Quasitriangular Hopf algebras

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

Last updates: 17 November 2011

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 $$\begin{array}{rcll}\tau :& A\otimes A& \to & A\otimes A\\ & a\otimes b& \mapsto & b\otimes a\end{array}.$$ Let ${\Delta }^{\mathrm{op}}=\tau \circ \Delta$ so that, if $a\in A$ and $$\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}\text{then}{\Delta }^{\mathrm{op}}\left(a\right)=\sum _a{a}_{\left(2\right)}\otimes a_{\left(1\right)}.$$ 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 $$\sum _a{a}_{\left(1\right)}S\left(a_{\left(2\right)}\right)=\sum _{a}S\left(a_{\left(1\right)}\right)a_{\left(2\right)}=\epsilon \left(a\right),$$ and using the fact that $S$ (and therefore $S^{-1}$) is an antihomomorphism.

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

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

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

$ℛ\in U\otimes U\text{such that}ℛ\Delta \left(a\right){ℛ}^{-1}={\Delta }^{\mathrm{op}}\left(a\right),\text{for}a\in U.$

(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

$(\Delta \otimes \mathrm{id})\left(ℛ\right)={ℛ}_{13}{ℛ}_{23}\text{and}(\mathrm{id}\otimes \Delta )\left(ℛ\right)={ℛ}_{13}{ℛ}_{12},$

(cab)

where, if $ℛ=\sum b_i\otimes b^i$ then $${ℛ}_{12}=\sum b_i\otimes b^i\otimes 1,{ℛ}_{13}=\sum b_i\otimes 1\otimes b^i,\text{and}{ℛ}_{23}=\sum 1\otimes b_i\otimes b^i.$$ 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

$(\epsilon \otimes \mathrm{id})\left(ℛ\right)=1=(\mathrm{id}\otimes \epsilon )\left(ℛ\right),(S\otimes \mathrm{id})\left(ℛ\right)={ℛ}^{-1}=(\mathrm{id}\otimes S^{-1})\left(ℛ\right)\text{and}(S\otimes S)\left(ℛ\right)=ℛ.$

(eSR)

If $(U,ℛ)$ is a quasitriangular Hopf algebra then $ℛ$ satisfies the quantum Yang-Baxter equation,

${ℛ}_{12}{ℛ}_{13}{ℛ}_{23}={ℛ}_{12}(\Delta \otimes \mathrm{id})\left(ℛ\right)=({\Delta }^{\mathrm{op}}\otimes \mathrm{id})\left(ℛ\right){ℛ}_{12}={ℛ}_{23}{ℛ}_{13}{ℛ}_{12}.$

(QYBE)

For any two $U$-modules $M$ and $N$, the map $$\begin{array}{rcll}{\check{R}}_{MN}:& M\otimes N& \to & N\otimes M\\ & m\otimes n& \mapsto & \sum b^{i}n\otimes b_{i}m\end{array}\text{INSERT INKSCAPE}$$ is a $U$-module isomorphism since $${\check{R}}_{MN}\left(a\right((m\otimes n))={\check{R}}_{MN}(\Delta \left(a\right)(m\otimes n))=\tau ℛ\Delta (a\left)\right(m\otimes n)=\tau {\Delta }^{\mathrm{op}}(a\left)\tau {\tau }^{-1}ℛ\right(m\otimes n)=\Delta (a\left){\check{R}}_{MN}\right(m\otimes n).$$ In order to be consistent with the graphical calculus the operators ${\check{R}}_{MN}$ should be written on the right.

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

Ribbon Hopf algebras and the quantum Casimir

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

$u=\sum _{ℛ}S\left(R_2\right)R_1\text{satisfies}ux{u}^{-1}=S^2\left(x\right),\text{for}x\in U.$

(udf)

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

$(ℛ\otimes 1)(\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}\otimes a_{\left(3\right)})=(\sum _a{a}_{\left(2\right)}\otimes a_{\left(1\right)}\otimes a_{\left(3\right)})(ℛ\otimes 1)$

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

$\sum _{ℛ,a}S\left(a_{\left(2\right)}S\right(a_{\left(3\right)}\left)\right)S\left(R_2\right)R_1{a}_{\left(1\right)}=\sum _{ℛ,a}{S}^2\left(a_{\left(3\right)}\right)S\left(R_2\right)S\left(a_{\left(1\right)}\right)a_{\left(2\right)}{R}_1$

(*)

