A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras

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

Last update: 13 March 2014

Quasitriangular Hopf Algebras, Ribbon Hopf Algebras and Quantum Groups

If $𝔘$ is a Hopf algebra, we shall denote the coproduct by $\Delta ,$ the counit by $\epsilon$ and the antipode by $S\text{.}$ We shall always assume that both the antipode $S$ and the skew antipode $S^{-1}$ exist. If $a\in 𝔘$ and $\Delta \left(a\right)=\sum _a{a}_{\left(1\right)}\otimes a_{\left(2\right)},$ then the opposite coproduct is defined by ${\Delta }^{\text{op}}\left(a\right)=\sum _a{a}_{\left(2\right)}\otimes a_{\left(1\right)}\text{.}$ Recall that if $V$ and $W$ are $𝔘$ modules, then $𝔘$ acts on the tensor product $V\otimes W$ by $$a(v\otimes w)=\Delta \left(a\right)(v\otimes w)=\sum _a{a}_{\left(1\right)}v\otimes a_{\left(2\right)}w,$$ for all $a\in 𝔘,$ $v\in V,$ and $w\in W\text{.}$

A quasitriangular Hopf algebra is a pair $(𝔘,ℛ)$ consisting of a Hopf algebra $𝔘,$ and an invertible element $ℛ\in 𝔘\otimes 𝔘$ such that $$\begin{array}{cccc}ℛ\Delta \left(a\right){ℛ}^{-1}& =& {\Delta }^{\text{op}}\left(a\right),\text{for all}a\in U,& \text{(2.1)}\\ (\Delta \otimes \text{id})\left(ℛ\right)& =& {ℛ}_{13}{ℛ}_{23},& \text{(2.2)}\\ (\text{id}\otimes \Delta )\left(ℛ\right)& =& {ℛ}_{13}{ℛ}_{12},& \text{(2.3)}\end{array}$$ where, if $ℛ=\sum a_i\otimes b_i,$ then $${ℛ}_{12}=\sum a_i\otimes b_i\otimes 1,{ℛ}_{13}=\sum a_i\otimes 1\otimes b_i,{ℛ}_{23}=\sum 1\otimes a_i\otimes b_i\text{.}$$

Let $(𝔘,ℛ)$ be a quasitriangular Hopf algebra, let $ℛ=\sum a_i\otimes b_i\in 𝔘\otimes 𝔘,$ ${ℛ}_{21}=\sum b_i\otimes a_i,$ and define $$\begin{array}{cc}u=\sum S\left(b_i\right)a_i\in 𝔘\text{and}z=uS\left(u\right)\text{.}& \text{(2.4)}\end{array}$$ Then, we have the following facts: $$\begin{array}{cc}(S\otimes \text{id})\left(ℛ\right)={ℛ}^{-1},& \text{(2.5)}\\ (S\otimes S)\left(ℛ\right)=ℛ,& \text{(2.6)}\\ u^{-1}=\sum _j{S}^{-1}\left(d_j\right)c_j,\hspace{0.17em}\text{where}\hspace{0.17em}{ℛ}^{-1}=\sum _j{c}_j\otimes d_j,& \text{(2.7)}\\ ua{u}^{-1}=S^2\left(a\right),\text{for all}a\in 𝔘,& \text{(2.8)}\\ \Delta \left(u\right)={\left({ℛ}_{21}ℛ\right)}^{-1}(u\otimes u),& \text{(2.9)}\\ z\hspace{0.17em}\text{is an invertible central element of}\hspace{0.17em}𝔘,& \text{(2.10)}\\ \Delta \left(z\right)={\left({ℛ}_{21}ℛ\right)}^{-2}(z\otimes z),& \text{(2.11)}\end{array}$$ These facts are proved in [Dri1990, Propositions 2.1, 3.1, 3.2, and the remarks immediately preceding Proposition 3.2]. The proofs are calculations involving only the definition of a quasitriangular Hopf algebra.

A ribbon Hopf algebra is a triple $(𝔘,ℛ,v)$ consisting of a quasitriangular Hopf algebra $(𝔘,ℛ),$ and an invertible element $v$ in the center of $𝔘,$ such that $$\begin{array}{cc}\begin{array}{c}{v}^2=uS\left(u\right),S\left(v\right)=v,\epsilon \left(v\right)=1,\\ \Delta \left(v\right)={\left({ℛ}_{21}{ℛ}_{12}\right)}^{-1}(v\otimes v)\text{.}\end{array}& \text{(2.12)}\end{array}$$ It is important to note that the element $v^{-1}u\in 𝔘$ is grouplike, i.e., $\Delta \left(v^{-1}u\right)=v^{-1}u\otimes v^{-1}u\text{.}$

Quantum Groups

Let $ℂ\left[\left[h\right]\right]$ be the ring of formal power series in an indeterminate $h\text{.}$ The notation $e^x$ shall always denote the formal exponential $$e^x=\sum _{k\ge 0}\frac{x^k}{k!},$$ and define $q=e^{h/2}\text{.}$ For each positive integer $n$ define $$\begin{array}{ccc}\left[n\right]& =& \frac{q^n-q^{-n}}{q-q^{-1}},\left[n\right]!=\left[n\right][n-1]\cdots \left[2\right]\left[1\right],\left[0\right]!=1,\\ \left[\genfrac{}{}{0ex}{}{n}{k}\right]& =& \frac{\left[n\right]!}{\left[k\right]![n-k]!},\text{for}0\le k\le n\text{.}\end{array}$$

Let $𝔤$ be a finite dimensional complex semisimple Lie algebra. Let $𝔥$ be the Cartan subalgebra of $𝔤\text{.}$ Let ${\alpha }_i\in {𝔥}^{*}$ be the simple roots and let $H_i={\alpha }_i^{\vee }\in 𝔥$ be the simple coroots so that the Cartan matrix is given by $$\left(⟨{\alpha }_i,{\alpha }_j^{\vee }⟩\right)=\left(a_{ij}\right)=A\text{.}$$

