## Quasitriangular Hopf algebras

Let $A=\left(A,m,\Delta ,\epsilon ,i,S\right)$ be a Hopf algebra and let $\tau$ be the $𝔽$-linear map $τ: A⊗A → A⊗A a⊗b ↦ b⊗a .$ Let ${\Delta }^{\mathrm{op}}=\tau \circ \Delta$ so that, if $a\in A$ and $Δ(a) =∑a a(1) ⊗ a(2) then Δop(a) = ∑a a(2) ⊗ a(1).$ Then $\left(A,m,{\Delta }^{\mathrm{op}},i,\epsilon ,{S}^{-1}\right)$ is also a Hopf algebra. This follows by applying ${S}^{-1}$ to the defining relation for the antipode $∑a a(1) S(a(2)) = ∑a S(a(1)) a(2) = ε(a) ,$ and using the fact that $S$ (and therefore ${S}^{-1}$) is an antihomomorphism.

With the algebra structure on $A\otimes A$ given by $\left(a\otimes b\right)\left(c\otimes d\right)=ac\otimes bd$, the map $\tau :A\otimes A\to A\otimes A$ is an algebra automorphism of $A\otimes A$ and the following diagram commutes

 $\begin{array}{lcl}A& \stackrel{\Delta }{\to }& A\otimes A\\ ↓\mathrm{id}& & ↓\tau \\ A& \stackrel{{\Delta }^{\mathrm{op}}}{\to }& A\otimes A\end{array}$                Sometimes we are lucky and can replace $\tau$ by an inner automorphism.

Let $U$ be a Hopf algebra with an invertible element

 $ℛ\in U\otimes U\phantom{\rule{2em}{0ex}}\text{such that}\phantom{\rule{2em}{0ex}}ℛ\Delta \left(a\right){ℛ}^{-1}={\Delta }^{\mathrm{op}}\left(a\right),\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}a\in U.$ (acc)
The pair $\left(U,ℛ\right)$ is a quasitriangular Hopf algebra if $ℛ\Delta \left(a\right){ℛ}^{-1}={\Delta }^{\mathrm{op}}\left(a\right)$, for $a\in U$, and
 $\left(\Delta \otimes \mathrm{id}\right)\left(ℛ\right)={ℛ}_{13}{ℛ}_{23}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left(\mathrm{id}\otimes \Delta \right)\left(ℛ\right)={ℛ}_{13}{ℛ}_{12},$ (cab)
where, if $ℛ=\sum {b}_{i}\otimes {b}^{i}$ then $ℛ12= ∑bi⊗ bi⊗1, ℛ13 =∑ bi⊗1⊗ bi, and ℛ23 = ∑1⊗ bi⊗ bi.$ The identities in (acc) and (cab) relate the $ℛ$-matrix to coproduct and the relations between the $ℛ$ matrix and the counit and antipode are given by
 $\left(\epsilon \otimes \mathrm{id}\right)\left(ℛ\right)=1=\left(\mathrm{id}\otimes \epsilon \right)\left(ℛ\right),\phantom{\rule{2em}{0ex}}\left(S\otimes \mathrm{id}\right)\left(ℛ\right)={ℛ}^{-1}=\left(\mathrm{id}\otimes {S}^{-1}\right)\left(ℛ\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left(S\otimes S\right)\left(ℛ\right)=ℛ.$ (eSR)
If $\left(U,ℛ\right)$ is a quasitriangular Hopf algebra then $ℛ$ satisfies the quantum Yang-Baxter equation,
 ${ℛ}_{12}{ℛ}_{13}{ℛ}_{23}={ℛ}_{12}\left(\Delta \otimes \mathrm{id}\right)\left(ℛ\right)=\left({\Delta }^{\mathrm{op}}\otimes \mathrm{id}\right)\left(ℛ\right){ℛ}_{12}={ℛ}_{23}{ℛ}_{13}{ℛ}_{12}.$ (QYBE)

For any two $U$-modules $M$ and $N$, the map $RˇMN : M⊗N → N⊗M m⊗n ↦ ∑bin ⊗bim INSERT INKSCAPE$ is a $U$-module isomorphism since $RˇMN (a((m⊗n)) = RˇMN ( Δ(a)(m⊗n) )= τℛΔ(a) (m⊗n) = τΔop(a)τ τ-1ℛ (m⊗n) =Δ(a) RˇMN (m⊗n).$ In order to be consistent with the graphical calculus the operators ${\stackrel{ˇ}{R}}_{MN}$ should be written on the right.

