## 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.

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

The h.w. of those algebras can be integer or halfinteger vectors. For $𝔤=𝔰𝔬\left(2n+1\right),$ $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{n}\right),$ ${\lambda }_{1}\ge {\lambda }_{2}\ge \dots \ge {\lambda }_{n},$ ${\lambda }_{i}\in \frac{1}{2}{ℤ}_{+},$ on the h.w. vector ${e}_{\lambda }:{H}_{i}{e}_{\lambda }=\left({\lambda }_{i}-?\right),$ $i=1,\dots ,n-1,$ ${H}_{n}{e}_{\lambda }={\lambda }_{n}{e}_{\lambda }\text{.}$ For $𝔤=𝔰𝔬\left(2n\right),$ $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{n-1},{\lambda }_{n}\right),$ ${\lambda }_{1}\ge \dots \ge {\lambda }_{n-1}\ge {\lambda }_{n},$ ${\lambda }_{i}\in \frac{1}{2}{ℤ}_{+},$ $i=1,\dots ,n-1,$ ${\lambda }_{n}\in \frac{1}{2}ℤ,$ ${H}_{i}{e}_{\lambda }=\left({\lambda }_{i}-{\lambda }_{i+1}\right){e}_{\lambda },$ $i=1,\dots ,n-1,$ ${H}_{n}{e}_{\lambda }=\left({\lambda }_{n-1}+{\lambda }_{n}\right)\text{.}$ 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. ${\omega }_{1}=\left(1,0,\dots ,0\right)\text{.}$ In the category of finite dimensional representations basic representation of $𝔰𝔬\left(2n+1\right)$ have h.w. ${\omega }_{n}=\left(\frac{1}{2},\dots ,\frac{1}{2}\right)\text{.}$ 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 ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ from the category of a tensorial representation.

### Tensorial representations of ${U}_{q}\left(𝔰𝔬\left(2n\right)\right)$ and ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$

The vectorial representation ${V}^{{\omega }_{1}}$ of ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ and ${U}_{q}\left(𝔰𝔬\left(2n\right)\right)$ do not depend on $q\text{:}$ $Uq(𝔰𝔬(2n+1))$ $π(Xj+) = Ejj+1- EN-j,N+1-j, j=1,…, π(Xn+) = 12- (En+1,n+2+Enn+1) π(Hj) = Ejj- Ej+1,j+1- EN+1-j,N+1-j +EN-j,N-j, j=1,…,n-1 π(Hn) = Enn -En+2,n+2 π(Xj-) = π(Xj+)t (5.1)$ $Uq(𝔰𝔬(2n))$ $π(Xj+) = Ejj+1- EN-j,N+1-j, j=1,…,n-1 π(Xn+) = Enn-1- Enn+2, π(Hj) = Ejj- Ej+1j+1- EN+1-j+ EN-j N-j, j=1,…,n-1 π(Hn) = Enn +En-1,n-1- En+1,n+1- En+2,n+2, π(Xj-) = π(Xj+)t. (5.2)$ 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 ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ are self-adjoint: ${\omega }_{1}^{*}={\omega }_{1}\text{.}$ The tensor product of two vectorial representations is decomposed into the sum of three irreducible components $(Vω1)⊗2= V2ω1⊕ Vω2⊕V0. (5.3)$

