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 solutions of Yang-Baxter equation connected with the $q\text{-deformations}$ of universal enveloping algebras of simple Lie algebras

The $q\text{-deformation}$ of the universal enveloping algebras (or quantum universal enveloping algebras QUEA) $U_q\left(𝔤\right)$ arose when studying some special class of integrable systems. The definition of $U_q\left(𝔤\right)$ for any simple Lie algebra $𝔤$ was given in [Dri1985,Dri1986-3,Jim1985]. One can define these algebras by generators and relations.

Definition 1.1 [Dri1985,Dri1986-3,Jim1985]. QUEA $U_q\left(𝔤\right)$ is the algebra with generators $X_i^{\pm },$ $H_i,$ $i=1,\dots ,r=\text{rank}\hspace{0.17em}𝔤$ and with the relations: $$\begin{array}{cc}{\displaystyle [H_i,H_j]=0,[H_j,X_i^{\pm }]=\pm \left({\alpha }_j{\alpha }_i\right)X_i^{\pm }}& \text{(1.1)}\\ {\displaystyle [X_i^{+},X_j^{-}]={\delta }_{ij}\frac{\text{sh}\left(\frac{h}{2}{H}_i\right)}{\text{sh}\left(\frac{h}{2}\right)},q=e^h}& \text{(1.2)}\\ {\displaystyle i\ne j,\hspace{0.17em}\sum _{k=0}^n{(-1)}^k{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}_{q_i}{q}_i^{-\frac{k(n-k)}{2}}{\left(X_i^{\pm }\right)}^k{X}_j^{\pm }{\left(X_i^{\pm }\right)}^{n-k}=0,q_i=q^{\frac{\left({\alpha }_i{\alpha }_i\right)}{2}}}& \text{(1.3)}\end{array}$$ where $A_{ij}$ is the Cartan matrix of $𝔤,$ $(\alpha ,\beta )$ is the scalar product of the roots $\text{(}{A}_{ij}=\left({\alpha }_i{\alpha }_j\right)\left({\alpha }_j{\alpha }_j\right)\text{)},$ $n=1-A_{ij},$ $i\ne j\text{.}$

The generators $X_i^{\pm },$ $H_i$ play the role of Chevalley basis in $U\left(𝔤\right)\text{.}$ The elements $H_i$ form the commutative subalgebra $U\left(f\right)\subset U_q\left(𝔤\right)\text{.}$ This subalgebra is the analog of Cartan subalgebra in $U\left(𝔤\right)\text{.}$ The elements $H_i$ correspond to the simple roots of $𝔤\text{.}$ Let $\alpha =\sum _{i=1}^r{n}_i{\alpha }_i$ be the root of $𝔤,$ a corresponding element of $f$ we denote as $H_{\alpha }={\sum }_{i=1}^r{n}_i{H}_i\text{.}$

Theorem 1.1 [Dri1985,Dri1986-3,Jim1985]. The algebra $U_q\left(𝔤\right)$ is the Hopf algebra [Abe1980] (see also Appendix A) with comultiplication and with the antipode $\gamma$ defined on the generators by the following formulae: $$\begin{array}{cc}\mathrm{\Delta }\left(H_i\right)=H_i\otimes 1+1\otimes H_i,& \text{(1.4)}\\ \mathrm{\Delta }\left(X_i^{\pm }\right)=X_i^{\pm }\otimes e^{\frac{h}{4}{H}_i}+e^{-\frac{h}{4}{H}_i}\otimes X_i^{\pm },& \text{(1.5)}\\ \gamma \left(H_i\right)=-H_i,\gamma \left(X_i^{\pm }\right)=-e^{\frac{h}{2}\rho }{X}_i^{\pm }{e}^{-\frac{h}{2}\rho },& \text{(1.6)}\end{array}$$ where $\rho =\frac{1}{2}\sum _{\alpha \in {\Delta }_{+}}{H}_{\alpha }$ is the element of $f$ and ${\Delta }_{+}$ is the positive roots of $𝔤\text{.}$

The parameter $q$ is the deformation parameter. At $q=1$ the algebra $U_q\left(𝔤\right)$ is the universal enveloping algebra of $𝔤\text{.}$ For simple Lie algebras of the algebras $U_q\left(𝔤\right)$ are simple for general $q\text{.}$ The special values of $q$ are situated in the rational points on the unit circle.

