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.

$𝔤=𝔰𝔬(2n+1)$ and $𝔤=𝔰𝔬\left(2n\right)$

The h.w. of those algebras can be integer or halfinteger vectors. For $𝔤=𝔰𝔬(2n+1),$ $\lambda =({\lambda }_1,\dots ,{\lambda }_n),$ ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_n,$ ${\lambda }_i\in \frac{1}{2}{\mathbb{Z}}_{+},$ on the h.w. vector $e_{\lambda }:H_i{e}_{\lambda }=({\lambda }_i-?),$ $i=1,\dots ,n-1,$ $H_n{e}_{\lambda }={\lambda }_n{e}_{\lambda }\text{.}$ For $𝔤=𝔰𝔬\left(2n\right),$ $\lambda =({\lambda }_1,\dots ,{\lambda }_{n-1},{\lambda }_n),$ ${\lambda }_1\ge \dots \ge {\lambda }_{n-1}\ge {\lambda }_n,$ ${\lambda }_i\in \frac{1}{2}{\mathbb{Z}}_{+},$ $i=1,\dots ,n-1,$ ${\lambda }_n\in \frac{1}{2}\mathbb{Z},$ $H_i{e}_{\lambda }=({\lambda }_i-{\lambda }_{i+1})e_{\lambda },$ $i=1,\dots ,n-1,$ $H_n{e}_{\lambda }=({\lambda }_{n-1}+{\lambda }_n)\text{.}$ The representation with integers h.w. $\lambda$ form a category of tensoriol representations of $𝔤\text{.}$ The basic representation in this category is the representation with h.w. ${\omega }_1=(1,0,\dots ,0)\text{.}$ In the category of finite dimensional representations basic representation of $𝔰𝔬(2n+1)$ have h.w. ${\omega }_n=(\frac{1}{2},\dots ,\frac{1}{2})\text{.}$ For $𝔰𝔬\left(2n\right)$ there are two basic representations in this category. These representations are ${\omega }_1$ and ${\omega }_n$ or ${\omega }_1$ and ${\omega }_{n-1}$ or ${\omega }_n$ and ${\omega }_{n-1}\text{.}$ All three possibilities are equivalent. Let us start to study the algebras $C_N^{\omega }\left(𝔤\right)$ and $R\text{-matrices}$ connected with algebras $U_q\left(𝔰𝔬\left(2n\right)\right)$ and $U_q\left(𝔰𝔬(2n+1)\right)$ from the category of a tensorial representation.

Tensorial representations of $U_q\left(𝔰𝔬\left(2n\right)\right)$ and $U_q\left(𝔰𝔬(2n+1)\right)$

The vectorial representation $V^{{\omega }_1}$ of $U_q\left(𝔰𝔬(2n+1)\right)$ and $U_q\left(𝔰𝔬\left(2n\right)\right)$ do not depend on $q\text{:}$ $$U_q\left(𝔰𝔬(2n+1)\right)$$ $$\begin{array}{cc}\begin{array}{ccc}\pi \left(X_j^{+}\right)& =& E_{jj+1}-E_{N-j,N+1-j},j=1,\dots ,\\ \pi \left(X_n^{+}\right)& =& \frac{1}{\sqrt{2}}-(E_{n+1,n+2}+E_{nn+1})\\ \pi \left(H_j\right)& =& E_{jj}-E_{j+1,j+1}-E_{N+1-j,N+1-j}+E_{N-j,N-j},j=1,\dots ,n-1\\ \pi \left(H_n\right)& =& E_{nn}-E_{n+2,n+2}\\ \pi \left(X_j^{-}\right)& =& \pi {\left(X_j^{+}\right)}^t\end{array}& \text{(5.1)}\end{array}$$ $$U_q\left(𝔰𝔬\left(2n\right)\right)$$ $$\begin{array}{cc}\begin{array}{ccc}\pi \left(X_j^{+}\right)& =& E_{jj+1}-E_{N-j,N+1-j},j=1,\dots ,n-1\\ \pi \left(X_n^{+}\right)& =& E_{nn-1}-E_{nn+2},\\ \pi \left(H_j\right)& =& E_{jj}-E_{j+1j+1}-E_{N+1-j}+E_{N-j\hspace{0.17em}N-j},j=1,\dots ,n-1\\ \pi \left(H_n\right)& =& E_{nn}+E_{n-1,n-1}-E_{n+1,n+1}-E_{n+2,n+2},\\ \pi \left(X_j^{-}\right)& =& \pi {\left(X_j^{+}\right)}^t\text{.}\end{array}& \text{(5.2)}\end{array}$$ Here $E_{ij}$ are the basic matrices in $V^{{\omega }_1}\simeq {ℂ}^N,$ $N=2n+1,$ for $U_q\left(𝔰𝔬\left(2n\right)\right)$ and $N=2n$ for $U_q\left(𝔰𝔬\left(2n\right)\right),$ $t$ is the transposition. The vector representations of $U_q\left(𝔰𝔬\left(2n\right)\right)$ and $U_q\left(𝔰𝔬(2n+1)\right)$ are self-adjoint: ${\omega }_1^{*}={\omega }_1\text{.}$ The tensor product of two vectorial representations is decomposed into the sum of three irreducible components $$\begin{array}{cc}{\left(V^{{\omega }_1}\right)}^{\otimes 2}=V^{2{\omega }_1}\oplus V^{{\omega }_2}\oplus V^0\text{.}& \text{(5.3)}\end{array}$$

