Quasitriangular Hopf algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 11 July 2011

Quasitriangular Hopf algebras

1.1 Let $A$ be a Hopf algebra with coproduct $\Delta$ and antipode $S$. Let $\sigma :A\otimes A\to A\otimes A$ be the map given by $\sigma \left(a\otimes b\right)=b\otimes a$ for all $a,b\in A$. Define $\Delta \prime$ to be the opposite coproduct given by $$\Delta \prime =\sigma \circ \Delta .$$ Then $A$ with coproduct $\Delta \prime$ and antipode $S^{-1}$ is also a Hopf algebra. This follows by applying $S^{-1}$ to the defining relation for the antipode $$\sum _a{a}_{\left(1\right)}S\left(a_{\left(2\right)}\right)=\sum _{a}S\left(a_{\left(1\right)}\right)a_{\left(2\right)}=ϵ\left(a\right),$$ for all $a\in A$ and using the fact that $S$ (and therefore $S^{-1}$) is an antihomomorphism.

1.2 A pair $\left(A,R\right)$ consisting of a Hopf algebra $A$ and an invertible element $R\in A\otimes A$ is called quasitriangular if

1. $\Delta \prime \left(a\right)=R\Delta \left(A\right)R^{-1},$ for all $a\in A$,
2. $\left(\Delta \otimes id\right)\left(R\right)=R^{12}{R}^{23},$
3. $\left(id\otimes R\right)=R^{13}{R}^{12},$

where, if $R=\sum _i{a}_i\otimes b_i$ then $$R^{12}=\sum _i{a}_i\otimes b_i\otimes 1,R^{13}=\sum _i{a}_i\otimes 1\otimes b_i,R^{23}=\sum _{i}1\otimes a_i\otimes b_i\otimes 1,\text{etc.}$$

([D] Prop. 3.1) If $\left(A,R\right)$ is a quasitriangular Hopf algebra then

1. $R^{12}{R}^{13}{R}^{23}=R^{23}{R}^{13}{R}^{12}.$
2. ${\check{R}}_{12}{\check{R}}_{23}{\check{R}}_{12}={\check{R}}_{23}{\check{R}}_{12}{\check{R}}_{23},$
   where ${\check{R}}_{ij}=\sigma \circ L_{R^{ij}}\in End\left(A\otimes A\right),$ and $\sigma ,L_R\in End\left(A\otimes A\right)$ are given by $\sigma \left(a\otimes b\right)=b\otimes a$ and left multiplication by $R$ respectively.
3. $\left(ϵ\otimes id\right)\left(R\right)=1=\left(id\otimes ϵ\right)\left(R\right)$
4. $\left(S\otimes id\right)\left(R\right)=\left(id\otimes S^{-1}\right)\left(R\right)=R^{-1}.$
5. $\left(S\otimes S\right)\left(R\right)=R.$

