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.

$𝔤=𝔰𝔭\left(2n\right)$

All formulae in thin caae are very similar to those in part one of the previous section.

The h.w. of finite dimensional representation of are parametrized by the numbers $\lambda =({\lambda }_1,\dots ,{\lambda }_n),$ ${\lambda }_1\ge \dots \ge {\lambda }_n,$ ${\lambda }_i\in {\mathbb{Z}}_{+}\text{.}$ If $e_{\lambda }$ is the h.w. vector, $H_i{e}_{\lambda }=({\lambda }_i-{\lambda }_{i+1})e_{\lambda },$ $i=1,\dots ,n-1,$ $H_n{e}_{\lambda }={\lambda }_n{e}_{\lambda },$ $X_i^{+}{e}_{\lambda }=0\text{.}$

The basic representation of $U_q\left(𝔰𝔭\left(2n\right)\right)$ have the h.w. ${\omega }_1=(1,0,\dots ,0),$ $V^{{\omega }_1}\simeq {ℂ}^{2n}$ $$\begin{array}{cc}\begin{array}{ccc}\pi \left(X_i^{+}\right)& =& E_{ii+1}-E_{N-i,N-i+1},i=1,\dots ,n-1\\ \pi \left(X_n^{-}\right)& =& E_{nn+1},\pi \left(X_i^{-}\right)=\pi {\left(X_i^{+}\right)}^t\\ \pi \left(H_i\right)& =& E_{ii}-E_{i+1,i+1}-E_{N-i+1,N-i+1}+E_{N-i,N-i}\\ \pi \left(H_n\right)& =& 2E_{nn}-2E_{n+1,n+1}\text{.}\end{array}& \text{(6.1)}\end{array}$$ The tensor square of the basic representation is the sum of three irreducible components: $$\begin{array}{cc}{V}^{{\omega }_1}\otimes V^{{\omega }_1}=V^{2{\omega }_1}\oplus V^{{\omega }_2}\oplus V^{\left(0\right)}& \text{(6.2)}\end{array}$$ and we have the following spectral decomposition of $R^{{\omega }_1{\omega }_1}\text{:}$ $$\begin{array}{cc}{R}^{{\omega }_1{\omega }_1}=q^{\frac{1}{2}}{P}_{2{\omega }_1}^{{\omega }_1{\omega }_1}-q^{-\frac{1}{2}}{P}_{{\omega }_2}^{{\omega }_1{\omega }_1}-q^{-\frac{2n+1}{2}}{P}_0^{{\omega }_1{\omega }_1}\text{.}& \text{(6.3)}\end{array}$$

As in the cases of $𝔤=𝔰𝔬(2n+1)$ and $𝔰𝔬\left(2n\right)$ one can calculate the matrices $K_{2{\omega }_1}^{{\omega }_1{\omega }_1},$ $K_{{\omega }_2}^{{\omega }_1{\omega }_1},$ and $K_0^{{\omega }_1{\omega }_1}\text{.}$ Substituting these matrices into (6.3) we obtain the matrix $R^{{\omega }_1{\omega }_1}\text{:}$ $$\begin{array}{cc}\begin{array}{ccc}{R}^{{\omega }_1{\omega }_1}& =& {\displaystyle q^{\frac{1}{2}}\sum _i{E}_{ii}\otimes E_{ii}+\sum _{i\ne j,j\prime }{E}_{ji}\otimes E_{ij}}\\ & & {\displaystyle +q^{-\frac{1}{2}}\sum _{i\ne i\prime }{E}_{ii\prime }\otimes E_{i\prime i}+(q^{\frac{1}{2}}-q^{-\frac{1}{2}})\sum _{i>j}{E}_{ii}\otimes E_{jj}}\\ & & {\displaystyle -(q^{\frac{1}{2}}-q^{-\frac{1}{2}})\sum _{i>j}{E}_i{E}_j{q}^{\frac{\bar{j}-\bar{i}}{2}}{E}_{i\prime j}\otimes E_{ij\prime }}\end{array}& \text{(6.4)}\end{array}$$ where $i\prime =2n+1-i\text{;}$ ${ϵ}_i=1,$ $i=1,\dots ,n,$ ${ϵ}_j=-1,$ $i=n+1,\dots ,2n\text{;}$ $\bar{i}=i-\frac{1}{2},$ $i\le n,$ $\bar{i}=i+\frac{1}{2},$ $i\ge n+1\text{.}$ This matrix can also be extracted from [Jim1986-2].