Since

$\sum _{a}S\left(a_{\left(1\right)}\right)a_{\left(2\right)}\otimes a_{\left(3\right)}=1\otimes a\text{and}\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)}S\left(a_{\left(3\right)}\right)=a\otimes 1$

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:

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

(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 $$(x_1\otimes x_2)*(y_1\otimes y_2\otimes y_3\otimes y_4)=S\left(y_3\right)x_1{y}_1\otimes S\left(y_4\right)x_2{y}_2.$$ Then, since $u=\sum _{ℛ}S\left(R_2\right)R_1$ and $\Delta \left(a\right){ℛ}_{21}ℛ={ℛ}_{21}ℛ\Delta \left(a\right)$, $$\begin{array}{rl}\Delta \left(u\right){ℛ}_{21}ℛ& =\sum _{ℛ}(S\otimes S)\left({\Delta }^{\mathrm{op}}\right(R_2\left)\right)\Delta \left(R_1\right){ℛ}_{21}ℛ\\ & =\sum _{ℛ}(S\otimes S)\left({\Delta }^{\mathrm{op}}\right(R_2\left)\right){ℛ}_{21}ℛ\Delta \left(R_1\right)={ℛ}_{21}*{ℛ}_{12}(\Delta \otimes {\Delta }^{\mathrm{op}})\left(ℛ\right).\end{array}$$ Since ${ℛ}_{12}(\Delta \otimes {\Delta }^{\mathrm{op}})\left(ℛ\right)={ℛ}_{12}{ℛ}_{13}{ℛ}_{23}{ℛ}_{14}{ℛ}_{24}={ℛ}_{23}{ℛ}_{13}{ℛ}_{12}{ℛ}_{14}{ℛ}_{24}$, $$\begin{array}{rl}\Delta \left(u\right){ℛ}_{21}ℛ& ={ℛ}_{21}*{ℛ}_{12}(\Delta \otimes {\Delta }^{\mathrm{op}})\left(ℛ\right)={ℛ}_{21}*{ℛ}_{23}{ℛ}_{13}{ℛ}_{12}{ℛ}_{14}{ℛ}_{24}\\ & =(1\otimes 1)*{ℛ}_{13}{ℛ}_{12}{ℛ}_{14}{ℛ}_{24}=(u\otimes 1)*{ℛ}_{12}{ℛ}_{14}{ℛ}_{24}=(u\otimes 1)*{ℛ}_{24}=(u\otimes u),\end{array}$$ where we have used $$\begin{array}{rl}{ℛ}_{21}*{ℛ}_{23}& =\sum _{i,j}S\left(b_j\right)b_i\otimes a_i{a}_j=(S\otimes \mathrm{id})\left((\sum _i{S}^{-1}\left(b_i\right)\otimes a_i)(\sum _j{b}_j\otimes a_j)\right)\\ & =(S\otimes \mathrm{id})\left((S^{-1}\otimes \mathrm{id})\left({ℛ}_{21}\right){ℛ}_{21}\right)=(S\otimes \mathrm{id})\left({\left({ℛ}_{21}\right)}^{-1}{ℛ}_{21}\right)=(S\otimes \mathrm{id})(1\otimes 1)=1\otimes 1,\\ (u\otimes 1)*{ℛ}_{12}{ℛ}_{14}& =\sum _{i,j}u{a}_i{a}_j\otimes S\left(b_j\right)b_i=(u\otimes 1)(1\otimes 1)=u\otimes 1,\end{array}$$ $(1\otimes 1)*{ℛ}_{13}=u\otimes 1,$ and $(u\otimes 1)*{ℛ}_{24}=u\otimes u.$ $\square$

Ribbon Hopf algebras and the quantum trace

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

$v\in Z\left(U\right),v^2=uS\left(u\right),S\left(v\right)=v,\epsilon \left(v\right)=1,\Delta \left(v\right)={\left({ℛ}_{21}ℛ\right)}^{-1}(v\otimes v),$

(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, $$\Delta \left(v^{-1}u\right)=v^{-1}u\otimes v^{-1}u.$$ If $M$ is a $U$-module and

$$\begin{array}{cccc}{C}_M:& M& \to & M\\ & m& \mapsto & vm\end{array}\text{so that}{C}_{M\otimes N}={\left({\check{R}}_{MN}{\check{R}}_{NM}\right)}^{-1}(C_M\otimes C_N).$$

(casR)

by the last identity in (rbH).