Let ${𝔟}_{\pm }\subset 𝔤$ be the Borel subalgebras in $𝔤\text{.}$ The algebras $U_q\left({𝔟}_{+}\right)$ and $U_q\left({𝔟}_{-}\right)$ with generators $(H_i,X_i^{+})$ and $(H_i,X_i^{-})$ are the Hopf subalgebras in $U_q\left(𝔤\right)\text{.}$ There is a duality between $U_q\left({𝔟}_{+}\right)$ and $U_q\left({𝔟}_{-}\right)\text{.}$ Let $\left\{e_s\right\}$ is the basis in $U_q\left({𝔟}_{-}\right)$ and $\left\{e^s\right\}$ be the dual basis in $U_q\left({𝔟}_{+}\right)\text{.}$

It is not difficult to prove that the map $\sigma \circ \mathrm{\Delta }:U_q\left(𝔤\right)\to U_q\left(𝔤\right)\otimes U_q\left(𝔤\right),$ where $\sigma$ is the permutation in $U_q\left(𝔤\right)\otimes U_q\left(𝔤\right)$ also defines the structure of Hopf algebra on $U_q\left(𝔤\right)$ with antipode $\gamma \prime ={\gamma }^{-1}\text{.}$

Theorem 1.2 [Dri1985,Dri1986-3]. The comultiplication $\mathrm{\Delta }$ and $\sigma \circ \mathrm{\Delta }$ are connected by the following relation $$\begin{array}{cc}\sigma \circ \mathrm{\Delta }\left(a\right)=R\mathrm{\Delta }\left(a\right)R^{-1}\forall a\in U_q\left(𝔤\right)& \text{(1.7)}\end{array}$$ where $\sigma$ is the permutation operator in $U_q\left(𝔤\right)\otimes U_q\left(𝔤\right)$ and $R=\sum _s{e}_s\otimes e^s\in U_q\left(𝔤\right)\otimes U_q\left(𝔤\right)\text{.}$

The proof of this theorem the following one is based on the construction of double Hopf algebra [Jim1985].

Theorem 1.3. The element $R$ satisfies the relations $$\begin{array}{cc}(\mathrm{\Delta }\otimes \text{id})R=R_{13}{R}_{23},& \text{(1.8a)}\\ (\text{id}\otimes \mathrm{\Delta })R=R_{13}{R}_{12},& \text{(1.8b)}\\ R_{12}{R}_{13}{R}_{23}=R_{23}{R}_{13}{R}_{12},& \text{(1.8c)}\\ (\text{id}\otimes \gamma )R=R^{-1}\text{.}& \text{(1.8d)}\end{array}$$ Here $R_{12},R_{13},R_{23}\in U_q{\left(𝔤\right)}^{\otimes 3},$ $R_{12}=e_s\otimes e^s\otimes 1,$ $R_{13}=e_s\otimes 1\otimes e^s,$ $R_{23}=1\otimes e_s\otimes e^s\text{.}$

		Proof.
	Let $m,\mu$ and $\gamma$ be the matrices of multiplication, comultiplication and antipode in algebra $U_q\left({𝔟}_{-}\right)\text{:}$ $$e_s{e}_t=m_{st}^{\tau }{e}_{\tau },\mathrm{\Delta }\left(e_s\right)={\mu }_s^{t\tau }{e}_t\otimes e_{\tau },\gamma \left(e_s\right)={\gamma }_s^t{e}_t\text{.}$$ The matrices $\mu ,m\prime$ and ${\gamma }^{-1}$ are the matrices of multiplication, comultiplication and antipode in $U_q\left({𝔟}_{+}\right)\text{:}$ $$e^s{e}^t={\mu }_{\tau }^{st}{e}^{\tau },\mathrm{\Delta }\left(e^s\right)=m_{t\tau }^s{e}^{\tau }\otimes e^t,\gamma \left(e^s\right)={\left({\gamma }^{-1}\right)}_t^s{e}^t\text{.}$$ From (1.7) we have the commutation relations between the elements $e^s$ and $e_t$ in $U_q\left(𝔤\right)\text{:}$ $${\mu }_s^{pq}{m}_{tp}^{\tau }{e}^t{e}_q={\mu }_s^{pq}{m}_{qt}^{\tau }{e}_p{e}^t\text{.}$$ From this commutation relations and from the definition of $R$ follows (1.8c). To prove (1.8d) we use the definition of antipode (A.): $$R(\text{id}\otimes \gamma )R=e_s{e}^t\otimes e^s\gamma \left(e^t\right)=m_{st}^{\tau }{e}_{\tau }\otimes {\gamma }_p^t{\mu }_q^{sp}{e}^q=m_{st}^{\tau }{\gamma }_p^t{\mu }_q^{sp}{e}_{\tau }\otimes e^q=e\otimes e\text{.}$$ Similarly $$(\text{id}\otimes \gamma )\left(R\right)·R=m_{st}^{\tau }{\gamma }_p^s{\mu }_q^{pt}{e}_{\tau }\otimes e^q=e\otimes e\text{.}$$ The following formulae prove the relations (1.8a) and (1.8b) $$\begin{array}{c}(\mathrm{\Delta }\otimes \text{id})R=\mathrm{\Delta }\left(e_s\right)\otimes e^s={\mu }_s^{\tau t}{e}_{\tau }\otimes e_t\otimes e^s=e_{\tau }\otimes e_t\otimes e^{\tau }{e}^t=R_{13}{R}_{23},\\ (\text{id}\otimes \mathrm{\Delta })R=e_s\otimes \mathrm{\Delta }\left(e^s\right)=e_s\otimes m_{\tau t}^s{e}^t\otimes e^{\tau }=e_{\tau }{e}_t\otimes e^t\otimes e^{\tau }=R_{13}{R}_{12}\text{.}\end{array}$$ The theorem is proved. $\square$