The matrices $R^{\lambda \mu }$ and $P_{\lambda }$ for any h.w. $\lambda ,\mu$ can be found from the theorems 2. - 3. and from the ramification rule: $$\begin{array}{cc}{\displaystyle V^{\lambda }\otimes V^{{\omega }_1}=\sum _{k=1}^{2n}\oplus V^{{\lambda }^{\left(k\right)}}}& \text{(6.5)}\end{array}$$ where ${\lambda }^{\left(k\right)}=({\lambda }_1,\dots ,{\lambda }_k+1,\dots ,{\lambda }_n),$ ${\lambda }^{(n+k)}=({\lambda }_1,\dots ,{\lambda }_k-1,\dots ,{\lambda }_k),$ $k=1,\dots ,n\text{.}$

As in the case $𝔤=𝔰𝔬(2n+1),𝔰𝔬\left(2n\right)$ we have the following propositions.

Proposition 6.1. The algebra $C_N^{{\omega }_1}\left(𝔰𝔭\left(2n\right)\right)$ is the factor of B.-W. algebra with $m=\left(i(q^{1/2}-q^{-1/2})\right),$ $\ell =i{q}^{-\frac{2n+1}{2}}$ over the ideal formed by the elements $$\begin{array}{cc}J=P_{{\omega }_{n+1}}{\left(q\right)}^{1\dots n+1}& \text{(6.6)}\end{array}$$ acting nontrivially only in the multipliers of ${\left(V^{{\omega }_1}\right)}^{\otimes N}$ with numbers $i\le n+1\text{.}$ The elements $P_k\left(q\right)$ are defined by (5.11) with the matrix $$\begin{array}{cc}Q=\left(\begin{array}{ccc}1& 1& 1\\ q& q^{-k}& q^{k-2-2n}\\ q^2& q^{-2k}& q^{2(k-2-2n)}\end{array}\right)& \text{(6.7)}\end{array}$$ and $C_M^{{\omega }_1}\left(𝔰𝔭\left(2n\right)\right)\simeq B{W}_M$ for $n+1>M\text{.}$

If we consider the block basis in $V^{{\omega }_1}$ connected with the embedding $U_q\left(𝔰𝔭\left(2n\right)\right)\supset U_q\left(𝔰𝔩\left(n\right)\right),$ $$\begin{array}{cc}{V}^{{\omega }_1}={\stackrel{\sim }{V}}^{{\omega }_1}\oplus {\stackrel{\sim }{V}}^{{\omega }_1^{*}},\text{dim}\hspace{0.17em}{\stackrel{\sim }{V}}^{{\omega }_1}=\text{dim}\hspace{0.17em}{\stackrel{\sim }{V}}^{{\omega }_1^{*}}=n& \text{(6.8)}\end{array}$$ we obtain the representation of $R^{{\omega }_1{\omega }_1}$ in the block form with the blocks constructed from $U_q\left(𝔰𝔩\left(n\right)\right)$ $R\text{-matrices.}$

Proposition 6.3. The matrices ${\left(R^{{\omega }_1{\omega }_1}\right)}^{\pm 1}$ have the following block structure in the basis (6.9): $$\begin{array}{cc}\begin{array}{c} \end{array}=\begin{array}{|cccc|}\hline \begin{array}{c} \end{array}& 0& 0& 0\\ 0& a(\begin{array}{c} \end{array}+q^{-\frac{n+1}{2}}\begin{array}{c} \end{array})& \begin{array}{c} \end{array}& 0\\ 0& \begin{array}{c} \end{array}& 0& 0\\ 0& 0& 0& \begin{array}{c} \end{array}\\ \hline\end{array}& \text{(6.9)}\\ \begin{array}{c} \end{array}=\begin{array}{|cccc|}\hline \begin{array}{c} \end{array}& 0& 0& 0\\ 0& 0& \begin{array}{c} \end{array}& 0\\ 0& \begin{array}{c} \end{array}& -a(\begin{array}{c} \end{array}+q^{\frac{n+1}{2}}\begin{array}{c} \end{array})& 0\\ 0& 0& 0& \begin{array}{c} \end{array}\\ \hline\end{array}& \text{(6.10)}\end{array}$$ $$a=q^{\frac{1}{2}}-q^{-\frac{1}{2}}$$ where we use the notations of sections 2 and 5.

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