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

$U=U𝔤\text{with}ℛ=1\otimes 1\text{and}v=1,\text{and}U=U_h𝔤\text{with}v=e^{-h\rho }u,$

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

$E_V:V\otimes V^{*}\stackrel{v^{-1}\otimes 1}{⟶}V\otimes V^{*}\stackrel{{\check{R}}_{V{V}^{*}}}{⟶}{V}^{*}\otimes V\stackrel{\mathrm{ev}}{⟶}1\stackrel{\mathrm{coev}}{⟶}V\otimes V^{*}$,

(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 $\psi \in \mathrm{End}(M\otimes V)$. Then, as operators on $M\otimes V\otimes V^{*}$,

$(1\otimes E_V)(\psi \otimes \mathrm{id})(1\otimes E_V)=(\mathrm{id}\otimes {\mathrm{qtr}}_V)\left(\psi \right)\otimes E_V$,

(qtr)

where the quantum trace $(\mathrm{id}\otimes {\mathrm{qtr}}_V)\left(\psi \right):M\to M$ is the composition

$M\otimes 1\stackrel{\mathrm{id}\otimes \mathrm{coev}}{⟶}M\otimes V\otimes V^{*}\stackrel{\psi \otimes \mathrm{id}}{⟶}M\otimes V\otimes V^{*}\stackrel{\mathrm{id}\otimes v^{-1}\otimes \mathrm{id}}{⟶}M\otimes V\otimes V^{*}\stackrel{\mathrm{id}\otimes {\check{R}}_{V{V}^{*}}}{⟶}M\otimes V^{*}\otimes V\stackrel{\mathrm{id}\otimes \mathrm{ev}}{⟶}M\otimes 1$.

The special case when $M=1$ and $\psi ={\mathrm{id}}_V$ is the quantum dimension of $V$,

${\dim }_q\left(V\right)={\mathrm{qtr}}_V\left({\mathrm{id}}_V\right)$.

(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 $\psi \in {\mathrm{End}}_U(M\otimes V)$. Then

$E_V(x\otimes \phi )=⟨\phi ,u{v}^{-1}x⟩(\sum _{i=1}^n{e}_i\otimes e^i),(\mathrm{id}\otimes {\mathrm{qtr}}_V)\left(\psi \right)=(\mathrm{id}\otimes {\mathrm{tr}}_V)\left(\right(1\otimes u{v}^{-1}\left)\psi \right)$,

(twr)

$E_V^2={\dim }_q\left(V\right)E_V,\text{and}(\mathrm{id}\otimes {\mathrm{qtr}}_V)\left({\check{R}}_{VV}\right)=C_V^{-1}$.

(fan)

Proof. Computing the action of $E_V$ on $x\otimes \phi$,

$\begin{array}{rcl}{E}_V(x\otimes \phi )& =& (\mathrm{coev}\circ \mathrm{ev}\circ {\check{R}}_{V{V}^{*}}(v^{-1}\otimes \mathrm{id}))(x\otimes \phi )=(\mathrm{coev}\circ \mathrm{ev})(\sum _{ℛ}{R}_2\phi \otimes R_1{v}^{-1}x)\\ & =& \sum _{ℛ}⟨R_2\phi ,R_1{v}^{-1}x⟩\sum _{i=1}^n{e}_i\otimes e^i=\sum _{ℛ}⟨\phi ,S\left(R_2\right)R_1{v}^{-1}x⟩\sum _{i=1}^n{e}_i\otimes e^i\\ & =& ⟨\phi ,u{v}^{-1}x⟩(\sum _{i=1}^n{e}_i\otimes e^i),\end{array}$

which establishes the first identity in (twr). If ${\psi }_{ji}\in \mathrm{End}\left(M\right)$ are such that

$\psi (m\otimes e_i)=\sum _{j=1}^n{\psi }_{ji}\left(m\right)\otimes e_j,\text{then}$