Let ${𝔘}_h\left(g\right)$ be the associative algebra with $1$ over $ℂ\left[\left[h\right]\right]$ generated (as an algebra complete in the $h\text{-adic}$ topology) by the space $𝔥$ and the elements $X_1,\dots ,X_r,$ $Y_1,\dots ,Y_r$ with relations $$\begin{array}{c}[a_1,a_2]=0,\text{for all}{a}_1,a_2\in 𝔥,\\ [a,X_j]=⟨{\alpha }_j,a⟩X_j,[a,Y_j]=⟨-{\alpha }_j,a⟩Y_j,\text{for all}a\in 𝔥,\\ X_i{Y}_j-Y_j{X}_i={\delta }_{ij}\frac{e^{(h/2)H_i}-e^{-(h/2)H_i}}{h},\\ \sum _{s+t=1-a_{ji}}{(-1)}^t\left[\genfrac{}{}{0ex}{}{1-a_{ji}}{s}\right]X_i^s{X}_j{X}_i^t=0,i\ne j,\\ \sum _{s+t=1-a_{ji}}{(-1)}^t\left[\genfrac{}{}{0ex}{}{1-a_{ji}}{s}\right]Y_i^s{Y}_j{Y}_i^t=0,i\ne j\text{.}\end{array}$$ There is a Hopf algebra structure on ${𝔘}_h\left(g\right)$ given by $$\begin{array}{ccc}\Delta \left(X_i\right)& =& X_i\otimes e^{(h/4)H_i}+e^{-(h/4)H_i}\otimes X_i,\\ \Delta \left(Y_i\right)& =& Y_i\otimes e^{(h/4)H_i}+e^{-(h/4)H_i}\otimes Y_i,\\ \epsilon \left(X_i\right)& =& \epsilon \left(Y_i\right)=\epsilon \left(a\right)=0,& \text{for all}a\in 𝔥,\\ S\left(X_i\right)& =& -e^{h/2}{X}_i,S\left(Y_i\right)=-e^{-h/2}{Y}_i,S\left(a\right)=-a,& \text{for all}a\in 𝔥\text{.}\end{array}$$ Given the definition of the coproduct $\Delta$ one can easily show that the formulas for the counit $\epsilon$ and the antipode $S$ are forced by the definitions of a Hopf algebra.

There is a $\mathbb{Z}$ grading on the algebra ${𝔘}_h\left(g\right)$ determined by defining $$\begin{array}{cccc}\text{deg}\left(h\right)& =& 0,& \text{for all}h\in 𝔥,\\ \text{deg}\left(E_i\right)& =& 1,\text{deg}\left(F_i\right)=-1,& \text{for all}1\le i\le r\text{.}\end{array}$$ Let ${𝔘}_h{\left(𝔤\right)}^{\ge 0}$ be the subalgebra of ${𝔘}_h\left(𝔤\right)$ generated by $𝔥$ and the elements $X_i,$ $1\le i\le r\text{.}$ Similarly let ${𝔘}_h{\left(𝔤\right)}^{\le 0}$ be the subalgebra generated by $𝔥$ and the elements $Y_i,$ $1\le i\le r\text{.}$ Let ${\stackrel{\sim }{H}}_1,\dots ,{\stackrel{\sim }{H}}_r$ be an orthonormal basis of $𝔥$ and let $t_0={\sum }_{i=1}^r{\stackrel{\sim }{H}}_i\otimes {\stackrel{\sim }{H}}_i\text{.}$ The algebra ${𝔘}_h\left(𝔤\right)$ is a quasitriangular Hopf algebra and the element $ℛ$ can be written in the form, See [Dri1990, Sect. 4], $$\begin{array}{cc}ℛ=\text{exp}\left(\frac{h}{2}{t}_0\right)+\sum a_i^{+}\otimes b_i^{-},& \text{(2.13)}\end{array}$$ where the elements $a_i^{+}\in {𝔘}_h{\left(𝔤\right)}^{\ge 0},$ $b_i^{-}\in {𝔘}_h{\left(𝔤\right)}^{\le 0}$ are homogeneous elements of degrees $\ge 1$ and $\le -1$ respectively.

As in the classical case, each finite dimensional ${𝔘}_h\left(𝔤\right)\text{-module,}$ $M,$ is a direct sum of its weight spaces, i.e., $$M=\underset{\lambda \in {𝔥}^{*}}{⨁}{M}^{\lambda },\text{where}{M}^{\lambda }=\{m\in M\hspace{0.17em}|\hspace{0.17em}am=⟨\lambda ,a⟩m,\hspace{0.17em}\text{for all}\hspace{0.17em}a\in 𝔥\}\text{.}$$ Every finite dimensional module is completely reducible and the finite dimensional irreducible modules ${\Lambda }_{\lambda },$ of ${𝔘}_h\left(𝔤\right)$ are labeled by the dominant integral weights $\lambda \text{.}$ Each of these modules is a highest weight module of highest weight $\lambda ,$ i.e., there is a unique vector $m\in {\Lambda }_{\lambda }$ (up to constant multiples) such that $$\begin{array}{cccc}am& =& ⟨\lambda ,a⟩m,& \text{for all}a\in 𝔥,\text{and}\\ X_{i}m& =& 0,& \text{for all}\hspace{0.17em}i\text{.}\end{array}$$ All of the facts in this paragraph can be proved, see [Dri1990, remarks after Proposition 4.2], by showing that since $H^2(𝔤,𝔘𝔤)=0,$ the enveloping algebra $𝔘𝔤$ of a finite dimensional complex simple Lie algebra $𝔤$ has no nontrivial deformations as an algebra and thus there must be an algebra isomorphism ${𝔘}_h\left(𝔤\right)\simeq 𝔘𝔤\text{.}$ Note that this is only on the level of algebras, ${𝔘}_h\left(𝔤\right)$ and $𝔘𝔤$ are not isomorphic as Hopf algebras. Thus, the representation theory of ${𝔘}_h\left(𝔤\right),$ provided we are not considering questions of tensor products of representations, depends only on its structure as an algebra and is the same as the representation theory of $𝔘𝔤\text{.}$

Quantum Groups are Ribbon Hopf Algebras

[Dri1990]. Let ${𝔘}_h\left(𝔤\right)$ be a Drinfel'd-Jimbo quantum group and let $\rho$ be the element of $𝔥$ such that $⟨{\alpha }_i,\rho ⟩=1$ for all simple roots ${\alpha }_i\text{.}$ Let $u$ be as given in (2.4). Then