The algebras $U_q\left(𝔤\right)$ and $𝔤$ have a common Cartan subalgebra $f$ generated by the elements $H_i\text{.}$ This means that the theories of the finite dimensional representations of $U_q\left(𝔤\right)$ and $𝔤$ are quite parallel. The following two statements will play the main role in the text.

Proposition 1.1. The finite dimensional representations of $U_q\left(𝔤\right)$ are completely irreducible.

Proposition 1.2.

(a)	The irreducible finlte-dimensional representations (IFR) of $U_q\left(𝔤\right)$ are parametrized by the highest weights $\lambda$ of the algebra $𝔤\text{.}$
(b)	Any irreducible finite dimensional module $V^{\lambda }$ is decomposed into the sum of the weight spaces $V^{\lambda }=\underset{\mu }{⨁}{V}^{\lambda }\left(\mu \right)$ and the dimensions of $V^{\lambda }\left(\mu \right)$ are the same as for IFR of $𝔤\text{.}$

From the complete irreducibility of finite dimensional representations of $U_q\left(𝔤\right)$ it follows that any IFR can be considered as some irreducible component of corresponding tensorial power of basic representations. The list of basic representation is given in table 1. $$\begin{array}{c}\begin{array}{cccccccccc}𝔤\hspace{0.17em}\text{category}& A_n& B_n& C_n& D_n& G_2& E_6& E_7& E_8& F_4\\ \text{finite dimensional}& {\omega }_1& {\omega }_1& {\omega }_1& {\omega }_1& {\omega }_1& {\omega }_1& {\omega }_7& {\omega }_8& {\omega }_4\\ \text{tensorial represent}& -& {\omega }_n& -& {\omega }_1,{\omega }_n& -& -& -& -& -\end{array}\\ \text{Table 1}\end{array}$$

For algebras $B_n$ and $D_n$ it is natural to separate from the finite dimensional representations the category of tensorial representations (with integers highest weights). In this category the basic representation is vectorial representation with h.w. ${\omega }_1$ in both cases.

It is very important for our purposes that all finite dimensional representations of $U_q\left(𝔤\right)$ are contained in tensorial powers of the basic representations.

Below we study the restrictions of universal $R\text{-matrices}$ on the IFR's. If ${\rho }_{\lambda }$ and ${\rho }_{\mu }$ are IFR of $U_q\left(𝔤\right)$ and $P^{\lambda \mu }$ is the permutation operator on $V^{\lambda }\otimes V^{\mu }$ $\text{(}{P}^{\lambda \mu }:V^{\lambda }\otimes V^{\mu }\to V^{\mu }\otimes V^{\lambda },$ $P^{\lambda \mu }(f\otimes g)=g\otimes f\text{)}$ then we denote the matrix $P^{\lambda \mu }({\rho }_{\lambda }\otimes {\rho }_{\mu })\left(R\right)$ as $R^{\lambda \mu }$ and we call it the $R\text{-matrix}$ in $\lambda ,\mu$ representation (or $\lambda ,\mu$ $R\text{-matrix).}$ This matrix map $V^{\lambda }\otimes V^{\mu }$ on $V^{\mu }\otimes V^{\lambda }$ and satisfies the following relations: $$\begin{array}{cc}(R^{\lambda \mu }\otimes I)(I\otimes R^{\nu \mu })(R^{\nu \lambda }\otimes I)=(I\otimes R^{\nu \lambda })(R^{\nu \mu }\otimes I)(I\otimes R^{\lambda \mu })\text{.}& \text{(1.9)}\end{array}$$ The left and the right sides of these relations act from $V^{\nu }\otimes V^{\lambda }\otimes V^{\mu }$ to $V^{\mu }\otimes V^{\lambda }\otimes V^{\nu }\text{.}$

