Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I

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

Last update: 2 June 2014

Notes and References

This is a typed copy of the LOMI preprint Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I by N.Yu. Reshetikhin.

Recommended for publication by the Scientific Council of Steklov Mathematical Institute, Leningrad Department (LOMI) 6, June, 1987.

The $q\text{-analog}$ of Brauer-Weyl duality

In this seotion we study the structure of centralizater of $U_q\left(𝔤\right)$ in ${\left(V^{\omega }\right)}^{\otimes N},$ where $V^{\omega }$ is the basic representation.

Let $V^{\left\{\lambda \right\}}=V^{{\lambda }_1}\otimes \dots \otimes V^{{\lambda }_N}$ and $V^{\left\{{\lambda }_{\sigma }\right\}}=V^{{\lambda }_{{\sigma }_1}}\otimes \dots \otimes V^{{\lambda }_{{\sigma }_N}},$ where $\sigma \in S_N$ is the permutation of the set $(1,\dots ,N)\text{.}$

Definition 3.1. An algebra of the maps $𝒴:V^{\left\{{\lambda }_{\sigma }\right\}}\to V^{\left\{{\lambda }_{\sigma \prime }\right\}}$ commuting with the action of $V_q\left(𝔤\right)$ in these spaces is called the centralizer of $U_q\left(𝔤\right)$ in $V^{{\lambda }_1}\otimes \dots \otimes V^{{\lambda }_N}$ and denoted by $C^{{\lambda }_1\dots {\lambda }_N}\left(𝔤\right)\text{.}$

Here we will consider only the algebras $C^{\omega \dots \omega }(𝔤,q)\equiv C_n^{\omega }{(𝔤,q)}^{*\text{)}}$ for the representations $\omega ,$ when multiplicity of the irreducible components in the decomposition $$\begin{array}{cc}{\displaystyle V^{\lambda }\otimes V^{\omega }=\sum _{\mu }\oplus V^{\mu }}& \text{(3.1)}\end{array}$$ is equal to one for any IFR $V^{\lambda }\text{.}$

Proposition 3.1. The algebra $C_n^{\omega }(𝔤,q)$ is simple for general $q$ and is generated by the elements $$\begin{array}{cc}{𝔤}_i=1\otimes \dots \otimes \underset{(i,i+1)}{R^{\omega \omega }}\otimes \dots \otimes 1& \text{(3.2)}\end{array}$$ if the spectrum of the Casimir operator is simple in the decomposition (3.1).

The proof of this proposition is based on the spectral decomposition of the matrix $R^{\omega \omega }\text{.}$

The representations of $C_N^{\omega }\left(𝔤\right)$ are given by the decomposition of ${\left(V^{\omega }\right)}^{\otimes N}$ on the $U_q\left(𝔤\right)\text{-irreducible}$ components: $$\begin{array}{cc}{\displaystyle {\left(V^{\omega }\right)}^{\otimes N}=\sum _{\lambda }{W}_{\lambda }\otimes V^{\lambda }\text{.}}& \text{(3.3)}\end{array}$$

The space $W_{\lambda }$ describes the multiplicity of the h.w. $\lambda$ in (3.3): $\text{dim}\hspace{0.17em}{W}_{\lambda }=\text{mult}\left(V^{\lambda }\right)\text{.}$

Proposition 3.2. The representations $W_{\lambda }$ in (3.3) form the list of all irreducible representations of $C_N^{\omega }\left(𝔤\right)\text{.}$

Let us use now the graphical representation of the matrices $R^{\lambda \mu }$ and $K_{\nu }^{\lambda \mu }$ developed in section 2 to describe the basis in $W_{\lambda }\text{.}$

Proposition 3.3. The elements $$\begin{array}{cc}\begin{array}{c} ⋯ ⋯ ω ω ω ω ω λ2 λN-1 λ \end{array}⟷K_{\lambda }(q,a),& \text{(3.4)}\end{array}$$ where $a=(\omega ,{\lambda }_2,\dots ,{\lambda }_{N-1},\lambda )$ and $V^{{\lambda }_i}\subset V^{\omega }\otimes V^{{\lambda }_{i-1}}$ form the orthogonal basis in $W_{\lambda }\text{.}$ More precisely $$K_{\lambda }(q,a)=E_{\lambda }\left(a\right)\otimes I_{\lambda }$$ where $\left\{E_{\lambda }\left(a\right)\right\}$ is the basis in $W_{\lambda }$ and $I_{\lambda }$ is the unit matrix in $V^{\lambda }\text{.}$