(1)	$e^{h\rho }a{e}^{-h\rho }=S^2\left(a\right)$ for all $a\in {𝔘}_h\left(𝔤\right)\text{.}$
(2)	$e^{-h\rho }u=u{e}^{-h\rho }$ is a central element in ${𝔘}_h\left(g\right)\text{.}$
(3)	${\left(e^{-h\rho }\right)}^2=uS\left(u\right)=S\left(u\right)u\text{.}$
(4)	$e^{-h\rho }u$ acts in an irreducible representation ${\Lambda }_{\lambda }$ of ${𝔘}_h\left(𝔤\right)$ of highest weight $\lambda$ by the constant $\text{exp}(-(h/2)⟨\lambda ,\lambda +2\rho ⟩)=q^{-⟨\lambda ,\lambda +2\rho ⟩}\text{.}$
(5)	$\Delta \left(e^{-h\rho }u\right)={\left({ℛ}_{21}ℛ\right)}^{-1}(e^{-h\rho }u\otimes e^{-h\rho }u)\text{.}$
(6)	$S\left(e^{-h\rho }u\right)=e^{-h\rho }u\text{.}$
(7)	$\epsilon \left(e^{-h\rho }u\right)=1\text{.}$

		Proof.
	(1) Since both $S^2$ and conjugation by $e^{h\rho }$ are algebra homomorphisms it is sufficient to check this on generators. We shall show how this is done for the generator $X_j\text{.}$ It follows from the fact $[\rho ,X_j]=\rho X_j-X_j\rho =⟨{\alpha }_j,\rho ⟩X_j,$ that $$e^{h\rho }{X}_j{e}^{-h\rho }=e^{h\rho }{e}^{-h(\rho -⟨{\alpha }_j,\rho ⟩)}{X}_j=e^{h⟨{\alpha }_j,\rho ⟩}{X}_j=e^h{X}_j=q^2{X}_j=S^2\left(X_j\right)\text{.}$$ (2) This follows from (1) and (2.8), since $e^{-h\rho }ua{u}^{-1}{e}^{h\rho }=S^{-2}\left(S^2\left(a\right)\right)=a\text{.}$ (4) Let ${\stackrel{\sim }{H}}_1,\dots ,{\stackrel{\sim }{H}}_r$ be an orthonormal basis of $𝔥\text{.}$ For each element $\lambda \in {𝔥}^{*}$ let ${\lambda }_i=⟨\lambda ,{\stackrel{\sim }{H}}_i⟩\text{.}$ Note that if $m$ is a weight vector of weight $\lambda$ in a ${𝔘}_h\left(𝔤\right)\text{-module}$ then ${\stackrel{\sim }{H}}_{i}m={\lambda }_{i}m\text{.}$ Let ${\Lambda }_{\lambda }$ be an irreducible ${𝔘}_h\left(𝔤\right)\text{-module}$ of highest weight $\lambda$ and let $v_{\lambda }$ be a highest weight vector in ${\Lambda }_{\lambda }\text{.}$ Since elements of ${𝔘}_h{\left(𝔤\right)}^{\ge 0}$ which are of degree $\ge 1$ annihilate $v_{\lambda }$ it follows that $$\begin{array}{ccc}u{v}_{\lambda }& =& \text{exp}\left(\frac{h}{2}\sum _{i=1}^{r}S\left({\stackrel{\sim }{H}}_i\right){\stackrel{\sim }{H}}_i\right)v_{\lambda }\\ & =& \prod _{i=1}^r\left(\sum _{k\ge 0}{\left(\frac{h}{2}\right)}^k\frac{S{\left({\stackrel{\sim }{H}}_i\right)}^k{\stackrel{\sim }{H}}_i^k}{k!}\right)v_{\lambda }\\ & =& \prod _{i=1}^r\left(\sum _{k\ge 0}{(-\frac{h}{2})}^k\frac{{\stackrel{\sim }{H}}_i^k{\stackrel{\sim }{H}}_i^k}{k!}\right)v_{\lambda }\\ & =& \prod _{i=1}^r\left(\sum _{k\ge 0}{(-\frac{h}{2})}^k\frac{{\lambda }_i^{2k}}{k!}\right)v_{\lambda }\\ & =& \text{exp}(-\frac{h}{2}\sum _{i=1}^r{\lambda }_i^2)v_{\lambda }\\ & =& \text{exp}(-\frac{h}{2}⟨\lambda ,\lambda ⟩)v_{\lambda }\end{array}$$ The result follows since $e^{-h\rho }{v}_{\lambda }=e^{-h⟨\lambda ,\rho ⟩}{v}_{\lambda }\text{.}$ (5) This follows from (2.9), since $$\begin{array}{ccc}\Delta \left(e^{-h\rho }u\right)& =& \Delta \left(u{e}^{-h\rho }\right)=\Delta \left(u\right)\Delta \left(e^{-h\rho }\right)={\left({ℛ}_{21}ℛ\right)}^{-1}(u\otimes u)(e^{-h\rho }\otimes e^{-h\rho })\\ & =& {\left({ℛ}_{21}ℛ\right)}^{-1}(e^{-h\rho }u\otimes e^{-h\rho }u)\text{.}\end{array}$$ (3) and (6) and (7) follow from equality $e^{h\rho }S\left(u\right)=e^{-h\rho }u$ which is proved as follows. Clearly, $e^{h\rho }S\left(u\right)=S\left(u{e}^{-h\rho }\right)$ is a central element of ${𝔘}_h\left(𝔤\right),$ so it is sufficient to check that both $e^{h\rho }S\left(u\right)$ and $u{e}^{-h\rho }$ act by the same constant on an irreducible representation ${\Lambda }_{\lambda }$ of ${𝔘}_h\left(𝔤\right)\text{.}$ But $e^{h\rho }S\left(u\right)=S\left(u{e}^{-h\rho }\right)$ acts on the representation ${\Lambda }_{\lambda }$ in the same way that $u{e}^{-h\rho }$ acts on the irreducible module ${\Lambda }_{\lambda }^{*}$ which has highest weight $-w_0\lambda$ where $w_0$ is the longest element of the Weyl group. Thus, $u{e}^{-h\rho }$ acts on the irreducible module ${\Lambda }_{\lambda }^{*}$ by the constant $$e^{-(h/2)⟨-w_0\lambda ,-w_0\lambda ⟩}{e}^{-h⟨-w_0\lambda ,\rho ⟩}=e^{-(h/2)⟨\lambda ,\lambda +2\rho ⟩}=q^{-⟨\lambda ,\lambda +\rho ⟩}$$ since $w_0\rho =-\rho$ and the inner product is invariant under the action of $w_0\text{.}$ $\square$

The Drinfel'd-Jimbo quantum group $({𝔘}_h\left(𝔤\right),ℛ,e^{-h\rho }u)$ is a ribbon Hopf algebra.