After some calculations one can find the matrices $K_{2{\omega }_1}^{{\omega }_1{\omega }_1},$ $K_{{\omega }_2}^{{\omega }_1{\omega }_1},$ $K_0^{{\omega }_1{\omega }_1}\text{.}$ They are determined by the following formulae $$\begin{array}{cc}\begin{array}{c}i\ne j\prime ,\hspace{0.17em}i<j,\hspace{0.17em}{e}_{ij}^{2{\omega }_1}=\frac{1}{\sqrt{1+q}}\{e_i\otimes e_j+q^{1/2}{e}_j\otimes e_i\},\\ i\ne i\prime ,\hspace{0.17em}{e}_{ii}^{2{\omega }_1}=e_i\otimes e_i,\\ {\displaystyle e_{ii\prime }^{2{\omega }_1}=n_i\{q^{1/2}·e_i\otimes e_{i\prime }+q^{-1/2}{e}_{i\prime }\otimes e_i-\frac{(q^{-1/2}-q^{1/2})}{q^{-1/2}-q^{\frac{N-1}{2}}}(q^{\frac{N-1}{2}}\sum _{k>i}{q}^{\frac{\bar{i}-\bar{k}}{2}}{e}_{k\prime }\otimes e_k+q^{-1/2}\sum _{k\le i}{q}^{\frac{\bar{i}-\bar{k}}{2}}{e}_{k\prime }\otimes e_k)\}}\end{array}& \text{(5.4)}\\ \begin{array}{c}{\displaystyle e_{ij}^{{\omega }_2}=\frac{e_i\otimes e_j-q^{1/2}{e}_j\otimes e_i}{1+q},i<j}\\ {\displaystyle e_{ii\prime }^{{\omega }_2}=n_i\{e_i\otimes e_{i\prime }-e_{i\prime }\otimes e_i+\frac{(q^{-\frac{1}{2}}-q^{\frac{1}{2}})}{q^{\frac{1}{2}}+q^{\frac{N-1}{2}}}(q^{\frac{N}{2}}\sum _{k>i}{q}^{\frac{\bar{i}-\bar{k}}{2}}{e}_{k\prime }\otimes e_k+q\sum _{k\le i}{q}^{\frac{\bar{i}-\bar{k}}{2}}{e}_{k\prime }\otimes e_k)\}\text{.}}\end{array}& \text{(5.5)}\\ {\displaystyle e_{ij}^0=\sqrt{\frac{(q^{\frac{1}{2}}-q^{-\frac{1}{2}})}{(q^{-\frac{N}{4}}-q^{-\frac{N}{4}})(q^{\frac{N-2}{4}}+q^{-\frac{N-2}{4}})}}{\delta }_{ij\prime }\sum _k{q}^{\frac{\bar{i}-\bar{k}}{2}}{e}_{k\prime }\otimes e_k}& \text{(5.6)}\end{array}$$ where $n_i$ are normalization constants, $j\prime =N+1-j,$ $e_{ij}^{2{\omega }_1},$ $e_{ij}^{{\omega }_2},$ $e_{ij}^0$ are the bases in $V^{2{\omega }_2},$ $V^{{\omega }_2},$ $V^0$ respectively, $$\bar{i}=\{\begin{array}{cc}i+\frac{1}{2},& i\le n\\ i,& i=n+1\\ i-\frac{1}{2},& i>n+1\end{array}\text{for}\hspace{0.17em}{U}_q\left(𝔰𝔬(2n+1)\right),\bar{i}=\{\begin{array}{cc}i+\frac{1}{2},& i\le n\\ i-\frac{1}{2},& i>n\end{array}\text{for}\hspace{0.17em}{U}_q\left(𝔰𝔬\left(2n\right)\right)\text{.}$$ From the theorem 1.5 we obtain the matrix $R^{{\omega }_1{\omega }_1}\text{:}$ $$\begin{array}{cc}{R}^{{\omega }_1{\omega }_1}=P_{2{\omega }_1}^{{\omega }_1{\omega }_1}{q}^{-1/2}-P_{{\omega }_2}^{{\omega }_1{\omega }_1}{q}^{1/2}+P_0^{{\omega }_1{\omega }_1}{q}^{-\frac{N-1}{2}}\text{.}& \text{(5.7)}\end{array}$$ Substituting the expressions (5.4)-(5.6) in this formula we obtain the representation of $R^{{\omega }_1{\omega }_1}$ as $N^2\times N^2$ matrix: $$\begin{array}{cc}\begin{array}{ccc}{R}^{{\omega }_1{\omega }_1}& =& {\displaystyle q^{1/2}\sum _{i\ne \frac{N-1}{2}}{E}_{ii}\otimes E_{ii}+E_{\frac{N+1}{2},\frac{N+1}{2}}\otimes E_{\frac{N+1}{2},\frac{N+1}{2}}}\\ & & {\displaystyle +\sum _{i\ne j,j\prime }{E}_{ji}\otimes E_{ij}+q^{-1/2}\sum _{i\ne i\prime }{E}_{ii\prime }\otimes E_{i\prime i}}\\ & & {\displaystyle -(q^{-1/2}-q^{1/2})\sum _{i>j}{E}_{jj}\otimes E_{ii}+(q^{-1/2}-q^{1/2})\sum _{i>j}{q}^{\frac{\bar{j}-\bar{i}}{2}}{E}_{i\prime j}\otimes E_{ij\prime },}\end{array}& \text{(5.8)}\end{array}$$ where $E_{ij}$ are the basic matrices in ${ℂ}^N$ and the second term is present only for $U_q\left(𝔰𝔬(2n+1)\right)\text{.}$