To find the matrices $R^{\lambda \mu }$ one can use two ways. The first one is the explicit projection of the universal $R\text{-matrix}$ by factorising both copies of $U_q\left(𝔤\right)$ over corresponding ideals. The second way uses the complete irreducibility of IFR of $U_q\left(𝔤\right)\text{.}$ Because all of IFR are contained in tensorial powers of basic representations of $U_q\left(𝔤\right)$ one can construct the matrices $R^{\lambda \mu }$ from a tensorial product of the $R\text{-matrices}$ acting in basic representations. The second way gives also the important relations between the matrices $R^{\lambda \mu }$ and we will use this way below.

Consider a Cartan antiinvolution in $U_q\left(𝔤\right)$ $$\theta \left(X_i^{\pm }\right)=X_i^{\mp },\theta \left(H_i\right)=H_i$$ and automorphism $\delta$ corresponding to on authomorphism of Dynkin diagram: $$\delta \left(X_i^{\pm }\right)=X_{i^{*}}^{\pm },\delta \left(H_i\right)=H_{i^{*}}\text{.}$$

We define a $q\text{-analog}$ $C$ of the element with the maximal length of the Weyl group by the formula $$\gamma =\delta \circ {\text{Ad}}_C\otimes \theta \text{.}$$

It is not difficult to check that in basic representations the element $C$ has the form $$C=S{q}^{\rho /2}$$ where $S$ is the corresponding element of the Weyl group for $U\left(𝔤\right)\text{.}$

(b) The element $R$ satisfies the relation $$(\text{id}\otimes \delta )\left(R^{-1}\right)=(1\otimes C)(\text{id}\otimes \theta )\left(R\right)(1\otimes C^{-1})\text{.}$$

Definition 1.2. If $V^{\nu }\subset V^{\lambda }\otimes V^{\mu },$ the projection matrix $K_{\nu }^{\lambda \mu }(q,a):V^{\lambda }\otimes V^{\mu }\to V^{\nu }$ is called the matrix of Klebsch-Gordan coefficient (KGC) (here index $a$ numbers the components $V^{\nu }$ in $V^{\lambda }\otimes V^{\mu }$ if the multiplicity of $V^{\nu }$ is more than one).

The matrices $K_{\nu }^{\lambda \mu }$ and $K_{\nu }^{\mu \lambda }$ are orthogonal if $\nu \ne \nu \prime \text{.}$ They are normalized when $\text{mult}\left(V^{\nu }\right)=\text{mult}\left(V^{\nu }\right)=1$ $$\begin{array}{cc}{K}_{\nu }^{\lambda \mu }{\left(K_{\nu \prime }^{\mu \lambda }\right)}^{\top }={\delta }_{\nu \nu \prime }{I}_{\nu }\text{.}& \text{(1.14)}\end{array}$$

Theorem 1.4. The matrices $R^{\lambda \mu }$ and $K_{\nu }^{\lambda \mu }$ for $\text{mult}\left(V^{\nu }\right)=1$ satisfy the following relations $$\begin{array}{cccc}{P}^{\mu \lambda }{R}_{\left(q\right)}^{\lambda \mu }{P}^{\mu \lambda }& =& S\otimes S{R}_{\left(q\right)}^{\mu \lambda }{s}^{-1}\otimes s^{-1}=R^{\lambda \mu }{\left(q^{-1}\right)}^{-1},& \text{(1.15)}\\ C^{-1}{K}_{\nu }^{\lambda \mu }\left(q\right)(C\otimes C)& =& {(-1)}^{\bar{\nu }}{K}_{{\nu }^{*}}^{{\lambda }^{*}{\mu }^{*}}\left(q^{-1}\right),& \text{(1.16)}\\ K_{\nu }^{\lambda \mu }\left(q\right)P^{\mu \lambda }& =& {(-1)}^{\bar{\nu }}{K}_{\nu }^{\mu \lambda }\left(q^{-1}\right),& \text{(1.17)}\\ R^{\lambda \mu }{\left(q\right)}^{\top }& =& R^{\lambda \mu }\left(q\right)\text{.}& \text{(1.18)}\end{array}$$

