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.

$𝔤=𝔰𝔩\left(n\right)$

The basic representation of $V_q\left(𝔰𝔩\left(n\right)\right)$ have the h.w. ${\omega }_1$ and $V^{{\omega }_1}\simeq {ℂ}^n\text{.}$ The matrix elements of the generators do not depend on $q$ in this representation. $$\begin{array}{cc}\pi \left(X_i^{+}\right)=E_{ii+1},\pi \left(X_i^{-}\right)=E_{i+1i},\pi \left(H_i\right)=E_{ii}-E_{i+1,i+1}& \text{(4.1)}\end{array}$$ where ${\left(E_{ij}\right)}_{\alpha \beta }={\delta }_{i\alpha }{\delta }_{j\beta }$ are the basic matrices in ${ℂ}^n\text{.}$

Tho Tensorial square of $V^{{\omega }_1}$ is decomposed into the sum of two irreducible components $$\begin{array}{cc}{\left(V^{{\omega }_1}\right)}^{\otimes 2}=V^{2{\omega }_1}\oplus V^{{\omega }_2}\text{.}& \text{(4.2)}\end{array}$$

The KGC of this decomposition are determined by the relations: $$\begin{array}{cc}{e}_{ij}^{2{\omega }_1}=\frac{1}{\sqrt{1+q^{-1}}}(e_i\otimes e_j+q^{-\frac{1}{2}}{e}_j\otimes e_i),\hspace{0.17em}i<j,e_{ii}^{2{\omega }_1}=e_i\otimes e_i& \text{(4.3)}\\ e_{ij}^{{\omega }_2}=\frac{1}{\sqrt{1+q^{-1}}}(e_i\otimes e_j-q^{-1/2}{e}_j\otimes e_i),\hspace{0.17em}i<j& \text{(4.4)}\end{array}$$ where $e_{ij}^{2{\omega }_1}$ and $e_{ij}^{{\omega }_2}$ are the bases in $V^{2{\omega }_1}$ and in $V^{{\omega }_2}$ respectively.

From the theorem 1.5 we find the basic $R\text{-matrix}$ for $U_q\left(𝔤𝔩\left(n\right)\right)\text{:}$ $$\begin{array}{cc}{R}^{{\omega }_1{\omega }_1}=P_{2{\omega }_1}^{{\omega }_1{\omega }_1}{q}^{1/2}-P_{{\omega }_2}^{{\omega }_1{\omega }_1}{q}^{-1/2}& \text{(4.5)}\end{array}$$ or using (4.3), (4.4) $$\begin{array}{cc}{\displaystyle R^{{\omega }_1{\omega }_1}=q^{1/2}\sum _{i=1}^n{E}_{ii}\otimes E_{ii}+\sum _{i\ne j}^n{E}_{ij}\otimes E_{ji}-(q^{-1/2}-q^{1/2})\sum _{j>i}{E}_{jj}\otimes E_{ii}\text{.}}& \text{(4.6)}\end{array}$$ In the form (4.6) the matrix $R^{{\omega }_1{\omega }_1}$ was found by Jimbo [Jim1985].