The orthogonality of the elements $K_{\lambda }(q,a)$ means that $$\begin{array}{cc}{K}_{\lambda }(q,b)K_{\lambda }{(q,a)}^t={\delta }_{a,b}·I,& \text{(3.5)}\end{array}$$ where $a=(\omega ,{\lambda }_2,\dots ,{\lambda }_{N-1},\lambda ),$ $b=(\omega ,{\lambda }_2^{\prime },\dots ,{\lambda }_{N-1}^{\prime },\lambda )\text{.}$ $K_{\lambda }^t$ is the transposition of the matrix $K_{\lambda }\text{.}$ The relation (3.5) is proved by the following equalities founded on the orthogonality of CGC: $$\begin{array}{cc}\begin{array}{c} ω ω ω ⋯ ω λ2 λ λ2′ λ ⋯ ⋯ \end{array}={\delta }_{{\lambda }_2{\lambda }_2^{\prime }}\begin{array}{c} λ2 ω ω λ3 λ λ3′ λ \end{array}{\displaystyle =\dots =\prod _{j=2}^{N-1}{\delta }_{{\lambda }_j{\lambda }_j^{\prime }}}\begin{array}{c} λ \end{array}& \text{(3.6)}\end{array}$$ The completeness of this basis in $W_{\lambda }$ is evident.

Proposition 3.4. The action of the elements $g_i$ in basis have the following form: $$\begin{array}{cc}{\displaystyle g_i{K}_{\lambda }(q,a)=\sum _{{\lambda }_i^{\prime }}W({\lambda }_{i-1},{\lambda }_i,{\lambda }_{i+1},{\lambda }_i^{\prime })K_{\lambda }(q,a\prime )}& \text{(3.7)}\end{array}$$ where $a=(\omega ,{\lambda }_2,\dots ,{\lambda }_{N-1},\lambda ),$ $a\prime =(\omega ,{\lambda }_2,\dots ,{\lambda }_i^{\prime },\dots ,{\lambda }_{N-1},\lambda )$ and $w({\lambda }_{i-1},{\lambda }_i,{\lambda }_{i+1},{\lambda }_i^{\prime })$ are some constants calculated below.

The relations (3.7) have a simple graphical representation: $$\begin{array}{cc}\begin{array}{c} ⋯ ⋯ ω ω ω ω ω λ2 λi λ \end{array}{\displaystyle =\sum _{{\lambda }_i^{\prime }}}\begin{array}{c} ⋯ ⋯ ω ω ω ω ω λ2 λi λ \end{array}w({{\lambda }_i}_1,{\lambda }_i,{{\lambda }_i}_1,{\lambda }_i^{\prime })\text{.}& \text{(3.8)}\end{array}$$ Using the orthogonality of the basis $K_{\lambda }(q,a)$ we obtain a formula for the coefficients $W\text{:}$

Proposition 3.5. The coefficients $W$ in (3.6) are calculated by the following praphical rule: $$\begin{array}{cc}w({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_2^{\prime })=\begin{array}{c} λ1 λ2 λ2′ λ3* \end{array}\frac{1}{{\chi }_{{\lambda }_2}}\text{.}& \text{(3.9)}\end{array}$$ An exact meaning of this formula is given in section 8.

The action of $g_i^{-1}$ in $K_{\lambda }$ basis is similar to (3.7): $$\begin{array}{cc}{\displaystyle g_i^{-1}{K}_{\lambda }\left(a\right)=\sum _{{\lambda }_i^{\prime }}\bar{w}({\lambda }_{i-1},{\lambda }_i,{\lambda }_{i+1},{\lambda }_i^{\prime })K_{\lambda }\left(a\prime \right)}& \text{(3.10)}\end{array}$$ where the coefficient $\bar{w}$ has the following form: $$\begin{array}{cc}\bar{w}({\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_2^{\prime })=\begin{array}{c} λ1 λ2 λ2′ λ3* \end{array}\frac{1}{{\chi }_{{\lambda }_3}}& \text{(3.11)}\end{array}$$ From $g_i{g}_i^{-1}=1$ follows the relation $$\begin{array}{cc}{\displaystyle \sum _{{\lambda }_i^{\prime }}w({\lambda }_{i-1},{\lambda }_i,{\lambda }_{i+1},{\lambda }_i^{\prime })\bar{w}({\lambda }_{i-1}{\lambda }_i^{\prime },{\lambda }_{i+1}{\lambda }_i^{″})={\delta }_{{\lambda }_i{\lambda }_i^{″}}\text{.}}& \text{(3.12)}\end{array}$$