Here $\bar{\nu }=0,1$ is the parity of $V^{\nu }$ in $V^{\lambda }\otimes V^{\mu }$ and $\top$ is the transposition.

Following theorems are the keys ones to study the matrices $R^{\lambda \mu }\text{.}$

Theorem 1.5. Let $V^{\nu }\subset V^{\lambda }\otimes V^{\mu }$ and $\text{mult}\left(V^{\nu }\right)=1$ then $$\begin{array}{cccc}{R}^{\lambda \mu }\left(q\right){\left(K_{\nu }^{\lambda \mu }\left(q\right)\right)}^{\top }& =& {(-1)}^{\bar{\nu }}{q}^{\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}}{K}_{\nu }^{\mu \lambda }{\left(q\right)}^{\top }& \text{(1.19)}\\ K_{\nu }^{\mu \lambda }\left(q\right)R^{\lambda \mu }\left(q\right)& =& {(-1)}^{\bar{\nu }}{q}^{\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}}{K}_{\nu }^{\lambda \mu }\left(q\right)& \text{(1.20)}\end{array}$$ where $\bar{\nu }$ is the parity of $V^{\nu }$ in $V^{\lambda }\otimes V^{\mu }$ and $c\left(\nu \right)$ is the value of Casimir operator of $𝔤$ on the irreducible representation with highest weight $\nu \text{:}$ $$\begin{array}{cc}{\displaystyle c=\sum _{i=1}^r{H}^{i^2}+\sum _{\alpha \in \mathrm{\Delta }}{X}_{\alpha }{X}_{-\alpha },H^i=(H,{\omega }^i)}& \text{(1.21)}\\ c\left(\nu \right)={\nu }^2+2\rho \nu \text{.}& \text{(1.22)}\end{array}$$

