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

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 ${X}_{i}^{±},$ ${H}_{i},$ $i=1,\dots ,r=\text{rank} 𝔤$ and with the relations: $[Hi,Hj]=0, [Hj,Xi±]=± (αjαi)Xi± (1.1) [Xi+,Xj-]= δij sh(h2Hi) sh(h2) ,q=eh (1.2) i≠j, ∑k=0n (-1)k (nk)qi qi-k(n-k)2 (Xi±)k Xj± (Xi±)n-k =0,qi= q(αiαi)2 (1.3)$ where ${A}_{ij}$ is the Cartan matrix of $𝔤,$ $\left(\alpha ,\beta \right)$ 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 ${X}_{i}^{±},$ ${H}_{i}$ 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: $Δ(Hi)= Hi⊗1+1⊗Hi, (1.4) Δ(Xi±)= Xi±⊗eh4Hi +e-h4Hi⊗Xi±, (1.5) γ(Hi)=-Hi, γ(Xi±)=- eh2ρXi± e-h2ρ, (1.6)$ 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 ${𝔟}_{±}\subset 𝔤$ be the Borel subalgebras in $𝔤\text{.}$ The algebras ${U}_{q}\left({𝔟}_{+}\right)$ and ${U}_{q}\left({𝔟}_{-}\right)$ with generators $\left({H}_{i},{X}_{i}^{+}\right)$ and $\left({H}_{i},{X}_{i}^{-}\right)$ 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 $σ∘Δ(a)= RΔ(a)R-1 ∀a∈Uq(𝔤) (1.7)$ 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 $(Δ⊗id)R =R13R23, (1.8a) (id⊗Δ)R= R13R12, (1.8b) R12R13R23= R23R13R12, (1.8c) (id⊗γ)R= R-1. (1.8d)$ 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 ${U}_{q}\left({𝔟}_{-}\right)\text{:}$ $eset=mstτ eτ,Δ(es) =μstτet⊗eτ ,γ(es)= γstet.$ The matrices $\mu ,m\prime$ and ${\gamma }^{-1}$ are the matrices of multiplication, comultiplication and antipode in ${U}_{q}\left({𝔟}_{+}\right)\text{:}$ $eset=μτst eτ,Δ(es) =mtτseτ⊗et, γ(es)= (γ-1)ts et.$ From (1.7) we have the commutation relations between the elements ${e}^{s}$ and ${e}_{t}$ in ${U}_{q}\left(𝔤\right)\text{:}$ $μspqmtpτ eteq=μspq mqtτepet.$ From this commutation relations and from the definition of $R$ follows (1.8c). To prove (1.8d) we use the definition of antipode (A.): $R(id⊗γ)R=es et⊗esγ(et) =mstτeτ⊗ γptμqspeq= mstτγptμqsp eτ⊗eq=e⊗e.$ Similarly $(id⊗γ)(R)·R= mstτγps μqpteτ⊗ eq=e⊗e.$ The following formulae prove the relations (1.8a) and (1.8b) $(Δ⊗id)R= Δ(es)⊗ es=μsτt eτ⊗et⊗es =eτ⊗et⊗ eτet=R13 R23, (id⊗Δ)R= es⊗Δ(es) =es⊗mτts et⊗eτ=eτ et⊗et⊗eτ =R13R12.$ 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. $𝔤 category An Bn Cn Dn G2 E6 E7 E8 F4 finite dimensional ω1 ω1 ω1 ω1 ω1 ω1 ω7 ω8 ω4 tensorial represent - ωn - ω1,ωn - - - - - Table 1$

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 ${V}^{\lambda }\otimes {V}^{\mu }$ $\text{(}{P}^{\lambda \mu }:{V}^{\lambda }\otimes {V}^{\mu }\to {V}^{\mu }\otimes {V}^{\lambda },$ ${P}^{\lambda \mu }\left(f\otimes g\right)=g\otimes f\text{)}$ then we denote the matrix ${P}^{\lambda \mu }\left({\rho }_{\lambda }\otimes {\rho }_{\mu }\right)\left(R\right)$ 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: $(Rλμ⊗I) (I⊗Rνμ) (Rνλ⊗I)= (I⊗Rνλ) (Rνμ⊗I) (I⊗Rλμ). (1.9)$ 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 ${U}_{q}\left(𝔤\right)$ $θ(Xi±)= Xi∓, θ(Hi)=Hi$ and automorphism $\delta$ corresponding to on authomorphism of Dynkin diagram: $δ(Xi±)= Xi*±, δ(Hi)=Hi*.$

We define a $q\text{-analog}$ $C$ of the element with the maximal length of the Weyl group by the formula $γ=δ∘AdC⊗θ.$

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

(b) The element $R$ satisfies the relation $(id⊗δ) (R-1)= (1⊗C) (id⊗θ) (R)(1⊗C-1).$

Definition 1.2. If ${V}^{\nu }\subset {V}^{\lambda }\otimes {V}^{\mu },$ the projection matrix ${K}_{\nu }^{\lambda \mu }\left(q,a\right):{V}^{\lambda }\otimes {V}^{\mu }\to {V}^{\nu }$ 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 $\text{mult}\left({V}^{\nu }\right)=\text{mult}\left({V}^{\nu }\right)=1$ $Kνλμ (Kν′μλ)⊤ =δνν′Iν. (1.14)$

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 $Pμλ R(q)λμ Pμλ = S⊗S R(q)μλ s-1⊗s-1 =Rλμ (q-1)-1, (1.15) C-1Kνλμ (q)(C⊗C) = (-1)ν‾ Kν*λ*μ* (q-1), (1.16) Kνλμ(q) Pμλ = (-1)ν‾ Kνμλ(q-1), (1.17) Rλμ(q)⊤ = Rλμ(q). (1.18)$