For $U$-modules $M$ and $N$ and a $U$-module isomorphism ${\tau }_{M}:M\to M$, $INSERT INKSCAPE$ and the relations in (cab) imply that if $M,N$ and $P$ are $U$-modules then $INSERT INKSCAPE$ as operators on $M\otimes N\otimes P.$ The preceding relations together imply the braid relation $INKSCAPE HERE$ $\left({\stackrel{ˇ}{R}}_{MN}\otimes {\mathrm{id}}_{P}\right)\left({\mathrm{id}}_{N}\otimes {\stackrel{ˇ}{R}}_{MP}\right)\left({\stackrel{ˇ}{R}}_{NP}\otimes {\mathrm{id}}_{M}\right)=\left({\mathrm{id}}_{M}\otimes {\stackrel{ˇ}{R}}_{NP}\right)\left({\stackrel{ˇ}{R}}_{MP}\otimes {\mathrm{id}}_{N}\right)\left({\mathrm{id}}_{P}\otimes {\stackrel{ˇ}{R}}_{MN}\right).$

## Ribbon Hopf algebras and the quantum Casimir

By [Dr, Prop. 2.1], the element $u$ in $U$ defined by

 $u=\sum _{ℛ}S\left({R}_{2}\right){R}_{1}\phantom{\rule{2em}{0ex}}\text{satisfies}\phantom{\rule{2em}{0ex}}ux{u}^{-1}={S}^{2}\left(x\right),\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{0.5em}{0ex}}x\in U.$ (udf)

Proof. (This proof is taken from [Dr, proof of Prop. 2.1].) The identity

 $\left(ℛ\otimes 1\right)\left(\sum _{a}{a}_{\left(1\right)}\otimes {a}_{\left(2\right)}\otimes {a}_{\left(3\right)}\right)=\left(\sum _{a}{a}_{\left(2\right)}\otimes {a}_{\left(1\right)}\otimes {a}_{\left(3\right)}\right)\left(ℛ\otimes 1\right)$
is
 $\sum _{ℛ,a}{R}_{1}{a}_{\left(1\right)}\otimes {R}_{2}{a}_{\left(2\right)}\otimes {a}_{\left(3\right)}=\sum _{ℛ,a}{a}_{\left(2\right)}{R}_{1}\otimes {a}_{\left(1\right)}{R}_{2}\otimes {a}_{\left(3\right)}$
So
 $\sum _{ℛ,a}{S}^{2}\left({a}_{\left(3\right)}\right)S\left({R}_{2}{a}_{\left(2\right)}\right){R}_{1}{a}_{\left(1\right)}=\sum _{ℛ,a}{S}^{2}\left({a}_{\left(3\right)}\right)S\left({a}_{\left(1\right)}{R}_{2}\right){a}_{\left(2\right)}{R}_{1}$
So
 $\sum _{ℛ,a}S\left({a}_{\left(2\right)}S\left({a}_{\left(3\right)}\right)\right)S\left({R}_{2}\right){R}_{1}{a}_{\left(1\right)}=\sum _{ℛ,a}{S}^{2}\left({a}_{\left(3\right)}\right)S\left({R}_{2}\right)S\left({a}_{\left(1\right)}\right){a}_{\left(2\right)}{R}_{1}$ (*)
Since
 $\sum _{a}S\left({a}_{\left(1\right)}\right){a}_{\left(2\right)}\otimes {a}_{\left(3\right)}=1\otimes a\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\sum _{a}{a}_{\left(1\right)}\otimes {a}_{\left(2\right)}S\left({a}_{\left(3\right)}\right)=a\otimes 1$
the identity in (*) becomes $ua={S}^{2}\left(a\right)u$. If
 $v=\sum _{{ℛ}^{-1}}{S}^{-1}\left({\left({R}^{-1}\right)}_{2}\right){\left({R}^{-1}\right)}_{1}$
then
 $uv=u\sum _{{ℛ}^{-1}}{S}^{-1}\left({\left({R}^{-1}\right)}_{2}\right){\left({R}^{-1}\right)}_{1}=\sum _{{ℛ}^{-1}}S\left({\left({R}^{-1}\right)}_{2}\right)u{\left({R}^{-1}\right)}_{1}=\sum _{{ℛ}^{-1},ℛ}S\left({R}_{2}{\left({R}^{-1}\right)}_{2}\right){R}_{1}{\left({R}^{-1}\right)}_{1}=1,$
since
 $\sum _{{ℛ}^{-1},ℛ}{R}_{2}{\left({R}^{-1}\right)}_{2}\otimes {R}_{1}{\left({R}^{-1}\right)}_{1}=ℛ{ℛ}^{-1}=1.$
So ${S}^{2}\left(v\right)u=uv=1$ and $u$ has both a left inverse and a right inverse. Thus $u$ is invertible and ${u}^{-1}=v$. $\square$

