quasitriangular bialgebra in nLab

Contents

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 AA be monoid object in a symmetric monoidal category CC with braiding τ\tau; fix m,lm,l and D∈A ⊗kD\in A^{\otimes k} and let 1≤i r≤l1\leq i_r\leq l for 1≤r≤m1\leq r\leq m be different. Then denote D i 1,…,i m∈A ⊗nD_{i_1,\ldots,i_m}\in A^{\otimes n} as the image of R⊗1 ⊗(l−m)R\otimes 1^{\otimes (l-m)} under the permutation which is the composition of the mm transpositions (r,i r)(r,i_r) of tensor factors interchanging rr and i ri_r. In the following CC is the monoidal category k Vect k Vect of kk-vector spaces.

A kk-bialgebra (in particular kk-Hopf algebra) is quasitriangular if there is an invertible element R∈H⊗HR\in H\otimes H such that for any h∈Hh\in H

τ∘Δ(h)=RΔ(h)R −1 \tau\circ\Delta(h) \,=\, R\Delta(h)R^{-1}

where τ=τ H,H:H⊗H→H⊗H\tau=\tau_{H,H} \colon H\otimes H\to H\otimes H and

(Δ⊗id)(R)=R 13R 23 (\Delta\otimes id)(R) \,=\, R_{13} R_{23}

(id⊗Δ)(R)=R 13R 12 (id\otimes\Delta)(R) \,=\, R_{13} R_{12}

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

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

and the quantum Yang-Baxter equation holds in the form

R 12R 13R 23=R 23R 13R 12. R_{12} R_{13} R_{23} \,=\, R_{23} R_{13} R_{12} \,.

A quasitriangular HH is called triangular if R 21:=τ(R)=R −1R_{21}:=\tau(R) = R^{-1}.

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

• quasitriangular Hopf algebra

References

• 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