		Proof.
	a) $$\begin{array}{rcl}{R}^{12}{R}^{13}{R}^{23}& =& R^{12}\left(\Delta \otimes id\right)\left(R\right)\text{by (1.2b)}\\ & =& \left(\Delta \prime \otimes id\right)\left(R\right)R^{12}\text{by (1.2a)}\\ & =& R^{23}{R}^{13}{R}^{12}.\end{array}$$ b) $$\begin{array}{rcl}{\check{R}}_{12}{\check{R}}_{23}{\check{R}}_{12}& =& {\sigma }^{12}{L}_{R^{12}}{\sigma }^{23}{L}_{R^{23}}{\sigma }^{12}{L}_{R^{12}}\\ & =& \left({\sigma }^{12}{\sigma }^{23}{\sigma }^{12}\right)\left({\sigma }^{12}{\sigma }^{23}{L}_{R^{12}}{\sigma }^{23}{\sigma }^{12}\right)\left({\sigma }^{12}{L}_{R^{23}}{\sigma }^{12}\right)L_{R^{12}}\\ & =& {\sigma }^{13}{L}_{R^{23}}{L}_{R^{13}}{L}_{R^{13}},\end{array}$$ and $$\begin{array}{rcl}{\check{R}}_{23}{\check{R}}_{12}{\check{R}}_{23}& =& {\sigma }^{23}{L}_{R^{23}}{\sigma }^{12}{L}_{R^{12}}{\sigma }^{23}{L}_{R^{23}}\\ & =& \left({\sigma }^{23}{\sigma }^{12}{\sigma }^{23}\right)\left({\sigma }^{23}{\sigma }^{12}{L}_{R^{23}}{\sigma }^{12}{\sigma }^{23}\right)\left({\sigma }^{23}{L}_{R^{12}}{\sigma }^{23}\right)L_{R^{23}}\\ & =& {\sigma }^{13}{L}_{R^{12}}{L}_{R^{13}}{L}_{R^{23}}.\end{array}$$ c) By (1.2b) $$R=\left(id\otimes id\right)\left(R\right)=\left(ϵ\otimes id\otimes id\right)\left(\Delta \otimes id\right)\left(R\right)=\left(ϵ\otimes id\otimes id\right)R^{13}{R}^{23}=\left(ϵ\otimes id\right)\left(R\right)\cdot R.$$ Thus $\left(ϵ\otimes id\right)(R)=1.$ Similarly, by (1.2c), $$R=\left(id\otimes id\right)\left(R\right)=\left(id\otimes id\otimes ϵ\right)\left(id\otimes \Delta \right)\left(R\right)=\left(id\otimes id\otimes ϵ\right)R^{13}{R}^{23}=\left(id\otimes ϵ\right)\left(R\right)\cdot R.$$ Thus $\left(id\otimes ϵ\right)\left(R\right)=1.$ d) $$\begin{array}{ccc}R\cdot \left(S\otimes id\right)\left(R\right)& =& \left(m\otimes \mathrm{<mo>id</mo>}\right)\left(id\otimes S\otimes id\right)\left(R^{13}{R}^{23}\right)\\ & =& \left(m\otimes id\right)\left(id\otimes S\otimes id\right)\left(\Delta \otimes id\right)\left(R\right)\\ & =& \left(ϵ\otimes id\right)\left(R\right)=1.\end{array}$$ So $\left(S\otimes id\right)\left(R\right)=R^{-1}.$ Let $A^{opp}$ be the Hopf algebra which is the same as $A$ except with the opposite comultiplication and with antipode $S^{-1}$. It is clear from the defining relations of a quasitriangular Hopf algebra that $\left(A^{opp},R^{21}\right)$ is also a quasitriangular Hopf algebra. Thus, it follows by applying the identity already proved to $\left(A^{opp},R^{21}\right)$ that $$\left(S^{-1}\otimes id\right)\left(R^{21}\right)={\left(R^{-1}\right)}^{21}$$ which is equivalent to $\left(id\otimes S^{-1}\right)\left(R\right)=R^{-1}$. e) This follows by letting $\left(id\otimes S\right)$ act on both sides of the equation $\left(id\otimes S^{-1}\right)\left(R\right)=\left(S\otimes id\right)\left(R\right)$ from d).

$\square$

References

The quantum double seems to have appeared first in the following paper.

[D] V.G. Drinfeld, Quantum Groups, Vol. 1 of Proccedings of the International Congress of Mathematicians (Berkeley, Calif., 1986). Amer. Math. Soc., Providence, RI, 1987, pp. 198–820. MR0934283

Some further proofs and hints appear in the following.

[D1] V.G. Drinfelʹd, Almost cocommutative Hopf algebras, (Russian) Algebra i Analiz 1 (1989), 30–46; translation in Leningrad Math. J. 1 (1990), 321–342. MR1025154

[Re] N. Yu. Reshetikhin, Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links I, LOMI preprint no. E–4–87, (1987).

page history