The matrix (5.8) can be extracted from the work [Jim1986-2]: $R^{{\omega }_1{\omega }_1}=\underset{x\to \infty }{\text{lim}}{x}^{-1/2}Ř\left(x\right),$ where $Ř\left(x\right)$ is the solution of Yang-Baxter equation corresponding to the series $B_n^{\left(1\right)}$ and $D_n^{\left(1\right)}\text{.}$

The following decompositions are well known: $$\begin{array}{cc}{\displaystyle V^{{\omega }_1}\otimes V^{\lambda }=\sum _{k=1}^{2n+1}\oplus V^{{\lambda }^{\left(k\right)}},𝔤=𝔰𝔬(2n+1),}& \text{(5.9)}\\ {\displaystyle V^{{\omega }_1}\otimes V^{\lambda }=\sum _{k=1}^{2n}\oplus V^{{\lambda }^{\left(k\right)}},𝔤=𝔰𝔬\left(2n\right)\text{.}}& \text{(5.10)}\end{array}$$ Here ${\lambda }^{\left(k\right)}=({\lambda }_1,\dots ,{\lambda }_k+1,\dots ,{\lambda }_n),$ ${\lambda }^{(n+1+k)}=({\lambda }_1,\dots ,{\lambda }_k-1,\dots ,{\lambda }_n),$ ${\lambda }^{(k+1)}=\lambda$ in (5.9) and ${\lambda }^{(n+k)}=({\lambda }_1,\dots ,{\lambda }_k-1,\dots ,{\lambda }_n)$ in (5.10), $1\le k\le n\text{.}$

Using the theorem 3.1 and the decompositions (5.9) (5.10) we obtain the formulae for projectors $P_{\lambda }(q,a)\text{.}$ For $P_{{\omega }_k}\left(q\right)$ $\text{(mult}({\omega }_k\subset {\left({\omega }_1\right)}^{\otimes k})=1\text{)}$ we have the following representation $$\begin{array}{cc}{\displaystyle P_{{\omega }_k}\left(q\right)=\sum _{n_1,\dots ,n_k=0}^c\prod _{i=1}^k{q}_{n_i,1}}\begin{array}{c} ω ⋯ ω ω ω α(n1,…,nk) \end{array}& \text{(5.11)}\end{array}$$ where $$\begin{array}{cc}Q={\left(q_{i,j}\right)}_{\underset{j=0,1,2}{i=0,1,2}}={\left(\begin{array}{ccc}1& 1& 1\\ q& q^{-k}& q^{-N+k}\\ q^2& q^{-2k}& q^{-2N+2k}\end{array}\right)}^{-1}& \text{(5.12)}\end{array}$$ and ${\alpha }_{(n_1,\dots ,n_k)}$ is defined by (3.19).

The algebras $C_M^{{\omega }_1}\left(𝔰𝔬(2n-1)\right)$ and $C_M^{{\omega }_1}\left(𝔰𝔬(2n+1)\right)$ are connected with the Birman-Wenzl algebra. The last one is an associative algebra formed by the generators $1,$ $G_i,$ $E_i,$ $i=1,\dots ,M-1$ with the following relations $$\begin{array}{cc}\begin{array}{ccc}{G}_i{G}_j& =& G_j{G}_i,|i-j|\ge 2\\ G_i{G}_{i+1}{G}_i& =& G_{i+1}{G}_i{G}_{i+1},\\ G_i+G_i^{-1}& =& m(1+E_i),\\ E_i{E}_{i+1}{E}_i& =& E_i=E_i{E}_{i-1}{E}_i\\ G_{i+1}{G}_i{E}_{i+1}& =& E_i{G}_{i+1}{G}_i=E_i{E}_{i+1}\\ G_{i-1}{G}_i{E}_{i-1}& =& E_i{G}_{i-1}{G}_i=E_i{E}_{i-1}\\ G_{i+1}{E}_i{G}_{i+1}& =& G_i^{-1}{E}_{i+1}{G}_i^{-1}\\ G_{i-1}{E}_i{G}_{i-1}& =& G_i^{-1}{E}_{i-1}{G}_i^{-1}\\ G_{i+1}{E}_i{E}_{i+1}& =& G_i^{-1}{E}_{i+1}\\ G_{i-1}{E}_i{E}_{i-1}& =& G_i^{-1}{E}_{i-1}\\ E_{i+1}{E}_i{G}_{i+1}& =& E_{i+1}{G}_i^{-1}\\ E_{i-1}{E}_i{G}_{i+1}& =& E_{i-1}{G}_i^{-1}\\ G_i{E}_i& =& E_i{G}_i={\ell }^{-1}{E}_i\\ E_i{G}_{i+1}{E}_i& =& E_i{G}_{i-1}{E}_i=\ell E_i\text{.}\end{array}& \text{(5.13)}\end{array}$$