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 $i≠j′, ii qi‾-k‾2 ek′⊗ek+ q-1/2∑k≤i qi‾-k‾2 ek′⊗ek ) } (5.4) eijω2= ei⊗ej-q1/2ej⊗ei1+q ,ii qi‾-k‾2 ek′⊗ek+q ∑k≤i qi‾-k‾2 ek′⊗ek ) } . (5.5) eij0= (q12-q-12) (q-N4-q-N4) (qN-24+q-N-24) δij′∑k qi‾-k‾2 ek′⊗ek (5.6)$ 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, $i‾= { i+12, i≤n i, i=n+1 i-12, i>n+1 for Uq(𝔰𝔬(2n+1)), i‾= { i+12, i≤n i-12, i>n for Uq(𝔰𝔬(2n)).$ From the theorem 1.5 we obtain the matrix ${R}^{{\omega }_{1}{\omega }_{1}}\text{:}$ $Rω1ω1= P2ω1ω1ω1 q-1/2- Pω2ω1ω1 q1/2+ P0ω1ω1 q-N-12. (5.7)$ Substituting the expressions (5.4)-(5.6) in this formula we obtain the representation of ${R}^{{\omega }_{1}{\omega }_{1}}$ as ${N}^{2}×{N}^{2}$ matrix: $Rω1ω1 = q1/2 ∑i≠N-12 Eii⊗Eii+ EN+12,N+12 ⊗EN+12,N+12 +∑i≠j,j′ Eji⊗Eij +q-1/2 ∑i≠i′ Eii′⊗ Ei′i -(q-1/2-q1/2) ∑i>jEjj⊗ Eii+(q-1/2-q1/2) ∑i>j qj‾-i‾2 Ei′j⊗Eij′, (5.8)$ where ${E}_{ij}$ are the basic matrices in ${ℂ}^{N}$ and the second term is present only for ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)\text{.}$

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: $Vω1⊗Vλ= ∑k=12n+1 ⊕Vλ(k), 𝔤=𝔰𝔬(2n+1), (5.9) Vω1⊗Vλ= ∑k=12n⊕ Vλ(k), 𝔤=𝔰𝔬(2n). (5.10)$ Here ${\lambda }^{\left(k\right)}=\left({\lambda }_{1},\dots ,{\lambda }_{k}+1,\dots ,{\lambda }_{n}\right),$ ${\lambda }^{\left(n+1+k\right)}=\left({\lambda }_{1},\dots ,{\lambda }_{k}-1,\dots ,{\lambda }_{n}\right),$ ${\lambda }^{\left(k+1\right)}=\lambda$ in (5.9) and ${\lambda }^{\left(n+k\right)}=\left({\lambda }_{1},\dots ,{\lambda }_{k}-1,\dots ,{\lambda }_{n}\right)$ 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 ${P}_{\lambda }\left(q,a\right)\text{.}$ For ${P}_{{\omega }_{k}}\left(q\right)$ $\text{(mult}\left({\omega }_{k}\subset {\left({\omega }_{1}\right)}^{\otimes k}\right)=1\text{)}$ we have the following representation $Pωk(q)= ∑n1,…,nk=0c ∏i=1kqni,1 \omega \cdots \omega \omega \omega {\alpha }_{\left({n}_{1},\dots ,{n}_{k}\right)} (5.11)$ where $Q=(qi,j)i=0,1,2j=0,1,2= ( 111 qq-kq-N+k q2q-2kq-2N+2k ) -1 (5.12)$ and ${\alpha }_{\left({n}_{1},\dots ,{n}_{k}\right)}$ is defined by (3.19).

The algebras ${C}_{M}^{{\omega }_{1}}\left(𝔰𝔬\left(2n-1\right)\right)$ and ${C}_{M}^{{\omega }_{1}}\left(𝔰𝔬\left(2n+1\right)\right)$ 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 $GiGj = GjGi, |i-j|≥2 GiGi+1Gi = Gi+1GiGi+1, Gi+Gi-1 = m(1+Ei), EiEi+1Ei = Ei=EiEi-1 Ei Gi+1GiEi+1 = EiGi+1Gi= EiEi+1 Gi-1GiEi-1 = EiGi-1Gi= EiEi-1 Gi+1EiGi+1 = Gi-1Ei+1 Gi-1 Gi-1EiGi-1 = Gi-1Ei-1 Gi-1 Gi+1EiEi+1 = Gi-1Ei+1 Gi-1EiEi-1 = Gi-1Ei-1 Ei+1EiGi+1 = Ei+1Gi-1 Ei-1EiGi+1 = Ei-1Gi-1 GiEi = EiGi= ℓ-1Ei EiGi+1Ei = EiGi-1Ei= ℓEi. (5.13)$

