal category of $k$ -vector spaces.

A $k$ -bialgebra (in particular $k$ -Hopf algebra) is quasitriangular if there is an invertible element $R \in H \otimes H$ such that for any $h \in H$

τ \circ Δ (h) = R Δ (h) R^{- 1}

where $τ = τ_{H, H} : H \otimes H \to H \otimes H$ and

(Δ \otimes id) Δ (R) = R_{13} R_{23}

(id \otimes Δ) Δ (R) = R_{13} R_{12}

An invertible element $R$ satisfying these 3 properties is called the universal $R$ -element. As a corollary

(ϵ \otimes id) R = 1, (id \otimes ϵ) R = id

and the quantum Yang-Baxter equation? holds in the form

R_{12} R_{13} R_{23} = R_{23} R_{13} R_{12}

A quasitriangular $H$ is called triangular if $R_{21} : = τ (R) = R^{- 1}$ .

The category of representations of a quasitrianguar bialgebra is a braided monoidal category. If $R$ is a universal $R$ -element, then $R_{21}^{- 1}$ is as well. If $H$ is quasitriangular, $H^{cop}$ and $H_{op}$ are as well, with the universal $R$ -element being $R_{21}$ , or instead, $R_{12}^{- 1}$ . Any twisting of a quasitriangular bialgebra by a bialgebra 2-cocycle twists the universal $R$ -element as well; hence the twisted bialgebra is again quasitriangular. Often the $R$ -element does not exist as an element in $H \otimes H$ but rather in some completion of the tensor square; we say that $H$ is essentially quasitriangular, this is true for quantized enveloping algebras $U_{q} (G)$ in the rational form. The famous Sweedler’s Hopf algebra has a 1-parameter family of universal $R$ -matrices.

V. G. Drinfel’d, Quantum groups, Proc. ICM 1986, Vol. 1, 2 798–820, AMS 1987.
S. Majid, Quasitriangular Hopf algebras and Yang-Baxter equations, Int. j. mod. physics A, 5, 01, pp. 1-91 (1990) doi:10.1142/S0217751X90000027
S. Majid, Foundations of quantum group theory, Cambridge University Press 1995, 2000.
A. U. Klymik, K. Schmuedgen, Quantum groups and their representations, Springer 1997.
V. Chari, A. Pressley, A guide to quantum groups, Camb. Univ. Press 1994

Created on August 15, 2009 21:51:02 by Zoran Škoda (95.168.118.179)

nLab quasitriangular bialgebra

nLab
quasitriangular bialgebra