		Proof.
	From the definition of universal $R\text{-matrix}$ follows the equality $$\begin{array}{cc}{R}^{\lambda \mu }{\mathrm{\Delta }}^{\lambda \mu }\left(a\right)={\mathrm{\Delta }}^{\mu \lambda }\left(a\right)R^{\lambda \mu }& \text{(1.23)}\end{array}$$ where ${\mathrm{\Delta }}^{\lambda \mu }\left(a\right)=({\rho }^{\lambda }\otimes {\rho }^{\mu })\mathrm{\Delta }\left(a\right)\text{.}$ The representation ${\mathrm{\Delta }}^{\lambda \mu }$ is reducible and $K_{\nu }^{\lambda \mu }$ are the projectors on the irreducible components: $$\begin{array}{cccc}{\mathrm{\Delta }}^{\lambda \mu }\left(a\right){K_{\nu }^{\lambda \mu }}^{\top }& =& {K_{\nu }^{\lambda \mu }}^{\top }{\rho }^{\nu }\left(a\right)& \text{(1.24)}\\ {\mathrm{\Delta }}^{\mu \lambda }\left(a\right){K_{\nu }^{\mu \lambda }}^{\top }& =& {K_{\nu }^{\mu \lambda }}^{\top }{\rho }^{\nu }\left(a\right)\text{.}& \text{(1.25)}\end{array}$$ From (1.23) and (1.24) we have $$\begin{array}{cc}{R}^{\lambda \mu }{K_{\nu }^{\lambda \mu }}^{\top }{\rho }_{\nu }\left(a\right)={\mathrm{\Delta }}^{\mu \lambda }\left(a\right)R^{\lambda \mu }{K_{\nu }^{\lambda \mu }}^{\top }\text{.}& \text{(1.26)}\end{array}$$ Comparing with (1.25) we obtain $$\begin{array}{cc}{R}^{\lambda \mu }{K_{\nu }^{\lambda \mu }}^{\top }=R_{\nu }\left(q\right){K_{\nu }^{\mu \lambda }}^{\top }& \text{(1.27)}\end{array}$$ where $R_{\nu }\left(q\right)$ is some function of $q,\lambda ,\mu ,\nu \text{.}$ To find $R_{\nu }\left(q\right)$ we consider the limit $q\to 1\text{.}$ Let ${\phi }_{\nu }^{\lambda \mu }\left(q\right)$ be the highest weight vector in $V^{\nu }\subset V^{\lambda }\otimes V^{\mu }\text{.}$ From (1.27) follows the equality $$\begin{array}{cc}{R}^{\lambda \mu }\left(q\right){\phi }_{\nu }^{\lambda \mu }\left(q\right)=R_{\nu }\left(q\right){\phi }_{\nu }^{\mu \lambda }\left(q\right)\text{.}& \text{(1.28)}\end{array}$$ At $q\to 1$ we have $$\begin{array}{cccc}{R}^{\lambda \mu }& =& P^{\lambda \mu }(1+(q-1)\tau +O\left({(q-1)}^2\right)),& \text{(1.29)}\\ \tau & =& {\displaystyle \frac{1}{2}\sum _i{H}^i\otimes H^i+\sum _{\alpha \in {\Delta }_{+}}{X}_{\alpha }\otimes X_{-\alpha }}& \text{(1.30)}\\ {\phi }_{\nu }^{\lambda \mu }\left(q\right)& =& {\phi }^{\lambda \mu }+(q-1){\psi }_{\nu }^{\lambda \mu }+O\left({(q-1)}^2\right),& \text{(1.31)}\\ R_{\nu }\left(q\right)& =& {(-1)}^{\bar{\nu }}(1+(q-1)s_{\nu }+O\left({(q-1)}^2\right))\text{.}& \text{(1.32)}\end{array}$$ Here $\tau \in 𝔤\otimes 𝔤$ $\text{(}𝔤\hookrightarrow U\left(𝔤\right)\text{)}$ is the so-called classical $\tau \text{-matrix}$ [Bax1982,Fad1984]. ${\phi }_{\nu }^{\lambda \mu }$ is the h.w.v. of $𝔤,$ and ${\psi }_{\nu }^{\lambda \mu }$ and $s_{\nu }$ some unknown vectors and constants. Theorem 1.4 implies symmetry relations for ${\phi }_{\nu }^{\lambda \mu }$ and ${\psi }_{\nu }^{\lambda \mu }\text{:}$ $$\begin{array}{cc}{P}^{\lambda \mu }{\phi }_{\nu }^{\lambda \mu }={(-1)}^{\bar{\nu }}{\phi }_{\nu }^{\mu \lambda },P^{\lambda \mu }{\psi }_{\nu }^{\lambda \mu }={(-1)}^{\bar{\nu }+1}{\psi }_{\nu }^{\mu \lambda }\text{.}& \text{(1.33)}\end{array}$$ Comparing the coefficients at $(q-1)$ in (1.28) we obtain the equation for $s_{\nu }\text{:}$ $$P^{\lambda \mu }\tau {\phi }_{\nu }^{\lambda \mu }+P^{\lambda \mu }{\psi }_{\nu }^{\lambda \mu }={(-1)}^{\bar{\nu }}{\psi }_{\nu }^{\mu \lambda }+{(-1)}^{\bar{\nu }}{s}_{\nu }{\phi }_{\nu }^{\mu \lambda }\text{.}$$ Multiplying this equality by ${\left({\phi }_{\nu }^{\mu \lambda }\right)}^{\top }$ from the left and using the symmetry relations (1.33) we obtain the following expression for $s_{\nu }$ $$s_{\nu }={\left({\phi }_{\nu }^{\lambda \mu }\right)}^{\top }\tau {\phi }_{\nu }^{\lambda \mu }\text{.}$$ Using the symmetry relations (1.33) and the property of highest weight vectors the value of $s_{\nu }$ one can express through the Casimir operators $$\begin{array}{cc}{s}_{\nu }=\frac{1}{4}(c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right))\text{.}& \text{(1.34)}\end{array}$$ From the symmetry relations (1.15) and from the definition of $R^{\lambda \mu }$ we have $$\begin{array}{cc}{R}_{\nu }\left(q\right)R_{\nu }\left(q^{-1}\right)=1& \text{(1.35)}\\ R_{\nu }\left(q\right)\simeq {(-1)}^{\bar{\nu }}{q}^{\alpha },q\to \infty \text{.}& \text{(1.36)}\end{array}$$ Moreover, the function $R_{\nu }\left(q\right)$ has no poles and zeros for finite values of $q\text{.}$ From Liouville theorem follows that there is only one function $R_{\nu }\left(q\right)$ satisfying the relations (1.29), (1.34)-(1.36) $$R_{\nu }\left(q\right)={(-1)}^{\bar{\nu }}{q}^{\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}}\text{.}$$ The equality (1.20) is the transposition of (1.19). The Theorem is proved. $\square$