[Dr, Prop. 3.2] proves the following important result essentially due to Lyubashenko:

 $\Delta \left(u\right)={\left({ℛ}_{21}ℛ\right)}^{-1}\left(u\otimes u\right)=\left(u\otimes u\right){\left({ℛ}_{21}ℛ\right)}^{-1}.$ (Δu)

Proof. (This proof is taken from [Dr, proof of Prop. 3.2].) Let $A\otimes A$ be the right $A\otimes A\otimes A\otimes A$-module defined by $(x1⊗x2) * (y1⊗y2 ⊗y3⊗y4) = S(y3)x1 y1⊗ S(y4)x2 y2 .$ Then, since $u=\sum _{ℛ}S\left({R}_{2}\right){R}_{1}$ and $\Delta \left(a\right){ℛ}_{21}ℛ={ℛ}_{21}ℛ\Delta \left(a\right)$, $Δ(u)ℛ21ℛ = ∑ℛ (S⊗S) (Δop (R2)) Δ(R1) ℛ21ℛ = ∑ℛ (S⊗S) (Δop (R2)) ℛ21ℛ Δ(R1) = ℛ21* ℛ12 (Δ⊗Δop) (ℛ) .$ Since ${ℛ}_{12}\left(\Delta \otimes {\Delta }^{\mathrm{op}}\right)\left(ℛ\right)={ℛ}_{12}{ℛ}_{13}{ℛ}_{23}{ℛ}_{14}{ℛ}_{24}={ℛ}_{23}{ℛ}_{13}{ℛ}_{12}{ℛ}_{14}{ℛ}_{24}$, $Δ(u)ℛ21ℛ = ℛ21* ℛ12 (Δ⊗Δop) (ℛ) = ℛ21* ℛ23 ℛ13 ℛ12 ℛ14 ℛ24 = (1⊗1)* ℛ13 ℛ12 ℛ14 ℛ24 = (u⊗1)* ℛ12 ℛ14 ℛ24 = (u⊗1)* ℛ24 = (u⊗u),$ where we have used $ℛ21* ℛ23 = ∑i,j S(bj)bi ⊗aiaj = (S⊗id) ( (∑i S-1 (bi) ⊗ai ) (∑j bj⊗aj ) ) =(S⊗id) ( (S-1⊗id) (ℛ21) ℛ21 ) =(S⊗id) ( (ℛ21)-1 ℛ21 ) =(S⊗id) (1⊗1) =1⊗1 , (u⊗1)* ℛ12 ℛ14 =∑i,j uaiaj ⊗S(bj) bi =(u⊗1) (1⊗1) =u⊗1,$ $\left(1\otimes 1\right)*{ℛ}_{13}=u\otimes 1,$ and $\left(u\otimes 1\right)*{ℛ}_{24}=u\otimes u.$ $\square$

## Ribbon Hopf algebras and the quantum trace

A ribbon Hopf algebra $\left(U,ℛ,v\right)$ is a quasitriangular Hopf algebra $\left(U,ℛ\right)$ with an invertible element $v$ such that