Theorem 5.1. The algebras ${C}_{M}^{{\omega }_{1}}\left(𝔰𝔬\left(2n\right)\right)$ and ${C}_{M}^{{\omega }_{1}}\left(𝔰𝔬\left(2n+1\right)\right)$ are factors of Birman-Wenzl algebra with $m=i\left({q}^{1/2}-{q}^{-1/2}\right),$ $\ell =-i{q}^{\frac{N-1}{2}}$ over the ideals formed by the elements $J=Pωn+1 (q)1,…,n+1$ 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 $m=i\left({q}^{1/2}-{q}^{-1/2}\right),$ $\ell =-i{q}^{\frac{N-1}{2}},$ $\alpha \left(G\right)=i\left(1\otimes \dots \otimes {R}^{{\omega }_{1}{\omega }_{1}}\otimes \dots \otimes 1\right)$ defined in the space ${\left({V}^{{\omega }_{1}}\right)}^{\otimes M}\text{.}$

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

Theorem 5.2. The algebras ${U}_{q}\left(𝔰𝔬\left(2n\right)\right)$ and ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ 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 ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ on the subalgebra ${U}_{q}\left(𝔰𝔩\left(n\right)\right)\text{.}$ We have $Vω1 = V∼ω1 ⊕V∼ω1* ⊕V0forUq (𝔰𝔬(2n+1)) Vω1 = V∼ω1 ⊕V∼ω1* forUq(𝔰𝔬(2n)).$ 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: $, , , .$ 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 ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ have the following block structures at the restriction of ${U}_{q}\left(𝔰𝔬\left(2n\right)\right)$ and ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ on ${U}_{q}\left(𝔰𝔩\left(n\right)\right)$ for ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)\text{:}$ $= n×n n n×n n 1 n n×n n n×n n×n 0 0 0 0 0 0 0 0 n 0 a↑ 0 ↑ 0 0 0 0 0 n×n 0 0 a ( -q-n-32 ) 0 aq-n-34 0 0 0 n 0 ↑ 0 0 0 0 0 0 0 1 0 0 aq-n-34 0 1 0 0 0 0 n 0 0 0 0 0 a↓ 0 ↓ 0 n×n 0 0 0 0 0 0 0 0 n 0 0 0 0 0 ↓ 0 0 0 n×n 0 0 0 0 0 0 0 0 a=(q-12-q12) (5.16) = n×n n n×n n 1 n n×n n n×n n×n 0 0 0 0 0 0 0 0 n 0 0 0 ↑ 0 0 0 0 0 n×n 0 0 0 0 0 0 0 0 n 0 ↑ 0 -a↑ 0 0 0 0 0 1 0 0 0 0 1 0 -aqn-34 0 0 n 0 0 0 0 0 0 0 ↓ 0 n×n 0 0 0 0 -aqn-34 0 -a ( -qn-32 ) 0 0 n 0 0 0 0 0 ↓ 0 -a↓ 0 n×n 0 0 0 0 0 0 0 0 (5.17)$ for ${U}_{q}\left(𝔰𝔬\left(2n\right)\right)\text{:}$ $= n×n n×n n×n n×n n×n 0 0 0 n×n 0 a ( -q-n-12 ) 0 n×n 0 0 0 n×n 0 0 0 (5.18) = 0 0 0 0 0 0 0 -a ( -q-n-12 ) 0 0 0 0 (5.19)$

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

Corollary. There exist the following embeddings of the algebras ${C}_{N}^{{\omega }_{1}}\text{:}$ $CMω1 (𝔰𝔬(2n+1))⊃ CMω1 (𝔰𝔩(n))× CMω1(𝔰𝔩(n)).$