Consequence. When all irreducible components in $V^{\lambda }\otimes V^{\mu }$ have the multiplicity equal to one the matrices $R^{\lambda \mu }$ and ${\left(R^{\lambda \mu }\right)}^{-1}$ have the following decompositions: $$\begin{array}{cccc}{R}^{\lambda \mu }\left(q\right)& =& {\displaystyle \sum _{\nu \in \lambda \otimes \mu }{(-1)}^{\bar{\nu }}{q}^{\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}}{P}_{\nu }^{\lambda \mu }\left(q\right)}& \text{(1.37)}\\ {\left(R^{\lambda \mu }\left(q\right)\right)}^{-1}& =& {\displaystyle \sum _{\nu \in \lambda \otimes \mu }{(-1)}^{\bar{\nu }}{q}^{-\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}}{P}_{\nu }^{\mu \lambda }\left(q\right)}& \text{(1.38)}\end{array}$$ where $$\begin{array}{cc}{P}_{\nu }^{\lambda \mu }\left(q\right)=K_{\nu }^{\mu \lambda }{\left(q\right)}^{\top }{K}_{\nu }^{\lambda \mu }\left(q\right)\text{.}& \text{(1.39)}\end{array}$$ These $R\text{-matrices}$ will be called $R\text{-matrices}$ with simple spectrum.

Let us consider the space $V^{\tau }\otimes V^{\lambda }\otimes V^{\mu }$ and the matrices $R_{12}^{\tau \lambda }=R^{\tau \lambda }\otimes 1:V^{\tau }\otimes V^{\lambda }\otimes V^{\mu }\to V^{\lambda }\otimes V^{\tau }\otimes V^{\mu },$ $R_{23}^{\tau \mu }=1\otimes R^{\tau \mu }:V^{\lambda }\otimes V^{\tau }\otimes V^{\mu }\to V^{\lambda }\otimes V^{\mu }\otimes V^{\tau },$ ${\left(K_{\nu }^{\lambda \mu }\right)}_{23}=1\otimes K_{\nu }^{\lambda \mu }:V^{\tau }\otimes V^{\lambda }\otimes V^{\mu }\to V^{\tau }\otimes V^{\nu },$ ${\left(K_{\nu }^{\lambda \mu }\right)}_{12}=K_{\nu }^{\lambda \mu }\otimes 1:V^{\lambda }\otimes V^{\mu }\otimes V^{\tau }\to V^{\nu }\otimes V^{\tau },$ $R^{\tau \nu }:V^{\tau }\otimes V^{\nu }\to V^{\nu }\otimes V^{\tau }$ and similar matrices $R_{23}^{\mu \tau },$ $R_{12}^{\lambda \tau },$ ${\left(K_{\nu }^{\lambda \mu }\right)}_{},$ ${\left(K_{\nu }^{\lambda \mu }\right)}_{23},$ ${\left(K_{\nu }^{\lambda \mu }\right)}_{12},$ $R^{\nu \tau }\text{.}$

Theorem 1.6. The matrices introduced above satisfy the following relations $$\begin{array}{cccc}{R}^{\tau \nu }{\left(K_{\nu }^{\lambda \mu }\right)}_{23}& =& {\left(K_{\nu }^{\lambda \mu }\right)}_{12}{R}_{23}^{\tau \mu }{R}_{12}^{\tau \lambda },& \text{(1.40)}\\ R^{\nu \tau }{\left(K_{\nu }^{\lambda \mu }\right)}_{12}& =& {\left(K_{\nu }^{\lambda \mu }\right)}_{23}{R}_{12}^{\lambda \tau }{R}_{23}^{\mu \tau }\text{.}& \text{(1.41)}\end{array}$$

