## 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 $Δ′=σ∘Δ.$ 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 $∑ a a(1) S(a(2))= ∑ a S(a(1)) a(2) =ϵ(a),$ 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 $R12= ∑ i ai⊗bi⊗1, R13= ∑ i ai⊗1⊗bi, R23= ∑ i 1⊗ai⊗bi⊗1, etc.$

([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. ${\stackrel{ˇ}{R}}_{12}{\stackrel{ˇ}{R}}_{23}{\stackrel{ˇ}{R}}_{12}={\stackrel{ˇ}{R}}_{23}{\stackrel{ˇ}{R}}_{12}{\stackrel{ˇ}{R}}_{23},$
where ${\stackrel{ˇ}{R}}_{ij}=\sigma \circ {L}_{{R}^{ij}}\in End\left(A\otimes A\right),$ 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) $R12 R13 R23 = R12 (Δ⊗id)(R) by (1.2b) = (Δ′⊗id)(R) R12 by (1.2a) = R23 R13 R12.$ b) $R ˇ 12 R ˇ 23 R ˇ 12 = σ12 L R12 σ23 L R23 σ12 LR12 = ( σ12 σ23 σ12 ) ( σ12 σ23 LR12 σ23 σ12 ) ( σ12 LR23 σ12 ) LR12 = σ13 LR23 LR13 LR13,$ and $R ˇ 23 R ˇ 12 R ˇ 23 = σ23 LR23 σ12 LR12 σ23 LR23 = ( σ23 σ12 σ23 ) ( σ23 σ12 LR23 σ12 σ23 ) ( σ23 LR12 σ23 ) LR23 = σ13 LR12 LR13 LR23.$ c) By (1.2b) $R= (id⊗id)(R) = (ϵ⊗id⊗id) (Δ⊗id)(R) = (ϵ⊗id⊗id) R13R23 = (ϵ⊗id)(R)⋅R.$ Thus $\left(ϵ\otimes id\right)\left(R\right)=1.$ Similarly, by (1.2c), $R= (id⊗id)(R) = (id⊗id⊗ϵ) (id⊗Δ)(R)= (id⊗id⊗ϵ) R13R23 = (id⊗ϵ)(R)⋅R.$ Thus $\left(id\otimes ϵ\right)\left(R\right)=1.$ d) $R⋅(S⊗id)(R) = (m⊗id) (id⊗S⊗id) (R13R23) = (m⊗id) (id⊗S⊗id) (Δ⊗id)(R) = (ϵ⊗id)(R)=1.$ 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 $( S-1⊗id ) (R21)= (R-1) 21$ 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).