nLab quasitriangular bialgebra

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Higher algebras

• symmetric monoidal (∞,1)-category of spectra

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Contents

Idea

A quasi-triangular bialgebra / triangular bialgebra is a bialgebra equipped with just the right structure such as to make its category of modules into a braided monoidal category/symmetric monoidal category.

Definition

Let $A$ be an algebra in a symmetric monoidal category $C$ with symmetry $\tau$; 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 $D_{i_1,\ldots,i_m}\in A^{\otimes n}$ 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 $h\in H$

$\tau\circ\Delta(h) = R\Delta(h)R^{-1}$

where $\tau=\tau_{H,H}:H\otimes H\to H\otimes H$ and

$(\Delta\otimes id)(R)=R_{13} R_{23}$
$(id\otimes\Delta)(R)=R_{13} R_{12}$

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

$(\epsilon\otimes id) R = 1,\,\,\,\,\,(id\otimes\epsilon)R = id$

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 $R_{21}:=\tau(R) = R^{-1}$.

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.

Properties

Tannaka duality

A quasitriangular structure on a bialgebra corresponds to a braided monoidal category structure on the category of modules of the underlying algebra. (For instance chapter 1, section 2 of (Carroll)).

for over /

/
$A$$Mod_A$
$R$-$Mod_R$-
= with -preserving
strict : with
with
(correct version) (without fiber functor)
with generalized
with
with
() with
with
() with and Schur smallness
form form

2-Tannaka duality for over

$A$$Mod_A$
$R$-$Mod_R$-
(with some duality and strictness structure)

3-Tannaka duality for over

$A$$Mod_A$
$R$-$Mod_R$-
• 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

• Robert Carroll, Calculus revisited