For $𝔤=𝔤𝔩\left(n\right)$ the decomposition of $V^{\lambda }\otimes V^{{\omega }_1}$ is well known: $$\begin{array}{cc}{\displaystyle V^{\lambda }\otimes V^{{\omega }_1}=\sum _{k=1}^n\prime \oplus V^{{\lambda }^{\left(k\right)}}}& \text{(4.7)}\end{array}$$ where ${\lambda }^{\left(k\right)}=({\lambda }_1,\dots ,{\lambda }_k+1,{\lambda }_{k+1},\dots ,{\lambda }_n)$ if $\lambda =({\lambda }_1,\dots ,{\lambda }_n),$ $({\lambda }_i\in {\mathbb{Z}}_{+},{\lambda }_i\ge {\lambda }_{i+1})$ and the sum is taken only over $k$ satisfying the condition ${\lambda }_k>{\lambda }_{k+1}\text{.}$ Using the rule (4.7) and the theorem 3.1 one can calculate the elements $P_{\lambda }(q,a)\text{.}$ The representations $\lambda =k{\omega }_1$ and $\lambda ={\omega }_k$ are contained in ${\left(V^{{\omega }_1}\right)}^{\oplus k}$ with multiplicity equal to one. It is not difficult to calculate the corresponding projectors $P_{\lambda }\text{:}$ $$\begin{array}{cccc}{P}_{k{\omega }_1}& =& {\displaystyle \frac{q^{\frac{k(k-1)}{4}}}{\left[k\right]!}\sum _{w\in S_k}{q}^{\frac{\ell \left(w\right)}{2}}{\pi }_w,}& \text{(4.8)}\\ P_{{\omega }_k}& =& {\displaystyle \frac{q^{-\frac{k(k-1)}{4}}}{\left[k\right]!}\sum _{w\in S_k}{(-q)}^{-\frac{\ell \left(w\right)}{2}}{\pi }_w,}& \text{(4.9)}\end{array}$$ where the sum taken is over the element of symmetric group $S_k,$ ${\pi }_{s_i}=g_i$ if $S_i$ is the elementary transposition in $S_k,$ the elements $g_i$ are defined by (3.2) and ${\pi }_w={\pi }_{s_1}\dots {\pi }_{s_k}$ if $w=s_1\dots s_k$ is the representation of $w$ in the nonreducible product of the transpositions, $\left[k\right]!=\left[k\right]\dots \left[1\right],$ $\left[k\right]=(q^{k/2}-q^{-k/2})/(q^{1/2}-q^{-1/2})\text{.}$ The formulae (4.8), (4.9) were obtained in [Jim1985] as the $q\text{-analogs}$ of the Young sym (antisym) metrisators. The following theorem describes the connection between the algebra $C_N^{{\omega }_1}\left(𝔤𝔩\left(n\right)\right)$ and Hecke algebra ${\mathscr{H}}_N\left(q\right)\text{.}$ The last one is the assotiative algebra with units generated by the elements $1,{\hat{g}}_i,$ $i=1,\dots ,N-1$ satisfying the following relations: $$\begin{array}{cc}\begin{array}{ccc}{\hat{g}}_i{\hat{g}}_{i+1}{\hat{g}}_i& =& {\hat{g}}_{i+1}{\hat{g}}_i{\hat{g}}_{i+1}\\ {\hat{g}}_i{\hat{g}}_j& =& {\hat{g}}_j{\hat{g}}_i,|i-j|>1\\ {\hat{g}}_i^2& =& (q-1){\hat{g}}_i+q\text{.}\end{array}& \text{(4.10)}\end{array}$$

Theorem 4.1. The algebra $C_N^{{\omega }_1}\left(𝔤𝔩\left(n\right)\right)$ is the factor of Hecke algebra ${\mathscr{H}}_N\left(q\right)$ over the ideal $J_N$ formed by the elements $$\sum _{w\in S_{n+1}}{(-q)}^{-\frac{\ell \left(w\right)}{2}}{\pi }_w=0$$ where ${\pi }_w$ are the same as in (4.8) and (4.9). If $n+1>N,$ $C_N^{{\omega }_1}\left(𝔤𝔩\left(n\right)\right)\simeq {\mathscr{H}}_N\left(q\right)\text{.}$ The generators $g_i$ of $C_N^{{\omega }_1}\left(𝔤𝔩\left(n\right)\right)$ correspond to the generators ${\hat{g}}_i$ of Hecke algebra ${\hat{g}}_i=q^{1/2}{g}_i\text{.}$

This theorem in equivalent to the fact that in the space ${\left(V^{{\omega }_1}\right)}^{\otimes N}$ the representation of Hecke algebra $\alpha \left({\hat{g}}_i\right)=q^{1/2}{g}_i$ (this fact was noted by Jimbo [Jim1985]) is defined. The theorem 4.1 means that the irreducible components of this representation of ${\mathscr{H}}_N\left(q\right)$ have the Young diagrams $\lambda$ with no more than $n$ rows.

An explicit expression for matrix elements of $g_i$ in the basics (3.4) of the representation $W_{\lambda }$ can be found from (3.9). Omitting the details, we give the answer: $$\begin{array}{cccc}w(\lambda ,\lambda +e_k,\lambda +e_k+e_{\ell },\lambda +e_{\ell })& =& {\displaystyle \frac{\sqrt{(q^{\frac{d+1}{2}}-q^{-\frac{d+1}{2}})(q^{\frac{d-1}{2}}-q^{-\frac{d-1}{2}})}}{(q^{\frac{1}{2}}+q^{-\frac{1}{2}})(q^{\frac{d}{2}}-q^{-\frac{d}{2}})},}& \text{(4.11)}\\ w(\lambda ,\lambda +e_k,\lambda +e_k+e_{\ell },\lambda +e_k)& =& \{\begin{array}{cc}{\displaystyle \frac{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}{q^d-1},}& k\ne \ell \\ q^{-\frac{1}{2}},& k=\ell \\ -q^{\frac{1}{2}},& \ell =k+1,{\lambda }_k={\lambda }_{\ell }\end{array}& \text{(4.12)}\end{array}$$ where $d={\lambda }_k-k-{\lambda }_{\ell }+\ell \text{.}$ This expression was given in [Wen1985] and [Ker1987] following a different argumentation.

Footnotes

${}^{*\text{)}}$ From this point we will use the normalization and the notation of books [Bou1981] for the highest weights and the roots of simple Lie algebras.

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