$\begin{array}{rcl}(\mathrm{id}\otimes {\mathrm{qtr}}_V)\left(\psi \right)\left(m\right)& =& (\mathrm{id}\otimes \mathrm{ev})(\mathrm{id}\otimes {\check{R}}_{V{V}^{*}})(1\otimes v^{-1}\otimes 1)(\psi \otimes \mathrm{id})(\mathrm{id}\otimes \mathrm{coev})(m\otimes 1)\\ & =& (\mathrm{id}\otimes \mathrm{ev})(\mathrm{id}\otimes {\check{R}}_{V{V}^{*}})(1\otimes v^{-1}\otimes 1)(\sum _{i,k}{\psi }_{ki}\left(m\right)\otimes e_k\otimes e^i)\\ & =& (\mathrm{id}\otimes \mathrm{ev})(\sum _{ℛ,i,k}{\psi }_{ki}\left(m\right)\otimes R_2{e}^i\otimes R_1{v}^{-1}{e}_k)=\sum _{ℛ,i,k}{\psi }_{ki}\left(m\right)⟨R_2{e}^i,R_1{v}^{-1}{e}_k⟩\\ & =& \sum _{ℛ,i,k}{\psi }_{ki}\left(m\right)⟨e^i,S\left(R_2\right)R_1{v}^{-1}{e}_k⟩=\sum _{ℛ,i,k}{\psi }_{ki}\left(m\right)⟨e^i,u{v}^{-1}{e}_k⟩\\ & =& \sum _i⟨\mathrm{id}\otimes e^i,(1\otimes u{v}^{-1})\psi (m\otimes e_i)⟩=(\mathrm{id}\otimes {\mathrm{tr}}_V)\left(\right(1\otimes u{v}^{-1}\left)\psi \right)\left(m\right)\end{array}$

is the second identity in (twr). The identity $E_V^2={\dim }_q\left(V\right)E_V$ is the special case of (qtr) when $M=1$ and $\phi ={\mathrm{id}}_V$. Finally, if $x\in V$ then

$\begin{array}{rcl}(\mathrm{id}\otimes {\mathrm{qtr}}_V)\left({\check{R}}_{VV}\right)\left(x\right)& =& \sum _i⟨{\mathrm{id}}_V\otimes e^i,(1\otimes v^{-1}u){\check{R}}_{VV}(x\otimes e_i)⟩=\sum _{ℛ,i}⟨{\mathrm{id}}_V\otimes e^i,R_2{e}_i\otimes v^{-1}u{R}_{1}x⟩\\ & =& \sum _{ℛ,i}{R}_2{e}_i⟨e^i,v^{-1}u{R}_{1}x⟩=\sum _{ℛ}{R}_2{v}^{-1}u{R}_{1}x=\sum _{ℛ}{R}_2{S}^2\left(R_1\right)v^{-1}ux\\ & =& u^{-1}{v}^{-1}ux=v^{-1}x=C_V^{-1}\left(x\right)\end{array}$

$\square$

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 $$ℛ=(\epsilon \otimes \mathrm{id}\otimes \mathrm{id})(\Delta \otimes \mathrm{id})\left(ℛ\right)=(\epsilon \otimes \mathrm{id}\otimes \mathrm{id}){ℛ}_{13}{ℛ}_{23}=(\epsilon \otimes \mathrm{id})\left(ℛ\right)\cdot ℛ,$$ and $$ℛ=(\mathrm{id}\otimes \mathrm{id}\otimes \epsilon )(\mathrm{id}\otimes \Delta )\left(ℛ\right)=(\mathrm{id}\otimes \mathrm{id}\otimes \epsilon ){ℛ}_{13}{ℛ}_{12}=(\mathrm{id}\otimes \epsilon )\left(ℛ\right)\cdot ℛ,$$ and so $$(\epsilon \otimes \mathrm{id})\left(ℛ\right)=1\text{and}(\mathrm{id}\otimes \epsilon )\left(ℛ\right)=1.$$ Then, since $$ℛ\cdot (S\otimes \mathrm{id})\left(ℛ\right)=(m\otimes \mathrm{id})(\mathrm{id}\otimes S\otimes \mathrm{id})\left({ℛ}_{13}{ℛ}_{23}\right)=(m\otimes \mathrm{id})(\mathrm{id}\otimes S\otimes \mathrm{id})(\Delta \otimes \mathrm{id})\left(ℛ\right)=(\epsilon \otimes \mathrm{id})\left(ℛ\right)=1,$$ it follows that $$(S\otimes \mathrm{id})\left(ℛ\right)={ℛ}^{-1}.$$ Since $(A^{\mathrm{op}},{ℛ}_{21})$ is a quasitriangular Hopf algebra with antipode $S^{-1}$ it follows that $(S^{-1}\otimes \mathrm{id})\left({ℛ}_{21}\right)={\left({ℛ}_{21}\right)}^{-1}.$ So $$(\mathrm{id}\otimes S^{-1})\left(ℛ\right)={ℛ}^{-1}.$$ Finally, $$(S\otimes S)\left(ℛ\right)=(\mathrm{id}\otimes S)(S\otimes \mathrm{id})\left(ℛ\right)=(\mathrm{id}\otimes S)\left({ℛ}^{-1}\right)=(\mathrm{id}\otimes S)(\mathrm{id}\otimes S^{-1})\left(ℛ\right)=ℛ,$$ 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).

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