 $v\in Z\left(U\right),\phantom{\rule{2em}{0ex}}{v}^{2}=uS\left(u\right),\phantom{\rule{2em}{0ex}}S\left(v\right)=v,\phantom{\rule{2em}{0ex}}\epsilon \left(v\right)=1,\phantom{\rule{2em}{0ex}}\Delta \left(v\right)={\left({ℛ}_{21}ℛ\right)}^{-1}\left(v\otimes v\right),$ (rbH)
where ${ℛ}_{21}=\sum _{ℛ}{R}_{2}\otimes {R}_{1}$ if $ℛ=\sum _{ℛ}{R}_{1}\otimes {R}_{2}$, and $u$ is as in (udf). Note that ${v}^{-1}u$ is grouplike, $Δ( v-1u ) = v-1u ⊗ v-1u .$ If $M$ is a $U$-module and
 $CM: M → M m ↦ vm so that C M⊗N = ( Rˇ MN Rˇ NM ) -1 (CM ⊗ CN ).$ (casR)
by the last identity in (rbH).

Examples. Let $𝔤$ be a finite dimensional complex semisimple Lie algebra. Both

 $U=U𝔤\phantom{\rule{0.5em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}ℛ=1\otimes 1\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}v=1,\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}U={U}_{h}𝔤\phantom{\rule{0.5em}{0ex}}\text{with}\phantom{\rule{0.5em}{0ex}}v={e}^{-h\rho }u,$
are ribbon Hopf algebras (see [LR, §2]).

Let $V$ be a finite dimensional $U$-module and let ${V}^{*}$ be the dual module. Let ${E}_{V}$ be the composition

 ${E}_{V}:V\otimes {V}^{*}\stackrel{{v}^{-1}\otimes 1}{⟶}V\otimes {V}^{*}\stackrel{{\stackrel{ˇ}{R}}_{V{V}^{*}}}{⟶}{V}^{*}\otimes V\stackrel{\mathrm{ev}}{⟶}1\stackrel{\mathrm{coev}}{⟶}V\otimes {V}^{*}$, (cex)
so that ${E}_{V}$ is a $U$-module homomorphism with image a submodule of $V\otimes {V}^{*}$ isomorphic to the trivial representation of $U$.

Let $M$ be a $U$-module and let $\psi \in \mathrm{End}\left(M\otimes V\right)$. Then, as operators on $M\otimes V\otimes {V}^{*}$,

 $\left(1\otimes {E}_{V}\right)\left(\psi \otimes \mathrm{id}\right)\left(1\otimes {E}_{V}\right)=\left(\mathrm{id}\otimes {\mathrm{qtr}}_{V}\right)\left(\psi \right)\otimes {E}_{V}$, (qtr)
where the quantum trace $\left(\mathrm{id}\otimes {\mathrm{qtr}}_{V}\right)\left(\psi \right):M\to M$ is the composition
 $M\otimes 1\stackrel{\mathrm{id}\otimes \mathrm{coev}}{⟶}M\otimes V\otimes {V}^{*}\stackrel{\psi \otimes \mathrm{id}}{⟶}M\otimes V\otimes {V}^{*}\stackrel{\mathrm{id}\otimes {v}^{-1}\otimes \mathrm{id}}{⟶}M\otimes V\otimes {V}^{*}\stackrel{\mathrm{id}\otimes {\stackrel{ˇ}{R}}_{V{V}^{*}}}{⟶}M\otimes {V}^{*}\otimes V\stackrel{\mathrm{id}\otimes \mathrm{ev}}{⟶}M\otimes 1$.
The special case when $M=1$ and $\psi ={\mathrm{id}}_{V}$ is the quantum dimension of $V$,
 ${\mathrm{dim}}_{q}\left(V\right)={\mathrm{qtr}}_{V}\left({\mathrm{id}}_{V}\right)$. (qdm)

Let $V$ be a finite dimensional $U$-module, ${V}^{*}$ the dual module and let ${C}_{V}:V\to V$ be as defined in (casR). Let $x\in V$ and $\phi \in {V}^{*}$. Let ${e}_{1},\dots ,{e}_{n}$ be a basis of $V$ and ${e}^{1},\dots ,{e}^{n}$ the dual basis in ${V}^{*}$. Let $M$ be a $U$-module and $\psi \in {\mathrm{End}}_{U}\left(M\otimes V\right)$. Then