Here $\stackrel{‾}{\nu }=0,1$ 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 $Rλμ(q) (Kνλμ(q))⊤ = (-1)ν‾ qc(ν)-c(λ)-c(μ)4 Kνμλ(q)⊤ (1.19) Kνμλ(q) Rλμ(q) = (-1)ν‾ qc(ν)-c(λ)-c(μ)4 Kνλμ(q) (1.20)$ where $\stackrel{‾}{\nu }$ 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 $\nu \text{:}$ $c=∑i=1r Hi2+ ∑α∈ΔXα X-α,Hi =(H,ωi) (1.21) c(ν)=ν2+ 2ρν. (1.22)$

 Proof. From the definition of universal $R\text{-matrix}$ follows the equality $RλμΔλμ (a)=Δμλ (a)Rλμ (1.23)$ where ${\mathrm{\Delta }}^{\lambda \mu }\left(a\right)=\left({\rho }^{\lambda }\otimes {\rho }^{\mu }\right)\mathrm{\Delta }\left(a\right)\text{.}$ The representation ${\mathrm{\Delta }}^{\lambda \mu }$ is reducible and ${K}_{\nu }^{\lambda \mu }$ are the projectors on the irreducible components: $Δλμ(a) Kνλμ⊤ = Kνλμ⊤ ρν(a) (1.24) Δμλ(a) Kνμλ⊤ = Kνμλ⊤ ρν(a). (1.25)$ From (1.23) and (1.24) we have $Rλμ Kνλμ⊤ ρν(a)= Δμλ(a) Rλμ Kνλμ⊤. (1.26)$ Comparing with (1.25) we obtain $Rλμ Kνλμ⊤= Rν(q) Kνμλ⊤ (1.27)$ 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 $Rλμ(q) φνλμ(q) =Rν(q) φνμλ(q). (1.28)$ At $q\to 1$ we have $Rλμ = Pλμ (1+(q-1)τ+O((q-1)2)), (1.29) τ = 12∑iHi⊗ Hi+∑α∈Δ+ Xα⊗X-α (1.30) φνλμ(q) = φλμ+(q-1) ψνλμ+ O((q-1)2), (1.31) Rν(q) = (-1)ν‾ (1+(q-1)sν+O((q-1)2)). (1.32)$ Here $\tau \in 𝔤\otimes 𝔤$ $\text{(}𝔤↪U\left(𝔤\right)\text{)}$ 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 ${\psi }_{\nu }^{\lambda \mu }\text{:}$ $Pλμφνλμ =(-1)ν‾ φνμλ, Pλμψνλμ =(-1)ν‾+1 ψνμλ. (1.33)$ Comparing the coefficients at $\left(q-1\right)$ in (1.28) we obtain the equation for ${s}_{\nu }\text{:}$ $Pλμτφνλμ +Pλμψνλμ =(-1)ν‾ ψνμλ+ (-1)ν‾ sνφνμλ.$ 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 ${s}_{\nu }$ $sν=(φνλμ)⊤ τφνλμ.$ 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 $sν=14 (c(ν)-c(λ)-c(μ)). (1.34)$ From the symmetry relations (1.15) and from the definition of ${R}^{\lambda \mu }$ we have $Rν(q) Rν(q-1) =1 (1.35) Rν(q)≃ (-1)ν‾ qα,q→∞. (1.36)$ 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) $Rν(q)= (-1)ν‾ qc(ν)-c(λ)-c(μ)4.$ 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: $Rλμ(q) = ∑ν∈λ⊗μ (-1)ν‾ qc(ν)-c(λ)-c(μ)4 Pνλμ(q) (1.37) (Rλμ(q))-1 = ∑ν∈λ⊗μ (-1)ν‾ q-c(ν)-c(λ)-c(μ)4 Pνμλ(q) (1.38)$ where $Pνλμ(q)= Kνμλ(q)⊤ Kνλμ(q). (1.39)$ 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 $Rτν (Kνλμ)23 = (Kνλμ)12 R23τμR12τλ, (1.40) Rντ (Kνλμ)12 = (Kνλμ)23 R12λτR23μτ. (1.41)$

 Proof. The restriction or the relation (1.6) on the irreducible representation gives: $∑s ( ρν(es)⊗ ρτ(es) ) (Kνλμ⊗I) =(Kνλμ⊗I) ∑s,tρλ(et) ⊗ρμ(es)⊗ ρτ(etes). (1.42)$ 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 $Pντ (Kνλμ⊗I) P13P23=Iτ⊗ Kνλμ$ 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: $(PμλRλμ)? = (C⊗1) (PμλRλμ)t1 (C-1⊗I) (1.43) ∑j (Kνλμ)kij (C)jℓ = χλχν (Kλνμ*)ikℓ (1.44) ∑i(Kνλμ)kij (C)iℓ = χμχν (Kμλ*ν)jℓk (1.45)$ where ${t}_{1}$ is transposition over the first space, ${\lambda }^{*}$ is the highest weight (h.w.) vector conjugated with $\lambda$ by Carton automorphism, $C=Sq-ρ2, χλ=trVλ (q-ρ). (1.46)$

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 $(Kλλ*)ij =1χλCji (1.47)$ 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 $Rλλ* (Kλλ*)⊤ = q-c(λ)2 (-1)[λ] (Kλ*λ)⊤ (1.48) tr2 ( (I⊗q-ρ) (Rλλ)±1 ) = q±c(λ)2 ·I. (1.49)$ 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.