Centralizer Algebras of Tensor Power Representations and the ${ℛ}_{21}{ℛ}_{12}$ Matrix

Let $(𝔘,ℛ)$ be a quasitriangular Hopf algebra. Let $V$ be a $𝔘\text{-module}$ and let $R\in \text{End}(V\otimes V)$ be the linear transformation induced by the action of $ℛ$ on $V\otimes V\text{.}$ Let $$\begin{array}{cc}Ř=\sigma R,& \text{(2.16)}\end{array}$$ where $\sigma :V\otimes V\to V\otimes V$ is the linear transformation given by $\sigma (v\otimes w)=w\otimes v\text{.}$ For each $1\le i\le m-1$ define $$\begin{array}{cc}{Ř}_i=1\otimes \cdots \otimes 1\otimes Ř\otimes 1\otimes \cdots \otimes 1\in \text{End}\left(V^{\otimes m}\right)& \text{(2.17)}\end{array}$$ where the $Ř$ appears as a transformation on the $i\text{th}$ and $(i+1)\text{st}$ tensor factors.

The transformations ${Ř}_i$ are elements of the centralizer ${𝒵}_m={\text{End}}_{𝔘}\left(V^{\otimes m}\right)$ and satisfy the following relations $$\begin{array}{cccc}{Ř}_i{Ř}_j& =& {Ř}_j{Ř}_i,& |i-j|>1,\\ {Ř}_i{Ř}_{i+1}{Ř}_i& =& {Ř}_{i+1}{Ř}_i{Ř}_{i+1},& 1\le i\le m-2\end{array}$$

		Proof.
	Let $({\pi }^{\otimes 2},V\otimes V)$ be the representation of $𝔘$ on $V\otimes V\text{.}$ Let us abuse notation and denote the transformation on $V\otimes V$ induced by the action of $\Delta \left(a\right),$ $a\in 𝔘,$ also by $\Delta \left(a\right)\text{.}$ It follows from the equation $$Ř{\pi }^{\otimes 2}\left(a\right)=\sigma R\Delta \left(a\right)=\sigma {\Delta }^{\text{op}}\left(a\right)R=\sigma {\Delta }^{\text{op}}\left(a\right){\sigma }^{-1}\sigma R=\Delta \left(a\right)Ř={\pi }^{\otimes 2}\left(a\right)Ř,$$ that $Ř\in {\text{End}}_{𝔘}(V\otimes V)\text{.}$ It follows that each ${Ř}_i\in {𝒵}_m$ and that the algebra of transformations generated by the ${Ř}_i$ is contained in the centralizer ${𝒵}_m\text{.}$ The fact that the ${Ř}_i$ satisfy the first relation follows immediately from the definition of the ${Ř}_i\text{.}$ The second relation is derived from the relations (2.1) and (2.2) as follows. In the following calculations we abuse notations so that all factors in the computation are viewed as elements of $\text{End}\left(V^{\otimes 3}\right)\text{.}$ We shall let $R_{ij}$ denote the transformation of $V^{\otimes 3}$ induced by the action of ${ℛ}_{ij}\text{.}$ We shall let ${\sigma }_{ij}$ denote the transformation of $V^{\otimes 3}$ which transposes the $i\text{th}$ and the $j\text{th}$ tensor factors of $V^{\otimes 3}\text{.}$ Then, using the equation $$\begin{array}{cccc}{R}_{12}{R}_{13}{R}_{23}& =& R_{12}(\Delta \otimes \text{id})\left({ℛ}_{12}\right)& \text{by (2.2)}\\ & =& ({\Delta }^{\text{op}}\otimes \text{id})\left({ℛ}_{12}\right)R_{12}& \text{by (2.1)}\\ & =& R_{23}{R}_{12}{R}_{12},\end{array}$$ we have $$\begin{array}{ccc}{Ř}_1{Ř}_2{Ř}_1& =& {\sigma }_{12}{R}_{12}{\sigma }_{23}{R}_{23}{\sigma }_{12}{R}_{12}\\ & =& \underbrace{{\sigma }_{12}{\sigma }_{23}{\sigma }_{12}}\underbrace{{\sigma }_{12}{\sigma }_{23}{R}_{12}{\sigma }_{23}{\sigma }_{12}}\underbrace{{\sigma }_{12}{R}_{23}{\sigma }_{12}}{R}_{12}\\ & =& {\sigma }_{13}{R}_{23}{R}_{13}{R}_{12},\end{array}$$ and $$\begin{array}{ccc}{Ř}_2{Ř}_1{Ř}_2& =& {\sigma }_{23}{R}_{23}{\sigma }_{12}{R}_{12}{\sigma }_{23}{R}_{23}\\ & =& \underbrace{{\sigma }_{23}{\sigma }_{12}{\sigma }_{23}}\underbrace{{\sigma }_{23}{\sigma }_{12}{R}_{23}{\sigma }_{12}{\sigma }_{23}}\underbrace{{\sigma }_{23}{R}_{12}{\sigma }_{23}}{R}_{23}\\ & =& {\sigma }_{13}{R}_{12}{R}_{13}{R}_{23}\text{.}\end{array}$$ It follows that ${Ř}_1{Ř}_2{Ř}_1={Ř}_2{Ř}_1{Ř}_2\text{.}$ $\square$

The proof of the following proposition is similar to the proof of Lemma 3.3.1 in [Wen1993].