 ${E}_{V}\left(x\otimes \phi \right)=⟨\phi ,u{v}^{-1}x⟩\left(\sum _{i=1}^{n}{e}_{i}\otimes {e}^{i}\right),\phantom{\rule{2em}{0ex}}\left(\mathrm{id}\otimes {\mathrm{qtr}}_{V}\right)\left(\psi \right)=\left(\mathrm{id}\otimes {\mathrm{tr}}_{V}\right)\left(\left(1\otimes u{v}^{-1}\right)\psi \right)$, (twr)
 ${E}_{V}^{2}={\mathrm{dim}}_{q}\left(V\right){E}_{V},\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\left(\mathrm{id}\otimes {\mathrm{qtr}}_{V}\right)\left({\stackrel{ˇ}{R}}_{VV}\right)={C}_{V}^{-1}$. (fan)

Proof. Computing the action of ${E}_{V}$ on $x\otimes \phi$,

 $\begin{array}{rcl}{E}_{V}\left(x\otimes \phi \right)& =& \left(\mathrm{coev}\circ \mathrm{ev}\circ {\stackrel{ˇ}{R}}_{V{V}^{*}}\left({v}^{-1}\otimes \mathrm{id}\right)\right)\left(x\otimes \phi \right)=\left(\mathrm{coev}\circ \mathrm{ev}\right)\left(\sum _{ℛ}{R}_{2}\phi \otimes {R}_{1}{v}^{-1}x\right)\\ & =& \sum _{ℛ}⟨{R}_{2}\phi ,{R}_{1}{v}^{-1}x⟩\sum _{i=1}^{n}{e}_{i}\otimes {e}^{i}=\sum _{ℛ}⟨\phi ,S\left({R}_{2}\right){R}_{1}{v}^{-1}x⟩\sum _{i=1}^{n}{e}_{i}\otimes {e}^{i}\\ & =& ⟨\phi ,u{v}^{-1}x⟩\left(\sum _{i=1}^{n}{e}_{i}\otimes {e}^{i}\right),\end{array}$
which establishes the first identity in (twr). If ${\psi }_{ji}\in \mathrm{End}\left(M\right)$ are such that
 $\psi \left(m\otimes {e}_{i}\right)=\sum _{j=1}^{n}{\psi }_{ji}\left(m\right)\otimes {e}_{j},\phantom{\rule{2em}{0ex}}\text{then}$
 $\begin{array}{rcl}\left(\mathrm{id}\otimes {\mathrm{qtr}}_{V}\right)\left(\psi \right)\left(m\right)& =& \left(\mathrm{id}\otimes \mathrm{ev}\right)\left(\mathrm{id}\otimes {\stackrel{ˇ}{R}}_{V{V}^{*}}\right)\left(1\otimes {v}^{-1}\otimes 1\right)\left(\psi \otimes \mathrm{id}\right)\left(\mathrm{id}\otimes \mathrm{coev}\right)\left(m\otimes 1\right)\\ & =& \left(\mathrm{id}\otimes \mathrm{ev}\right)\left(\mathrm{id}\otimes {\stackrel{ˇ}{R}}_{V{V}^{*}}\right)\left(1\otimes {v}^{-1}\otimes 1\right)\left(\sum _{i,k}{\psi }_{ki}\left(m\right)\otimes {e}_{k}\otimes {e}^{i}\right)\\ & =& \left(\mathrm{id}\otimes \mathrm{ev}\right)\left(\sum _{ℛ,i,k}{\psi }_{ki}\left(m\right)\otimes {R}_{2}{e}^{i}\otimes {R}_{1}{v}^{-1}{e}_{k}\right)=\sum _{ℛ,i,k}{\psi }_{ki}\left(m\right)⟨{R}_{2}{e}^{i},{R}_{1}{v}^{-1}{e}_{k}⟩\\ & =& \sum _{ℛ,i,k}{\psi }_{ki}\left(m\right)⟨{e}^{i},S\left({R}_{2}\right){R}_{1}{v}^{-1}{e}_{k}⟩=\sum _{ℛ,i,k}{\psi }_{ki}\left(m\right)⟨{e}^{i},u{v}^{-1}{e}_{k}⟩\\ & =& \sum _{i}⟨\mathrm{id}\otimes {e}^{i},\left(1\otimes u{v}^{-1}\right)\psi \left(m\otimes {e}_{i}\right)⟩=\left(\mathrm{id}\otimes {\mathrm{tr}}_{V}\right)\left(\left(1\otimes u{v}^{-1}\right)\psi \right)\left(m\right)\end{array}$
is the second identity in (twr). The identity ${E}_{V}^{2}={\mathrm{dim}}_{q}\left(V\right){E}_{V}$ is the special case of (qtr) when $M=1$ and $\phi ={\mathrm{id}}_{V}$. Finally, if $x\in V$ then
 $\begin{array}{rcl}\left(\mathrm{id}\otimes {\mathrm{qtr}}_{V}\right)\left({\stackrel{ˇ}{R}}_{VV}\right)\left(x\right)& =& \sum _{i}⟨{\mathrm{id}}_{V}\otimes {e}^{i},\left(1\otimes {v}^{-1}u\right){\stackrel{ˇ}{R}}_{VV}\left(x\otimes {e}_{i}\right)⟩=\sum _{ℛ,i}⟨{\mathrm{id}}_{V}\otimes {e}^{i},{R}_{2}{e}_{i}\otimes {v}^{-1}u{R}_{1}x⟩\\ & =& \sum _{ℛ,i}{R}_{2}{e}_{i}⟨{e}^{i},{v}^{-1}u{R}_{1}x⟩=\sum _{ℛ}{R}_{2}{v}^{-1}u{R}_{1}x=\sum _{ℛ}{R}_{2}{S}^{2}\left({R}_{1}\right){v}^{-1}ux\\ & =& {u}^{-1}{v}^{-1}ux={v}^{-1}x={C}_{V}^{-1}\left(x\right)\end{array}$
$\square$