		Proof.
	The restriction or the relation (1.6) on the irreducible representation gives: $$\begin{array}{cc}{\displaystyle \sum _s({\rho }_{\nu }\left(e_s\right)\otimes \rho \tau \left(e^s\right))(K_{\nu }^{\lambda \mu }\otimes I)=(K_{\nu }^{\lambda \mu }\otimes I)\sum _{s,t}{\rho }_{\lambda }\left(e_t\right)\otimes {\rho }_{\mu }\left(e_s\right)\otimes {\rho }_{\tau }\left(e^t{e}^s\right)\text{.}}& \text{(1.42)}\end{array}$$ Multiplying this formula by the permutation operator $P^{\nu \tau }:V^{\nu }\otimes V^{\tau }\to V^{\tau }\otimes V^{\nu }$ from the left and using the equality $$P^{\nu \tau }(K_{\nu }^{\lambda \mu }\otimes I)P_{13}{P}_{23}=I_{\tau }\otimes K_{\nu }^{\lambda \mu }$$ we obtain (1.41). The relation (1.40) is proved similarly. $\square$

Crossing-symmetry of the universal $R\text{-matrices}$ (1.8d) implies crossing-symmetry of the matrices $R^{\lambda \mu }$ and $K_{\nu }^{\lambda \mu }\text{:}$

Theorem 1.7. The matrices $R^{\lambda \mu }$ and $K_{\nu }^{\lambda \mu }$ satisfy the following crossing-symmetry relations: $$\begin{array}{cccc}{\left(P^{\mu \lambda }{R}^{\lambda \mu }\right)}^{?}& =& (C\otimes 1){\left(P^{\mu \lambda }{R}^{\lambda \mu }\right)}^{t_1}(C^{-1}\otimes I)& \text{(1.43)}\\ {\displaystyle \sum _j{\left(K_{\nu }^{\lambda \mu }\right)}_k^{ij}{\left(C\right)}_j^{\ell }}& =& {\displaystyle \sqrt{\frac{{\chi }_{\lambda }}{{\chi }_{\nu }}}{\left(K_{\lambda }^{\nu {\mu }^{*}}\right)}_i^{k\ell }}& \text{(1.44)}\\ {\displaystyle \sum _i{\left(K_{\nu }^{\lambda \mu }\right)}_k^{ij}{\left(C\right)}_i^{\ell }}& =& {\displaystyle \sqrt{\frac{{\chi }_{\mu }}{{\chi }_{\nu }}}{\left(K_{\mu }^{{\lambda }^{*}\nu }\right)}_j^{\ell k}}& \text{(1.45)}\end{array}$$ where $t_1$ is transposition over the first space, ${\lambda }^{*}$ is the highest weight (h.w.) vector conjugated with $\lambda$ by Carton automorphism, $$\begin{array}{cc}C=S{q}^{-\frac{\rho }{2}},{\chi }_{\lambda }={\text{tr}}_{V^{\lambda }}\left(q^{-\rho }\right)\text{.}& \text{(1.46)}\end{array}$$

Using the explicit formula for coproduct in it is not difficult to prove the first statement of the following theorem.

Theorem 1.8.

(a)

The matrix $$\begin{array}{cc}{\left(K^{\lambda {\lambda }^{*}}\right)}^{ij}=\frac{1}{\sqrt{{\chi }_{\lambda }}}{C}^{ji}& \text{(1.47)}\end{array}$$ is the projector on $\omega$ the one-dimensional subrepresentation in $V^{\lambda }\otimes V^{{\lambda }^{*}}\text{.}$

(b)

The matrices $R^{\lambda {\lambda }^{*}}$ and $R^{\lambda \lambda }$ satisfy the relations $$\begin{array}{cccc}{R}^{\lambda {\lambda }^{*}}{\left(K^{\lambda {\lambda }^{*}}\right)}^{\top }& =& q^{-\frac{c\left(\lambda \right)}{2}}{(-1)}^{\left[\lambda \right]}{\left(K^{{\lambda }^{*}\lambda }\right)}^{\top }& \text{(1.48)}\\ {\text{tr}}_2\left((I\otimes q^{-\rho }){\left(R^{\lambda \lambda }\right)}^{\pm 1}\right)& =& q^{\pm \frac{c\left(\lambda \right)}{2}}·I\text{.}& \text{(1.49)}\end{array}$$ Here $\left[\lambda \right]=0,1,$ ${\text{tr}}_2$ is the matrix trace over the second space in $V^{\lambda }\otimes V^{\lambda }\text{.}$

The proof of the second statement of this theorem is given in the next section, where we introduce graphical representation for the $R\text{-matrices}$ and for KGC.

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