(1)	If $({\pi }_W,W)$ and $({\pi }_V,V)$ are two representations of $𝔘,$ then $({\pi }_W\otimes {\pi }_V)\left({ℛ}_{21}{ℛ}_{12}\right)\in {\text{End}}_{𝔘}(W\otimes V)\text{.}$
(2)	Let $(\pi ,V)$ be a representation of $𝔘\text{.}$ Then $$({\pi }^{\otimes (m-1)}\otimes \pi )\left({ℛ}_{21}{ℛ}_{12}\right)={Ř}_{m-1}{Ř}_{m-2}\cdots {Ř}_1{Ř}_1{Ř}_2\cdots {Ř}_{m-1}\in {\text{End}}_{𝔘}\left(V^{\otimes m}\right)\text{.}$$

		Proof.
	(1) The equality $ℛ\Delta \left(a\right){ℛ}^{-1}={\Delta }^{\text{op}}\left(a\right)$ is equivalent to ${ℛ}_{21}{\Delta }^{\text{op}}\left(a\right){ℛ}_{21}^{-1}=\Delta \left(a\right)$ which in turn implies ${\Delta }^{\text{op}}\left(a\right)={ℛ}_{21}^{-1}\Delta \left(a\right){ℛ}_{21}\text{.}$ Thus, we have ${ℛ}_{21}^{-1}\Delta \left(a\right){ℛ}_{21}=ℛ\Delta \left(a\right){ℛ}^{-1},$ which is the same as $${ℛ}_{21}ℛ\Delta \left(a\right)=\Delta \left(a\right){ℛ}_{21}ℛ\text{.}$$ (2) Using (2.2), we have, by induction, $$\begin{array}{ccc}({\Delta }^{(m-2)}\otimes \text{id})\left({ℛ}_{12}\right)& =& (\Delta \otimes {\text{id}}^{\otimes (m-2)})({\Delta }^{(m-3)}\otimes \text{id})\left(ℛ\right)\\ & =& (\Delta \otimes {\text{id}}^{\otimes (m-2)})\left({ℛ}_{1(m-1)}{ℛ}_{2(m-1)}\cdots {ℛ}_{(m-2)(m-1)}\right)\\ & =& {ℛ}_{1m}{ℛ}_{2m}\cdots {ℛ}_{(m-1)m}\text{.}& \text{(2.20)}\end{array}$$ Similarly, we get that $({\Delta }^{(m-2)}\otimes \text{id})\left({ℛ}_{21}\right)={ℛ}_{m(m-1)}{ℛ}_{m(m-2)}\cdots {ℛ}_{m2}{ℛ}_{m1}\text{.}$ Let $\sigma :V^{\otimes (m-1)}\otimes V\to V^{\otimes (m-1)}$ be the transformation which transposes the tensor factors $V^{\otimes (m-1)}$ and $V\text{.}$ As a transformation in the symmetric group $S_m$ acting on $V^{\otimes m}$ we have $\sigma ={\sigma }_{1\cdots m}={\sigma }_{12}{\sigma }_{23}\cdots {\sigma }_{(m-1)m}$ where ${\sigma }_{i(i+1)}$ is the permutation in $S_m$ that switches the $i\text{th}$ and the $(i+1)\text{st}$ tensor factors of $V^{\otimes m}\text{.}$ Let $R_{ij}$ denote the endomorphism of $V^{\otimes m}$ induced by multiplying by ${ℛ}_{ij}\in {𝔘}^{\otimes m}\text{.}$ Then, viewing $({\Delta }^{(m-2)}\otimes \text{id})\left(ℛ\right)$ as a transformation on $V^{\otimes m},$ we have $$\begin{array}{ccc}\sigma ({\Delta }^{(m-2)}\otimes \text{id})\left(ℛ\right)& =& {\sigma }_{1\cdots m}{R}_{1m}{R}_{2m}{R}_{3m}\cdots R_{(m-1)m}\\ & =& {\sigma }_{12}{\sigma }_{2\cdots m}{R}_{1m}{\sigma }_{2\cdots m}^{-1}{\sigma }_{2\cdots m}{R}_{2m}{\sigma }_{3\cdots m}^{-1}{\sigma }_{3\cdots m}\\ & & \times R_{3m}\cdots R_{(m-2)m}{\sigma }_{(m-1)m}^{-1}{\sigma }_{(m-1)m}{R}_{(m-1)m}\\ & =& {\sigma }_{12}{\sigma }_{2\cdots m}{R}_{1m}{\sigma }_{2\cdots m}^{-1}{\sigma }_{23}{\sigma }_{3\cdots m}{R}_{2m}{\sigma }_{3\cdots m}^{-1}\\ & & \times {\sigma }_{34}{\sigma }_{4\cdots m}{R}_{3m}\cdots {\sigma }_{(m-1)m}{R}_{(m-1)m}\\ & =& {\sigma }_{12}{R}_{12}{\sigma }_{23}{R}_{23}{\sigma }_{34}{R}_{34}\cdots {\sigma }_{(m-1)m}{R}_{(m-1)m}\\ & =& {Ř}_1{Ř}_2\cdots {Ř}_{m-1}\text{.}\end{array}$$ In a similar fashion one shows that $$\begin{array}{ccc}({\Delta }^{(m-2)}\otimes \text{id})\left({ℛ}_{21}\right){\sigma }^{-1}& =& R_{m1}{R}_{m2}{R}_{m3}\cdots R_{m(m-1)}{\sigma }_{m\cdots 1}\\ & =& {Ř}_{m-1}{Ř}_{m-2}\cdots {Ř}_1,\end{array}$$ where ${\sigma }^{-1}={\sigma }_{m\cdots 1}={\sigma }_{(m-1)m}\cdots {\sigma }_{23}{\sigma }_{12}\text{.}$ Thus, it follows that $$({\Delta }^{(m-2)}\otimes \text{id})\left({ℛ}_{21}ℛ\right)={Ř}_{m-1}{Ř}_{m-2}\cdots {Ř}_1{Ř}_1{Ř}_2\cdots {Ř}_{m-1}\text{.}$$ $\square$