The identity (qtr) is the reason that the Jones basic construction often arises in the study of modules for quasitriangular Hopf algebras.

## Notes and References

These notes follow Drinfeld [Dr]. Other references are [CP].

What do the identities (eSR) mean in terms of representations?

Following [Dr, Prop. 3.1], the identities in (eSR) are proved as follows: Since $ℛ =(ε⊗id⊗id) (Δ⊗id) (ℛ) =(ε⊗id⊗id) ℛ13 ℛ23 = (ε⊗id)(ℛ) ⋅ℛ,$ and $ℛ= (id⊗id⊗ε) (id⊗Δ)(ℛ) = (id⊗id⊗ε) ℛ13 ℛ12 = (id⊗ε)(ℛ) ⋅ℛ,$ and so $(ε⊗id)(ℛ) =1 and (id⊗ε)(ℛ) =1.$ Then, since $ℛ⋅ (S⊗id)(ℛ) = (m⊗id) (id⊗S⊗id) (ℛ13 ℛ23) = (m⊗id) (id⊗S⊗id) (Δ⊗id) (ℛ) = (ε⊗id)(ℛ) =1,$ it follows that $(S⊗id)(ℛ) =ℛ-1.$ Since $\left({A}^{\mathrm{op}},{ℛ}_{21}\right)$ is a quasitriangular Hopf algebra with antipode ${S}^{-1}$ it follows that $\left({S}^{-1}\otimes \mathrm{id}\right)\left({ℛ}_{21}\right)={\left({ℛ}_{21}\right)}^{-1}.$ So $(id⊗S-1) (ℛ) =ℛ-1.$ Finally, $(S⊗S)(ℛ) =(id⊗S) (S⊗id)(ℛ) =(id⊗S)( ℛ-1) =(id⊗S) (id⊗S-1) (ℛ) =ℛ,$ which completes the proofs of the identities in (eSR).

[Dr, §2 remark (1)] explains that Lyubashenko says that if $\gamma :U\to \mathrm{End}\left(M\right)$ is a representation of $U$ then $\gamma \left(u\right)$ is the map

 $M\stackrel{\sim }{⟶}M\otimes 1\stackrel{1\otimes \mathrm{coev}}{⟶}M\otimes {M}^{*}\otimes {M}^{**}\stackrel{\tau ℛ}{⟶}M\otimes {M}^{*}\otimes {M}^{**}\stackrel{\mathrm{ev}\otimes 1}{⟶}{M}^{**}$

## References

[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

[Dr] 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

[LR] R. Leduc and A. Ram, A ribbon Hopf algebra approach to the irreducible representations of centralizer algebras: The Brauer, Birman-Wenzl and type A Iwahori-Hecke algebras, Adv. Math. 125 (1997), 1-94. MR1427801.

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