Theorem 5.1. The algebras $C_M^{{\omega }_1}\left(𝔰𝔬\left(2n\right)\right)$ and $C_M^{{\omega }_1}\left(𝔰𝔬(2n+1)\right)$ are factors of Birman-Wenzl algebra with $m=i(q^{1/2}-q^{-1/2}),$ $\ell =-i{q}^{\frac{N-1}{2}}$ over the ideals formed by the elements $$J=P_{{\omega }_{n+1}}{\left(q\right)}^{1,\dots ,n+1}$$ acting nontrivially only in the multipliers of ${\left(V^{{\omega }_1}\right)}^{\otimes M}$ with numbers $i\le n+1\text{.}$ In other multipliers $J$ acts as a unit matrix. Here $P_{{\omega }_{n+1}}\left(q\right)$ is defined by (5.11), (5.12). The generators $G_i$ of B.W. algebra are connected with the generators of $C_M^{{\omega }_1}\left(𝔤\right)$ by the identification $G_j=i{g}_j,$ the generators $E_j$ are determined by (5.13). For $n+1>M,$ $C_M^{{\omega }_1}\simeq {BW}_M\text{.}$

Consequence. The representation $\alpha$ of B.W. algebra with $m=i(q^{1/2}-q^{-1/2}),$ $\ell =-i{q}^{\frac{N-1}{2}},$ $\alpha \left(G\right)=i(1\otimes \dots \otimes R^{{\omega }_1{\omega }_1}\otimes \dots \otimes 1)$ defined in the space ${\left(V^{{\omega }_1}\right)}^{\otimes M}\text{.}$

The structure of algebras $C_M^{{\omega }_1}\left(𝔰𝔬\left(2n\right)\right),$ $C_M^{{\omega }_1}\left(𝔰𝔬(2n+1)\right)$ and $C_M^{{\omega }_1}\left(𝔰𝔩\left(n\right)\right)$ is in agreement with the natural embeddings $𝔰𝔩\left(n\right)\subset 𝔰𝔬(2n+1),$ $𝔰𝔩\left(n\right)\subset 𝔰𝔬\left(2n\right)\text{.}$

Theorem 5.2. The algebras $U_q\left(𝔰𝔬\left(2n\right)\right)$ and $U_q\left(𝔰𝔬(2n+1)\right)$ contain $U_q\left(𝔰𝔩\left(n\right)\right)$ as a Hopf subalgebra.

This theorem follows from the commutation relations (1.1)-(1.3) and from the formulae (1.4) (1.5) for coproduct in $U_q\left(𝔤\right)\text{.}$

Let us consider the restriction of the vector representation of $U_q\left(𝔰𝔬\left(2n\right)\right)$ and $U_q\left(𝔰𝔬(2n+1)\right)$ on the subalgebra $U_q\left(𝔰𝔩\left(n\right)\right)\text{.}$ We have $$\begin{array}{ccc}{V}^{{\omega }_1}& =& {\stackrel{\sim }{V}}^{{\omega }_1}\oplus {\stackrel{\sim }{V}}^{{\omega }_1^{*}}\oplus V^0\text{for}{U}_q\left(𝔰𝔬(2n+1)\right)\\ V^{{\omega }_1}& =& {\stackrel{\sim }{V}}^{{\omega }_1}\oplus {\stackrel{\sim }{V}}^{{\omega }_1^{*}}\text{for}{U}_q\left(𝔰𝔬\left(2n\right)\right)\text{.}\end{array}$$ Here ${\stackrel{\sim }{V}}^{{\omega }_1}$ and ${\stackrel{\sim }{V}}^{{\omega }_1^{*}}$ are vector and conjugated to vector $\text{(}{\stackrel{\sim }{V}}^{{\omega }_1^{*}}={\stackrel{\sim }{V}}^{{\omega }_{n-1}}\text{)}$ representations of $U_q\left(𝔰𝔩\left(n\right)\right)\text{.}$ The $U_q\left(𝔰𝔩\left(n\right)\right)$ $R\text{-matrices}$ $R^{{\omega }_1{\omega }_1},$ $R^{{\omega }_1{\omega }_1^{*}},$ $R^{{\omega }_1^{*}{\omega }_1}$ and $R^{{\omega }_1^{*}{\omega }_1^{*}}$ in accordance with the notations of the section 2 we represent by the graphs: $$\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array},\begin{array}{c} \end{array}\text{.}$$ From the symmetries of $R\text{-matrices}$ we have $R^{{\omega }_1{\omega }_1}=R^{{\omega }_1^{*}{\omega }_1^{*}},$ $R^{{\omega }_1^{*}{\omega }_1}={R^{{\omega }_1^{*}{\omega }_1}}^T\text{.}$