So, we described the algebra $C_N^{\omega }\left(𝔤\right)$ and its irreducible representations. It is necessary to note that the basis (3.4) is exactly a $q\text{-analog}$ of the Young basis in irreducible representations of symmetric group $S_N\text{.}$ This basis is regular for the embeddings $$\begin{array}{cc}{C}_1^{\omega }\left(𝔤\right)\subset C_2^{\omega }\left(𝔤\right)\subset \dots \subset C_N^{\omega }\left(𝔤\right)& \text{(3.13)}\end{array}$$ where $C_n^{\omega }\left(𝔤\right)$ is formed by the elements $1,\dots ,g_{N-1}$ and $C_{N-1}^{\omega }\left(𝔤\right)$ is formed by the elements $1,\dots ,g_{N-2}\text{.}$ Indeed, if we consider the restriction $$\begin{array}{cc}{\displaystyle w_{\lambda }\left(C_N^{\omega }\right){|}_{C_{N-1}^{\omega }}=\sum _{\lambda \prime }\oplus w_{\lambda }\left(C_{N-1}^{\omega }\right)}& \text{(3.14)}\end{array}$$ the basis in $w_{\lambda \prime }\left(C_{N-1}^{\omega }\right)$ will be formed by the elements $K_{\lambda }\left(a_{\lambda \prime }\right),$ where $a_{\lambda \prime }=(\omega ,\dots ,{\lambda }_{N-2},\lambda \prime ,\lambda )\text{.}$

Let us also note that the decompositions (3.14) determine tho graph of the algebra $C_N^{\omega }\left(𝔤\right)$ [KVe1985]. It is evident that really this graph is fixed by decomposition (3.3).

Let us consider now the $q\text{-analog}$ of Young symmetrizers. For this purpose we introduce the matrices $P_{\lambda }(q,a):{\left(V^{\omega }\right)}^{\otimes N}\to {\left(V^{\omega }\right)}^{\otimes N},$ $\text{Im}\left(P_{\lambda }(q,a)\right)\simeq V^{\lambda }\text{:}$ $$\begin{array}{cc}{P}_{\lambda }(q,a)=K_{\lambda }^t(q,a)K_{\lambda }(q,a)=(E_{\lambda }\left(a\right)\otimes E_{\lambda }^t\left(a\right))\otimes I_{\lambda }\text{.}& \text{(3.15)}\end{array}$$ This matrix corresponds to the figure $$\begin{array}{cc}\begin{array}{c} λ a a ⋯ ⋯ ω ω ω ω ω ω \end{array}=\begin{array}{c} ⋯ ⋯ ⋯ ⋯ λ λN-1 λN-1 λ2 λ2 ω ω ω ω ω ω ω ω \end{array}& \text{(3.16)}\end{array}$$

Proposition 3.3 follows that the set $P_{\lambda }(q,a)$ forms an orthogonal set of projectors in ${\left(V^a\right)}^{\otimes N}\text{:}$ $$\begin{array}{cc}{P}_{\lambda }(q,a)P_{\mu }(q,b)={\delta }_{\lambda \mu }{\delta }_{a,b}\text{.}& \text{(3.17)}\end{array}$$

Theorem 3.1. The elements $P_{\lambda }$ satisfy the following relations: $$\begin{array}{cc}\begin{array}{c} λ a a ⋯ ⋯ ω ω ω ω ω ω \end{array}{\displaystyle =\sum _{n=0}^{r-1}{q}_{n,\lambda }}\begin{array}{c} a a ω ⋯ ω ω ω ⋯ ω ω } 2n crosses \end{array}& \text{(3.18)}\end{array}$$ where $a=(\omega ,{\lambda }_2,\dots ,{\lambda }_{N-1},\lambda ),$ $a\prime =(\omega ,{\lambda }_2,\dots ,{\lambda }_{N-1}),$ $q_{n,\lambda }$ are the matrix elements of the matrix: $$\begin{array}{cc}A={\left(q^{n\frac{c\left(\lambda \right)-c\left({\lambda }_{N-1}\right)-c\left(\omega \right)}{2}}\right)}_{\lambda \in {\lambda }_{N-1}\otimes \omega ,\hspace{0.17em}n=0,\dots ,r-1}& \text{(3.19)}\end{array}$$ $r$ is the number of irreducible components in ${\lambda }_{N-1}\otimes \omega ,$ the index $n$ numbers the rows of $A$ and $\lambda$ numbers the columns of $A\text{.}$