(1)	Let $(𝔘,ℛ)$ be a quasitriangular Hopf algebra and let $z=uS\left(u\right)$ be as given in (2.4). The element $z$ acts on each irreducible representation ${\Lambda }_{\lambda }$ of $𝔘$ by a scalar. Denote this scalar by $z\left(\lambda \right)\text{.}$ Then the element ${\left({ℛ}_{21}{ℛ}_{12}\right)}^2$ acts on the irreducible component of ${\Lambda }_{\nu }$ of ${\Lambda }_{\lambda }\otimes {\Lambda }_{\mu }$ by the scalar $$\frac{z\left(\lambda \right)z\left(\mu \right)}{z\left(\nu \right)}\text{.}$$
(2)	Let $(𝔘,ℛ,v)$ be a ribbon Hopf algebra. The element $v$ acts on each irreducible representation ${\Lambda }_{\lambda }$ of $𝔘$ by a scalar. Denote this scalar by $v\left(\lambda \right)\text{.}$ Then the element ${ℛ}_{21}{ℛ}_{12}$ acts on the irreducible component ${\Lambda }_{\nu }$ of ${\Lambda }_{\lambda }\otimes {\Lambda }_{\mu }$ by the scalar $$\frac{v\left(\lambda \right)v\left(\mu \right)}{v\left(\nu \right)}\text{.}$$
(3)	Let ${𝔘}_h\left(𝔤\right)$ be a Drinfel'd-Jimbo quantum group. The element ${ℛ}_{21}{ℛ}_{12}$ acts on the irreducible component ${\Lambda }_{\nu }$ of ${\Lambda }_{\lambda }\otimes {\Lambda }_{\mu }$ by the scalar $$q^{⟨\nu ,\nu +2\rho ⟩-⟨\lambda ,\lambda +2\rho ⟩-⟨\mu ,\mu +2\rho ⟩}\text{.}$$

		Proof.
	(1) Since $z$ is in the center of $𝔘,$ the element $z$ acts on each irreducible representation ${\Lambda }_{\lambda }$ of $𝔘$ by a scalar. The element $(z\otimes z)$ acts on ${\Lambda }_{\lambda }\otimes {\Lambda }_{\mu }$ by the constant $z\left(\lambda \right)z\left(\mu \right)\text{.}$ Similarly, $\Delta \left(z\right)$ acts on the irreducible component ${\Lambda }_{\nu }$ of ${\Lambda }_{\lambda }\otimes {\Lambda }_{\mu }$ by the scalar $z\left(\nu \right)\text{.}$ The result now follows from the identity $\Delta \left(z\right)={\left({ℛ}_{21}ℛ\right)}^{-2}(z\otimes z)\text{.}$ The proof of (2) is entirely similar to the proof of (1). Now, (3) follows from (2) by noting that the quantum group is a ribbon Hopf algebra with $v=e^{-h\rho }u$ and that the element $e^{-h\rho }u$ acts on each irreducible representation ${\Lambda }_{\lambda }$ of ${𝔘}_h\left(𝔤\right)$ by the scalar $q^{-⟨\lambda ,\lambda +2\rho ⟩}\text{.}$ $\square$

(1)	Let $(𝔘,ℛ)$ be a quasitriangular Hopf algebra and denote the constant given by the action of $z=uS\left(u\right)$ on an irreducible representation ${\Lambda }_{\nu }$ by $z\left(\nu \right)\text{.}$ Suppose that $V={\Lambda }_{\omega }$ is an irreducible representation of $𝔘\text{.}$ Let ${\widehat{𝒵}}_2$ be an index set for the irreducible $𝔘\text{-modules}$ appearing the decomposition of $V^{\otimes 2}\text{.}$ Then ${Ř}_i$ satisfies the equation $$\prod _{\nu \in {\widehat{𝒵}}_2}({Ř}_i^4-\frac{{z\left(\omega \right)}^2}{z\left(\nu \right)})=0\text{.}$$
(2)	Let $(𝔘,ℛ,v)$ be a ribbon Hopf algebra and denote the constant given by the action of $v$ on an irreducible representation ${\Lambda }_{\nu }$ by $v\left(\nu \right)\text{.}$ Suppose that $V={\Lambda }_{\omega }$ is an irreducible representation of $𝔘\text{.}$ Then ${Ř}_i$ satisfies the equation $$\prod _{\nu \in {\widehat{𝒵}}_2}({Ř}_i^2-\frac{{v\left(\omega \right)}^2}{v\left(\nu \right)})=0\text{.}$$
(3)	([Res1987], formula (1.38)) Suppose that $V={\Lambda }_{\omega }$ is an irreducible representation of a Drinfel'd-Jimbo quantum group ${𝔘}_h\left(𝔤\right)$ and that the Bratteli diagram for tensoring by $V$ is multiplicity free. Then ${Ř}_i$ satisfies the equation $$\prod _{\nu \in {\widehat{𝒵}}_2}({Ř}_i\pm q^{(1/2)⟨\nu ,\nu +2\rho ⟩-⟨\omega ,\omega +2\rho ⟩})=0,$$ where the sign in the factor $({Ř}_i\pm q^{(1/2)⟨\nu ,\nu +2\rho ⟩-⟨\omega ,\omega +2\rho ⟩})$ is negative if ${\Lambda }_{\nu }$ is an irreducible component of the symmetric part of $V^{\otimes 2}$ and positive if ${\Lambda }_{\nu }$ is an irreducible component of the antisymmetric part ${\bigwedge }^2\left(V\right)$ of $V^{\otimes 2}\text{.}$