Theorem 5.3. The basic $R\text{-matrices}$ of $U_q\left(𝔰𝔬\left(2n\right)\right)$ and $U_q\left(𝔰𝔬(2n+1)\right)$ have the following block structures at the restriction of $U_q\left(𝔰𝔬\left(2n\right)\right)$ and $U_q\left(𝔰𝔬(2n+1)\right)$ on $U_q\left(𝔰𝔩\left(n\right)\right)$ for $U_q\left(𝔰𝔬(2n+1)\right)\text{:}$ $$\begin{array}{cc}\begin{array}{c}\begin{array}{c} \end{array}=\begin{array}{|cccccccccc|}\hline & n\times n& n& n\times n& n& 1& n& n\times n& n& n\times n\\ n\times n& \begin{array}{c} \end{array}& 0& 0& 0& 0& 0& 0& 0& 0\\ n& 0& a\uparrow & 0& \uparrow & 0& 0& 0& 0& 0\\ n\times n& 0& 0& a(\begin{array}{c} \end{array}-q^{-\frac{n-3}{2}}\begin{array}{c} \end{array})& 0& a{q}^{-\frac{n-3}{4}}\begin{array}{c} \end{array}& 0& \begin{array}{c} \end{array}& 0& 0\\ n& 0& \uparrow & 0& 0& 0& 0& 0& 0& 0\\ 1& 0& 0& a{q}^{-\frac{n-3}{4}}\begin{array}{c} \end{array}& 0& 1& 0& 0& 0& 0\\ n& 0& 0& 0& 0& 0& a\downarrow & 0& \downarrow & 0\\ n\times n& 0& 0& \begin{array}{c} \end{array}& 0& 0& 0& 0& 0& 0\\ n& 0& 0& 0& 0& 0& \downarrow & 0& 0& 0\\ n\times n& 0& 0& 0& 0& 0& 0& 0& 0& \begin{array}{c} \end{array}\\ \hline\end{array}\\ a=(q^{-\frac{1}{2}}-q^{\frac{1}{2}})\end{array}& \text{(5.16)}\\ \begin{array}{c} \end{array}=\begin{array}{|cccccccccc|}\hline & n\times n& n& n\times n& n& 1& n& n\times n& n& n\times n\\ n\times n& \begin{array}{c} \end{array}& 0& 0& 0& 0& 0& 0& 0& 0\\ n& 0& 0& 0& \uparrow & 0& 0& 0& 0& 0\\ n\times n& 0& 0& 0& 0& 0& 0& \begin{array}{c} \end{array}& 0& 0\\ n& 0& \uparrow & 0& -a\uparrow & 0& 0& 0& 0& 0\\ 1& 0& 0& 0& 0& 1& 0& -a{q}^{\frac{n-3}{4}}\begin{array}{c} \end{array}& 0& 0\\ n& 0& 0& 0& 0& 0& 0& 0& \downarrow & 0\\ n\times n& 0& 0& 0& 0& -a{q}^{\frac{n-3}{4}}\begin{array}{c} \end{array}& 0& -a(\begin{array}{c} \end{array}-q^{\frac{n-3}{2}}\begin{array}{c} \end{array})& 0& 0\\ n& 0& 0& 0& 0& 0& \downarrow & 0& -a\downarrow & 0\\ n\times n& 0& 0& 0& 0& 0& 0& 0& 0& \begin{array}{c} \end{array}\\ \hline\end{array}& \text{(5.17)}\end{array}$$ for $U_q\left(𝔰𝔬\left(2n\right)\right)\text{:}$ $$\begin{array}{cc}\begin{array}{c} \end{array}=\begin{array}{|ccccc|}\hline & n\times n& n\times n& n\times n& n\times n\\ n\times n& \begin{array}{c} \end{array}& 0& 0& 0\\ n\times n& 0& a(\begin{array}{c} \end{array}-q^{-\frac{n-1}{2}}\begin{array}{c} \end{array})& \begin{array}{c} \end{array}& 0\\ n\times n& 0& \begin{array}{c} \end{array}& 0& 0\\ n\times n& 0& 0& 0& \begin{array}{c} \end{array}\\ \hline\end{array}& \text{(5.18)}\\ \begin{array}{c} \end{array}=\begin{array}{|cccc|}\hline \begin{array}{c} \end{array}& 0& 0& 0\\ 0& 0& \begin{array}{c} \end{array}& 0\\ 0& \begin{array}{c} \end{array}& -a(\begin{array}{c} \end{array}-q^{-\frac{n-1}{2}}\begin{array}{c} \end{array})& 0\\ 0& 0& 0& \begin{array}{c} \end{array}\\ \hline\end{array}& \text{(5.19)}\end{array}$$

This theorem follows from the formulae (5.8) and (4.6).

Corollary. There exist the following embeddings of the algebras $C_N^{{\omega }_1}\text{:}$ $$C_M^{{\omega }_1}\left(𝔰𝔬(2n+1)\right)\supset C_M^{{\omega }_1}\left(𝔰𝔩\left(n\right)\right)\times C_M^{{\omega }_1}\left(𝔰𝔩\left(n\right)\right)\text{.}$$

Using the theorem 5.3 and the action (4.11), (4.12) of $C_M^{{\omega }_1}\left(𝔰𝔩\left(n\right)\right)$ in $K_{\lambda }\text{-basis}$ in $W_{\lambda }$ one can find the action of $C_M^{{\omega }_1}\left(𝔰𝔬(2n+1)\right)$ and $C_M^{{\omega }_1}\left(𝔰𝔬\left(2n\right)\right)$ in $K_{\lambda }\text{-basis}$ in $W_{\lambda }\text{.}$ Explicit formulae will be given in a separate publication.