		Proof.
	Let us consider the matrix $R^{{\lambda }_{N-1}\omega }(q,a\prime )$ corresponding to the figure $$\begin{array}{c} a′ a′ ω ⋯ ω ω ω ⋯ ω λN-1 \end{array}$$ From the theorem 1.5 we have the decomposition or matrix $R^{{\lambda }_{N-1}\omega }(q,a\prime )R^{\omega {\lambda }_{N-1}}(q,a\prime )$ $$\begin{array}{c} a′ a′ ω ⋯ ω ω ω ⋯ ω λN-1 \end{array}=\sum _{\lambda \in {\lambda }_{N-1}\otimes \omega }{q}^{\frac{c\left(\lambda \right)-c\left({\lambda }_{N-1}\right)-c\left(\omega \right)}{2}}\begin{array}{c} a a ω ⋯ ω ω ⋯ ω λ \end{array}\text{.}$$ Taking powers of degrees $n=0,1,\dots ,r-1$ of this equality we obtain a system for the projectors $P_{\lambda }(q,a)\text{.}$ Solving this system we obtain the relation (3.18). Considering the relation (3.18) as an inductive process for calculating $P_{\lambda }(q,a)$ we obtain the explicit expression for $P_{\lambda }(q,a)\text{:}$ $$\begin{array}{cc}{\displaystyle P_{\lambda }(q,a)=\sum _{n_1=0}^{r_1-1}\dots \sum _{n_{N-1}=0}^{r_{N-1}-1}{q}_{n_1\lambda }{q}_{n_2{\lambda }_{N-1}}\dots q_{n_{N-1},\omega }\times \begin{array}{c} ω ⋯ ω ω ⋯ ω α(n1,…,nN-1) \end{array}\text{.}}& \text{(3.19)}\end{array}$$ Here ${\alpha }_{(n_1,\dots ,n_{N-1})}$ is a braid (see section 8) where the string with number $i$ turned round the strings with numbers $1,\dots ,i-1$ (from left to right) $2n_i$ times moving counter-clockwise and form the top to the bottom. $\square$

Now we can describe the structure of the matrices $R^{\lambda \mu }\text{.}$

Consider a decomposition of $V^{\lambda }\otimes V^{\mu }$ into the sum or $U_q\left(𝔤\right)$ irreducible components: $$\begin{array}{cc}{\displaystyle V^{\lambda }\otimes V^{\mu }=\sum _{\nu }{W}_{\nu }^{\lambda \mu }\otimes V^{\nu }}& \text{(3.20)}\end{array}$$

The matrix $R^{\lambda \mu }$ is an element of $\text{End}(V^{\lambda }\otimes V^{\mu })\to \text{End}(V^{\lambda }\otimes V^{\mu })$ commuting with the action of $U_q\left(𝔤\right)\text{.}$ Hence it has the following form: $$\begin{array}{cc}{\displaystyle R^{\lambda \mu }=\sum _{\nu }{R}_{\nu }^{\lambda \mu }\otimes I_{\nu }}& \text{(3.21)}\end{array}$$ where $R_{\nu }^{\lambda \mu }:W_{\nu }^{\lambda \mu }\to W_{\nu }^{\lambda \mu }\text{.}$

Let $W_{\nu }$ be the $C_{N+M}^{\omega }\text{-irreducible}$ module. The decomposition of this module into tho sum of $C_N^{\omega }\times C_M^{\omega }\text{-irreducible}$ components is: $$\begin{array}{cc}{\displaystyle W_{\nu }{|}_{C_N^{\omega }\times C_M^{\omega }}=\sum _{\lambda \mu }{W}_{\nu }^{\lambda \mu }\otimes W_{\lambda }\otimes W_{\mu }}& \text{(3.22)}\end{array}$$ where the spaces $W_{\nu }^{\lambda \mu }$ are the same as in (3.20).

Let $R^{(N,M)}$ be the matrix acting in ${\left(V^{\omega }\right)}^{\otimes (N+M)}$ of the form: $$\begin{array}{cc}{R}^{(N,M)}⟷\begin{array}{c} ⋯ ⋯ ⋯ ⋯ \end{array}& \text{(3.23)}\end{array}$$

In accordance with (3.3) this matrix has the following decomposition $$R^{(N,M)}=\sum _{\nu }{R}_{\nu }^{(N,M)}\otimes I_{\nu }$$ $$\begin{array}{cc}{\displaystyle R_{\nu }^{(N,M)}=\sum _{\lambda ,\mu }{R}_{\nu }^{\lambda \mu }\otimes {\sigma }^{\lambda \mu }}& \text{(3.25)}\end{array}$$ where ${\sigma }^{\lambda \mu }:W_{\lambda }\otimes W_{\mu }\to W_{\mu }\otimes W_{\lambda }$ is the the permutation operator ${\sigma }^{\lambda \mu }(f_{\lambda }\otimes g_{\mu })=g_{\mu }\otimes f_{\lambda }\text{.}$

Tho decomposition (3.24) and (3.25) is the generalization of the formula (1.37) on matrices $R^{\lambda \mu }$ with arbitrary $\lambda$ and $\mu \text{.}$ The formula (3.25) gives the method for calculating the matrices $R^{\lambda \mu }\text{.}$

Footnotes

${}^{*\text{)}}$ We will write $C_N^{\omega }\left(𝔤\right)$ when it will not be ambiguous.

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