		Proof.
	(1) By Proposition (2.19) part (2), ${Ř}_1^2={\pi }^{\otimes 2}\left({ℛ}_{21}ℛ\right)\text{.}$ Suppose that $V\otimes V=\underset{T\in {𝒯}^2}{⨁}{V}_T,$ is a decomposition of $V^{\otimes 2}$ into irreducibles. Then, by Proposition (2.21), ${Ř}_1^4$ acts on the irreducible $V_T$ by the constant $z{\left(\omega \right)}^2/z\left(\nu \right)$ if $V_T\cong {\Lambda }_{\nu }\text{.}$ It follows that ${Ř}_1^4$ is a central element of ${𝒵}_2$ and that the minimal polynomial of ${Ř}_1^4$ is $$\prod _{\nu \in {\stackrel{\sim }{𝒵}}_2}(t-\frac{z\left(\nu \right)}{{z\left(\omega \right)}^2})\text{.}$$ The proof of (2) is similar to the proof of (1). Let us complete the proof of (3). It follows from (2) that ${Ř}_1$ satisfies the polynomial $\prod _{\nu \in {\stackrel{\sim }{𝒵}}_2}(R_1^2-q^{⟨\nu ,\nu +2\rho ⟩-2⟨\omega ,\omega +2\rho ⟩})=0\text{.}$ Given that ${Ř}_1$ is a central element of ${\text{End}}_{{𝔘}_h\left(𝔤\right)}\left(V^{\otimes 2}\right)$ since the Bratteli diagram is multiplicity free, it follows that the eigenvalues of ${Ř}_1$ are $\pm q^{(1/2)⟨\nu ,\nu +2\rho ⟩-⟨\omega ,\omega +2\rho ⟩}\text{.}$ Since, ${Ř}_1$ is a deformation of the transposition which switches the two factors of $V^{\otimes 2}$ we know that if we specialize $q=1$ the eigenvalues of ${Ř}_1$ are $+1$ if ${\Lambda }_{\nu }$ is an irreducible component of the symmetric part of $V^{\otimes 2}$ and $-1$ if ${\Lambda }_{\nu }$ is an irreducible component of the antisymmetric part ${\bigwedge }^2\left(V\right)$ of $V^{\otimes 2}\text{.}$ This observation determines the signs of the eigenvalues of ${Ř}_1\text{.}$ $\square$

Let $V={\Lambda }_{\omega }$ be an irreducible representation of $𝔘$ and let ${𝒵}_m={\text{End}}_{𝔘}\left(V^{\otimes m}\right)\text{.}$ Recall, from Section 1, that there is a natural way of identifying the path algebras corresponding to the Bratteli diagram for tensor powers of $V$ with the centralizer algebras ${𝒵}_m\text{.}$ As stated in Section 1 we shall always assume that the Bratteli diagram for tensor powers of $V$ is multiplicity free. This is probably not necessary for part (1) of the following corollary but it is certainly necessary for part (2).

Let $(𝔘,ℛ)$ be a quasitriangular Hopf algebra and let $V={\Lambda }_{\omega }$ be an irreducible representation of $𝔘\text{.}$ Identify the path algebras corresponding to the Bratteli diagram for tensor powers of $V$ with the centralizer algebras ${𝒵}_m={\text{End}}_{𝔘}\left(V^{\otimes m}\right)$ as in Section 1.

(1)

Let $D_m={Ř}_{m-1}{Ř}_{m-2}\cdots {Ř}_1{Ř}_1{Ř}_2\cdots {Ř}_{m-1}\in {𝒵}_m$ be the element given in Proposition (2.19). Then $$D_m^2=\sum _{T\in {𝒯}^m}{\left(D_m^2\right)}_{TT}{E}_{TT},\text{where}{\left(D_m^2\right)}_{TT}=\frac{z\left({\tau }^{(m-1)}\right)z\left(\omega \right)}{z\left({\tau }^{\left(m\right)}\right)},$$ for each $T=({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)},{\tau }^{\left(m\right)})\in {𝒯}^m\text{.}$

(2)

Fix $T=({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)},{\tau }^{\left(m\right)})\in {𝒯}^m$ and let $T\prime =({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)})\in {𝒯}^{m-1}\text{.}$ Let ${\left(T\prime \right)}^{+}$ be the set of tableaux that are extensions of $T\prime ,$ i.e. the set of $S=({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)},{\sigma }^{\left(m\right)})\in {𝒯}^m\text{.}$ If the values ${\left(D_m^2\right)}_{SS}$ are all different as $S$ runs over all elements of ${\left(T\prime \right)}^{+}$ then $$E_{TT}=\prod _{\underset{S\ne T}{S\in {\left(T\prime \right)}^{+}}}\frac{E_{T\prime T\prime }{D}_m^2{E}_{T\prime T\prime }-{\left(D_m^2\right)}_{SS}{E}_{T\prime T\prime }}{{\left(D_m^2\right)}_{TT}-{\left(D_m^2\right)}_{SS}}$$

		Proof.
	(1) Recall that the identification of the path algebras with the centralizer algebras ${𝒵}_m$ is done so that for each $T=({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)},{\tau }^{\left(m\right)})\in {𝒯}^m$ we have that $E_{TT}{V}^{\otimes m}$ is an irreducible $𝔘$ module isomorphic to ${\Lambda }_{{\tau }^{\left(m\right)}}\text{.}$ Furthermore, if we let $T\prime =({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)})\in {𝒯}^{m-1}$ we know that $$E_{T\prime T\prime }{V}^{\otimes m}=E_{T\prime T\prime }{V}^{\otimes (m-1)}\otimes V\cong {\Lambda }_{{\tau }^{(m-1)}}\otimes {\Lambda }_{\omega }$$ and that, by Proposition (2.21), $D_m^2={\left({ℛ}_{12}{ℛ}_{21}\right)}^2$ acts on each irreducible component ${\Lambda }_{{\tau }^{\left(m\right)}}$ of the tensor product $E_{T\prime T\prime }{V}^{\otimes (m-1)}\otimes V$ by the constant $z\left({\tau }^{(m-1)}\right)z\left(\omega \right)/z\left({\tau }^{\left(m\right)}\right)\text{.}$ It follows that $$\begin{array}{ccc}{D}_m^2{V}^{\otimes m}& =& D_m^2\sum _{T\prime \in {𝒯}^{m-1}}{E}_{T\prime T\prime }(V^{\otimes (m-1)}\otimes V)\\ & =& \sum _{T\prime \in {𝒯}^{m-1}}{D}_m^2(E_{T\prime T\prime }{V}^{\otimes (m-1)}\otimes V)\\ & =& \sum _{T\prime \in {𝒯}^{m-1}}$ {\left({ℛ}_{21}{ℛ}_{12}\right)}^2$(E_{T\prime T\prime }{V}^{\otimes (m-1)}\otimes V)\\ & =& \sum _{T\prime \in {𝒯}^{m-1}}$ {\left({ℛ}_{21}ℛ\right)}^2$\left(\sum _{T\in {\left(T\prime \right)}^{+}}{E}_{TT}{V}^{\otimes m}\right)\\ & =& \sum _{T\in T^m}\frac{z\left({\tau }^{(m-1)}\right)z\left(\omega \right)}{z\left({\tau }^{\left(m\right)}\right)}{E}_{TT}{V}^{\otimes m}\text{.}\end{array}$$ The result follows as $D_m^2$ is determined by its action on $V^{\otimes m}\text{.}$ (2) It follows from part (1) that $$E_{T\prime T\prime }{D}_m^2{E}_{T\prime T\prime }=\sum _{T\in {\left(T\prime \right)}^{+}}\frac{z\left({\tau }^{(m-1)}\right)z\left(\omega \right)}{z\left({\tau }^{\left(m\right)}\right)}{E}_{TT}\text{.}$$ If the Bratteli diagram is multiplicity free and the eigenvalues $z\left({\tau }^{(m-1)}\right)z\left(\omega \right)/z\left({\tau }^{\left(m\right)}\right)$ are all different, then the result follows by taking the spectral projection of $E_{T\prime T\prime }{D}_m^2{E}_{T\prime T\prime }$ with respect to a particular eigenvalue. $\square$