The spinor representations

The spinor representations $V^{{\omega }_n}$ and $V^{{\omega }_{n-1}}$ of $U_q\left(𝔰𝔬\left(2n\right)\right)$ each have a dimension $2^{n-1}\text{.}$ For $U_q\left(𝔰𝔬(2n+1)\right)$ spinor representation $V^{{\omega }_n}$ has dimension $2^n\text{.}$ The bases in spinor representations are parametrized by the weights $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_n),$ ${\epsilon }_i=\pm \frac{1}{2}\text{.}$ For $U_q\left(𝔰𝔬\left(2n\right)\right)$ there are auxiliary restrictions on $\epsilon \text{:}$ $\prod _{i=1}^n{\epsilon }_i=2^n$ for $V^{{\omega }_n}$ and $\prod _{i=1}^n{\epsilon }_i=-2^{-n}$ for $V^{{\omega }_{n-1}}\text{.}$ The spinor representations of $U_q\left(𝔰𝔬(2n+1)\right)$ and $U_q\left(𝔰𝔬\left(2n\right)\right)$ are defined by the action of the generators on the basic vectors $e_{\epsilon }\text{:}$ $$U_q\left(𝔰𝔬(2n+1)\right)\text{:}$$ $$\begin{array}{cc}\begin{array}{ccc}{X}_i^{+}{e}_{\cdots {\epsilon }_i{\epsilon }_{i+1}\cdots }& =& -e_{\cdots {\epsilon }_i+1,{\epsilon }_i-1,\cdots },\\ X_i{e}_{\cdots {\epsilon }_n}& =& \frac{1}{\sqrt{2\text{ch}\left(\frac{h}{4}\right)}}{e}_{\cdots {\epsilon }_n+1},\\ H_i{e}_{\epsilon }& =& ({\epsilon }_i-{\epsilon }_{i+1})e_{\epsilon },i=1,\dots ,n-1\\ H_n{e}_{\epsilon }& =& {\epsilon }_n{e}_{\epsilon }\\ X_i^{-}& =& {\left(X_i^{+}\right)}^t\text{.}\end{array}& \text{(5.20)}\end{array}$$ $$U_q\left(𝔰𝔬\left(2n\right)\right)\text{:}$$ $$\begin{array}{cc}\begin{array}{ccc}{X}_i^{+}{e}_{\cdots {\epsilon }_i{\epsilon }_{i+1}\cdots }& =& -e_{\cdots {\epsilon }_i+1,{\epsilon }_{i+1}-1,\cdots },\\ X_n^{+}{e}_{\cdots {\epsilon }_{n-1}{\epsilon }_n}& =& -e_{\cdots {\epsilon }_{n-1}+1,{\epsilon }_n+1,\dots },\\ H_i{e}_{\epsilon }& =& ({\epsilon }_i-{\epsilon }_{i+1})e_{\epsilon },\\ H_n{e}_{\epsilon }& =& ({\epsilon }_{n-1}+{\epsilon }_n)e_{\epsilon },\\ X_i^{-}& =& {\left(X_i^{+}\right)}^t\text{.}\end{array}& \text{(5.21)}\end{array}$$ Here r.h.s. $=0$ if the weight $\epsilon \prime$ of $e_{\epsilon \prime }$ does not belong to the set of weights of the space $V^{{\omega }_n}$ (or $V^{{\omega }_{n-1}}\text{).}$ The tensor product of spinor representations has the following spectral decomposition $$U_q\left(𝔰𝔬\left(2n\right)\right)\text{:}$$ $$\begin{array}{cccc}{V}^{{\omega }_{n-1}}\otimes V^{{\omega }_{n-1}}\simeq V^{{\omega }_n}\otimes V^{{\omega }_n}& =& \{\begin{array}{cc}{\displaystyle (\sum _{s=0}^{\frac{n}{2}-1}\oplus V^{{\omega }_{2s}})\oplus V^{2{\omega }_n},}& n\hspace{0.17em}\text{even}\\ {\displaystyle (\sum _{s=1}^{\frac{n-1}{2}}\oplus V^{{\omega }_{2s-1}})\oplus V^{2{\omega }_n},}& n\hspace{0.17em}\text{odd}\end{array}& \text{(5.22)}\\ V^{{\omega }_n}\otimes V^{{\omega }_{n-1}}& =& \{\begin{array}{cc}{\displaystyle (\sum _{s=1}^{\frac{n}{2}-1}\oplus V^{{\omega }_{2s-1}})\oplus V^{{\omega }_n+{\omega }_{n-1}},}& n\hspace{0.17em}\text{even}\\ {\displaystyle (\sum _{s=0}^{\frac{n}{2}-2}\oplus V^{{\omega }_{2s}})\oplus V^{{\omega }_n+{\omega }_{n-1}},}& n\hspace{0.17em}\text{odd}\end{array}& \text{(5.23)}\\ V^{{\omega }_1}\otimes V^{{\omega }_n}& =& \{\begin{array}{cc}{\displaystyle V^{{\omega }_{n-1}}\oplus V^{{\omega }_n+{\omega }_1},}& n\hspace{0.17em}\text{even}\\ {\displaystyle V^{{\omega }_n}\oplus V^{{\omega }_n+\omega },}& n\hspace{0.17em}\text{odd}\end{array}& \text{(5.24)}\\ V^{{\omega }_1}\otimes V^{{\omega }_{n-1}}& =& \{\begin{array}{cc}{V}^{{\omega }_n}\oplus V^{{\omega }_{n-1}+{\omega }_1},& n\hspace{0.17em}\text{even}\\ V^{{\omega }_{n-1}}\oplus V^{{\omega }_{n-1}+{\omega }_1},& n\hspace{0.17em}\text{odd}\end{array}& \text{(5.25)}\end{array}$$ $$U_q\left(𝔰𝔬\left(2n\right)\right)\text{:}$$ $$\begin{array}{cc}\begin{array}{ccc}{V}^{{\omega }_n}\otimes V^{{\omega }_n}& =& {\displaystyle (\sum _{s=0}^{n-1}\oplus V^{{\omega }_s})\oplus V^{2{\omega }_n}}\\ V^{{\omega }_1}\otimes V^{{\omega }_n}& =& V^{{\omega }_n}\oplus V^{{\omega }_1+{\omega }_n}\text{.}\end{array}& \text{(5.26)}\end{array}$$

Let us describe the KFC of these decompositions. As in the previous part of this section we shall use the graphical representation of the matrices $R^{\lambda \mu }$ and $K_{\nu }^{\lambda \mu }$ given in section 2.

It is convenient to consider the space $V^s=V^{{\omega }_n}\oplus V^{{\omega }_{n-1}}$ for $U_q\left(𝔰𝔬\left(2n\right)\right)$ and the matrix $$K_s^{s{\omega }_1}=\left(\begin{array}{cc}{K}_{{\omega }_n}^{{\omega }_n{\omega }_1}& K_{{\omega }_{n-1}}^{{\omega }_n{\omega }_1}\\ K_{{\omega }_n}^{{\omega }_{n-1}{\omega }_1}& K_{{\omega }_{n-1}}^{{\omega }_{n-1}{\omega }_1}\end{array}\right)$$ For $U_q\left(𝔰𝔬(2n+1)\right)$ we shall write $V^s=V^{{\omega }_n}\text{.}$

Theorem 5.5. The matrices $K^{{\omega }_1}$ for $U_q\left(𝔰𝔬\left(2n\right)\right)$ and for $U_q\left(𝔰𝔬(2n+1)\right)$ satisfy the following relations: $$\begin{array}{cc}\begin{array}{c} s s s ω1 ω1 \end{array}=q^{-1/2}\begin{array}{c} s s s ω1 ω1 \end{array}+(1+q^{-\frac{N-2}{2}})\begin{array}{c} s s ω1 ω1 \end{array}& \text{(5.27)}\end{array}$$

The proof of this theorem follows from theorem 3.1.

It is interesting that the relations $\left(\right)$ between the matrix elements of $K_{{\omega }_n}^{{\omega }_n{\omega }_1}$ and $K_s^{s{\omega }_1}$ permit us to introduce $q\text{-analog}$ of Clifford algebra. If we write $$\begin{array}{cc}{\left({\gamma }_i\right)}_{\alpha \beta }=\begin{array}{c} s s ω1 α β i \end{array}& \text{(5.28)}\end{array}$$ the relations (5.27) give the $q\text{-anticommutation}$ relations for the matrices ${\gamma }_i\text{:}$ $$\begin{array}{cc}\begin{array}{ccc}{\gamma }_i{\gamma }_j& =& -q^{-i/2}{\gamma }_j{\gamma }_i,i<j,i\ne j\prime \\ {\gamma }_i^2& =& 0\\ {\gamma }_i{\gamma }_{i\prime }& =& -{\gamma }_{i\prime }{\gamma }_i-(q^{-1}-1)\sum _{k>i}{q}^{\frac{\bar{i}-\bar{k}}{2}}{\gamma }_{k\prime }{\gamma }_k+(1+q^{\frac{N-2}{2}})q^{\frac{\bar{i}}{2}}\end{array}& \text{(5.29)}\end{array}$$ with normalization condition $$\sum _i{\gamma }_i{\gamma }_{i\prime }={\chi }_{{\omega }_1}\text{.}$$ Using explicit form of the spinorial and vector representations the matrices $K_s^{s{\omega }_1}$ can be calculated exactly. But we will not present here these calculations because they take up too much space and are not very instructive for out purposes.

The matrices $K_{{\omega }_1}^{ss}$ are the crossing transformations of the matrices $K_s^{s{\omega }_1}$ (see (2.20)). Using the formulae of the section 5.1 and (5.27) we obtain an expression for the projector $P_{{\omega }_k}^{ss}\text{:}$ $$\begin{array}{cc}\begin{array}{c} s s* ωk s* s \end{array}=N_k^s\begin{array}{c} ⋯ ⋯ s s s* s* ω1 ω1 ω1 ω1 ω1 ω1 ωk \end{array}& \text{(5.30)}\end{array}$$ where $N_k^s$ are some normalisation constants and projector $P:{\left(V^{{\omega }_1}\right)}^{\otimes k}\to V^{{\omega }_k}$ was given by (5.11).