The following corollaries follow in exactly the same fashion.

Let $(𝔘,ℛ,v)$ be a ribbon Hopf algebra and let $V={\Lambda }_{\omega }$ be an irreducible representation of $𝔘\text{.}$ Identify the path algebras corresponding to the Bratteli diagram for tensor powers of $V$ with the centralizer algebras ${𝒵}_m$ as in Section 1.

(1)

Let $D_m={Ř}_{m-1}{Ř}_{m-2}\cdots {Ř}_1{Ř}_1{Ř}_2\cdots {Ř}_{m-1}\in {𝒵}_m$ be the element given in Proposition (2.19). Then $$D_m=\sum _{T\in {𝒯}^m}{\left(D_m\right)}_{TT}{E}_{TT},\text{where}{\left(D_m\right)}_{TT}=\frac{v\left({\tau }^{(m-1)}\right)v\left(\omega \right)}{v\left({\tau }^{\left(m\right)}\right)},$$ for $T=({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)},{\tau }^{\left(m\right)})\in {𝒯}^m\text{.}$

(2)

Fix $T=({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)},{\tau }^{\left(m\right)})\in {𝒯}^m$ and let $T\prime =({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)})\in {𝒯}^{m-1}\text{.}$ Let ${\left(T\prime \right)}^{+}$ be the set of tableaux that are extensions of $T\prime ,$ i.e. the set of $S=({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)},{\sigma }^{\left(m\right)})\in {𝒯}^m\text{.}$ If the values ${\left(D_m\right)}_{SS}$ are all different as $S$ runs over all elements of ${\left(T\prime \right)}^{+}$ then $$E_{TT}=\prod _{\underset{S\ne T}{S\in {\left(T\prime \right)}^{+}}}\frac{E_{T\prime T\prime }{D}_m{E}_{T\prime T\prime }-{\left(D_m\right)}_{SS}{E}_{T\prime T\prime }}{{\left(D_m\right)}_{TT}-{\left(D_m\right)}_{SS}}$$

[Res1987, formula (3.19)]. Let $({𝔘}_h\left(𝔤\right),ℛ,e^{-h\rho }u)$ be a Drinfel'd-Jimbo quantum group and let $V={\Lambda }_{\omega }$ be an irreducible representation of $𝔘\text{.}$ Identifying the path algebras $A_m$ corresponding to the Bratteli diagram for tensor powers of $V$ with the centralizer algebras ${𝒵}_m$ as in Section 1.

(1)

Let $D_m={Ř}_{m-1}{Ř}_{m-2}\cdots {Ř}_1{Ř}_1{Ř}_2\cdots {Ř}_{m-1}\in {𝒵}_m$ be the element given in Proposition (2.19). Then $$D_m=\sum _{T\in {𝒯}^m}{\left(D_m\right)}_{TT}{E}_{TT},\text{where}{\left(D_m\right)}_{TT}=q^{⟨{\tau }^{\left(m\right)},{\tau }^{\left(m\right)}+2\rho ⟩-⟨{\tau }^{(m-1)},{\tau }^{(m-1)}+2\rho ⟩-⟨\omega ,\omega +2\rho ⟩},$$ and ${\tau }^{\left(m\right)}$ and ${\tau }^{(m-1)}$ are determined from $T$ by $T=({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)},{\tau }^{\left(m\right)})\in {𝒯}^m\text{.}$

(2)

Fix $T=({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)},{\tau }^{\left(m\right)})\in {𝒯}^m$ and let $T\prime =({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)})\in {𝒯}^{m-1}\text{.}$ Let ${\left(T\prime \right)}^{+}$ be the set of tableaux that are extensions of $T\prime ,$ i.e. the set of $S=({\tau }^{\left(0\right)},\dots ,{\tau }^{(m-1)},{\sigma }^{\left(m\right)})\in {𝒯}^m\text{.}$ If the values ${\left(D_m\right)}_{SS}$ are all different as $S$ runs over all elements of ${\left(T\prime \right)}^{+}$ then $$E_{TT}=\prod _{\underset{S\ne T}{S\in {\left(T\prime \right)}^{+}}}\frac{E_{T\prime T\prime }{D}_m{E}_{T\prime T\prime }-{\left(D_m\right)}_{SS}{E}_{T\prime T\prime }}{{\left(D_m\right)}_{TT}-{\left(D_m\right)}_{SS}}\text{.}$$

Notes and References

This is a typed version of the paper A Ribbon Hopf Algebra Approach to the Irreducible Representations of Centralizer Algebras: The Brauer, Birman-Wenzl, and Type A Iwahori-Hecke Algebras by Robert ${\text{Leduc}}^{*}$ and Arun ${\text{Ram}}^{†}\text{.}$

The paper was received June 24, 1994; accepted September 12, 1994.

${}^{*}$Supported in part by National Science Foundation Grant DMS-9300523 to the University of Wisconsin.
${}^{†}$Supported in part by National Science Foundation Postdoctoral Fellowship DMS-9107863.

page history