So, we have the expression for spinor-spinorial $R\text{-matrices}$ through the matrices $K_{{\omega }_1}^{ss}$ and $R^{{\omega }_1{\omega }_1}\text{:}$ $$\begin{array}{cccc}\begin{array}{c} ωn ωn \end{array}& =& {\displaystyle \sum _{s=0}^{n-1}{q}^{\frac{1}{4}((2n+1)k-k^2-n^2-\frac{n}{2})}}\begin{array}{c} ωn ωn ωk ωn ωn \end{array}+q^{\frac{n}{8}}\begin{array}{c} ωn ωn 2ωk ωn ωn \end{array}& \text{(5.31)}\\ \begin{array}{c} ω(n-1n) ω(n-1n) \end{array}& =& \{\begin{array}{cc}{\displaystyle \sum _{s=0}^{\frac{n}{2}-1}{(-1)}^{\cdots }{q}^{(ns-s^2-\frac{n^2}{4}+\frac{n}{8})}}\begin{array}{c} ω(n-1n) ω(n-1n) ω2s ω(n-1n) ω(n-1n) \end{array}+q^{\frac{n}{8}}\begin{array}{c} ω(n-1n) ω(n-1n) 2ω(n-1n) ω(n-1n) ω(n-1n) \end{array},& n\hspace{0.17em}\text{even}\\ {\displaystyle \sum _{s=1}^{\frac{n-1}{2}}{(-1)}^{\cdots }{q}^{(ns-s^2-\frac{{(n+1)}^2}{4}+\frac{n}{8})}}\begin{array}{c} ωn-1 ωn-1 ω2s-1 ωn-1 ωn-1 \end{array}+q^{\frac{n}{8}}\begin{array}{c} ωn-1 ωn-1 2ωn-1 ωn-1 ωn-1 \end{array},& n\hspace{0.17em}\text{odd}\end{array}& \text{(5.32)}\\ \begin{array}{c} ωn ωn-1 \end{array}& =& \{\begin{array}{cc}{\displaystyle \sum _{s=1}^{\frac{n}{2}-1}(-1)q^{(ns-s^2-\frac{{(n+1)}^2}{4}+\frac{n}{8})}}\begin{array}{c} ωn ωn-1 ω2s-1 ωn-1 ωn \end{array}+q^{\frac{n}{8}-\frac{1}{4}}\begin{array}{c} ωn ωn-1 ωn+ωn-1 ωn-1 ωn \end{array},& n\hspace{0.17em}\text{even}\\ {\displaystyle \sum _{s=0}^{\frac{n}{2}-2}(-1)q^{(ns-s^2-\frac{n^2}{4}+\frac{n}{8})}}\begin{array}{c} ωn ωn-1 ω2s ωn-1 ωn \end{array}+q^{\frac{n}{8}-\frac{1}{4}}\begin{array}{c} ωn ωn-1 ωn+ωn-1 ωn-1 ωn \end{array}& n\hspace{0.17em}\text{odd}\end{array}& \text{(5.33)}\\ \begin{array}{c} ω1 ω(n-1n) \end{array}& =& \{\begin{array}{cc}q\begin{array}{c} ω1 ω(n-1n) ω1+ω(n-1n) ω(n-1n) ω1 \end{array}-q^{-\frac{2n-1}{4}}\begin{array}{c} ω1 ω(n-1n) ω(nn-1) ω(n-1n) ω1 \end{array},& n\hspace{0.17em}\text{even}\\ q\begin{array}{c}] ω1 ω(n-1n) ω1+ω(n-1n) ω(n-1n) ω1 \end{array}-q^{-\frac{2n-1}{4}}\begin{array}{c} ω1 ω(n-1n) ω(n-1n) ω(n-1n) ω1 \end{array},& n\hspace{0.17em}\text{odd.}\end{array}& \text{(5.34)}\end{array}$$ Here $P_{{\omega }_k}^{{\omega }_s{\omega }_s}$ are given by (5.30).

Using the commutation relations (5.27) it is easy to prove the following identities: $$\begin{array}{cc}\begin{array}{c}\begin{array}{ccc}\begin{array}{c} ω1 ω1 ⋯ ω1 ω1 ωs ⏞ 2n \end{array}& =& {\displaystyle \frac{{\left(q{q}^{\frac{1}{2}}\right)}^n}{(1+q^{n-\frac{1}{2}})\dots (1+q^{\frac{1}{2}})}\sum _{k_1=1}^{2n-1}\sum _{k_2\ne k_1,{\stackrel{\sim }{k}}_1}}\\ & & {\displaystyle \cdots \sum _{k_i\ne k_1,{\stackrel{\sim }{k}}_1,\dots ,k_{i-1},{\stackrel{\sim }{k}}_{i-1}}\cdots {(-q)}^{\frac{1}{2}(k_1+\dots +k_n)}\begin{array}{c} ω1 ω1 ⋯ ω1 ω1 [k1,…,kn] ⏞ 2n \end{array}}\\ {\stackrel{\sim }{k}}_i& =& \underset{j}{\text{max}}\{j\in (1,\dots ,2n)\backslash (k_1,{\stackrel{\sim }{k}}_1,\dots ,k_{i-1},{\stackrel{\sim }{k}}_{i-1})\}\text{.}\end{array}\end{array}& \text{(5.35)}\end{array}$$ Here $\left[k_1\dots k_n\right]$ is the joint in which the line numbered (the numeration being from left to right) is connected with the ${\stackrel{\sim }{k}}_i\text{-line}$ in such a way that the $k_i\text{-line}$ goes above $k_j\text{-line}$ if $i<j\text{.}$

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