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 ${C}_{M}^{{\omega }_{1}}\left(𝔰𝔬\left(2n+1\right)\right)$ 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 ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ spinor representation ${V}^{{\omega }_{n}}$ has dimension ${2}^{n}\text{.}$ The bases in spinor representations are parametrized by the weights $\epsilon =\left({\epsilon }_{1},\dots ,{\epsilon }_{n}\right),$ ${\epsilon }_{i}=±\frac{1}{2}\text{.}$ 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 ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ and ${U}_{q}\left(𝔰𝔬\left(2n\right)\right)$ are defined by the action of the generators on the basic vectors ${e}_{\epsilon }\text{:}$ $Uq(𝔰𝔬(2n+1)):$ $Xi+e⋯εiεi+1⋯ = -e⋯εi+1,εi-1,⋯, Xie⋯εn = 12ch(h4) e⋯εn+1, Hieε = (εi-εi+1) eε,i=1,…,n-1 Hneε = εneε Xi- = (Xi+)t. (5.20)$ $Uq(𝔰𝔬(2n)):$ $Xi+e⋯εiεi+1⋯ = -e⋯εi+1,εi+1-1,⋯, Xn+e⋯εn-1εn = -e⋯εn-1+1,εn+1,…, Hieε = (εi-εi+1) eε, Hneε = (εn-1+εn) eε, Xi- = (Xi+)t. (5.21)$ 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 $Uq(𝔰𝔬(2n)):$ $Vωn-1⊗ Vωn-1≃ Vωn⊗Vωn = { (∑s=0n2-1⊕Vω2s) ⊕V2ωn, n even (∑s=1n-12⊕Vω2s-1) ⊕V2ωn, n odd (5.22) Vωn⊗Vωn-1 = { (∑s=1n2-1⊕Vω2s-1) ⊕Vωn+ωn-1, n even (∑s=0n2-2⊕Vω2s) ⊕Vωn+ωn-1, n odd (5.23) Vω1⊗Vωn = { Vωn-1 ⊕Vωn+ω1, n even Vωn⊕ Vωn+ω, n odd (5.24) Vω1⊗Vωn-1 = { Vωn⊕ Vωn-1+ω1, n even Vωn-1⊕ Vωn-1+ω1, n odd (5.25)$ $Uq(𝔰𝔬(2n)):$ $Vωn⊗Vωn = (∑s=0n-1⊕Vωs) ⊕V2ωn Vω1⊗Vωn = Vωn⊕Vω1+ωn. (5.26)$

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 $Kssω1= ( Kωnωnω1 Kωn-1ωnω1 Kωnωn-1ω1 Kωn-1ωn-1ω1 )$ For ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ 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 ${U}_{q}\left(𝔰𝔬\left(2n+1\right)\right)$ satisfy the following relations: $s s s {\omega }_{1} {\omega }_{1} =q-1/2 s s s {\omega }_{1} {\omega }_{1} +(1+q-N-22) s s {\omega }_{1} {\omega }_{1} (5.27)$

The proof of this theorem follows from theorem 3.1.

It is interesting that the relations $\left(\phantom{\rule{1em}{0ex}}\right)$ 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 $(γi)αβ= s s {\omega }_{1} \alpha \beta i (5.28)$ the relations (5.27) give the $q\text{-anticommutation}$ relations for the matrices ${\gamma }_{i}\text{:}$ $γiγj = -q-i/2 γjγi, ii qi‾-k‾2 γk′γk+ (1+qN-22) qi‾2 (5.29)$ with normalization condition $∑iγiγi′= χω1.$ 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 ${P}_{{\omega }_{k}}^{ss}\text{:}$ $s {s}^{*} {\omega }_{k} {s}^{*} s =Nks \cdots \cdots s s {s}^{*} {s}^{*} {\omega }_{1} {\omega }_{1} {\omega }_{1} {\omega }_{1} {\omega }_{1} {\omega }_{1} {\omega }_{k} (5.30)$ 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 ${R}^{{\omega }_{1}{\omega }_{1}}\text{:}$ ${\omega }_{n} {\omega }_{n} = ∑s=0n-1 q14((2n+1)k-k2-n2-n2) {\omega }_{n} {\omega }_{n} {\omega }_{k} {\omega }_{n} {\omega }_{n} +qn8 {\omega }_{n} {\omega }_{n} 2{\omega }_{k} {\omega }_{n} {\omega }_{n} (5.31) {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} = { ∑s=0n2-1 (-1)⋯ q(ns-s2-n24+n8) {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{2s} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} +qn8 {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} 2{\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} , n even ∑s=1n-12 (-1)⋯ q(ns-s2-(n+1)24+n8) {\omega }_{n-1} {\omega }_{n-1} {\omega }_{2s-1} {\omega }_{n-1} {\omega }_{n-1} +qn8 {\omega }_{n-1} {\omega }_{n-1} 2{\omega }_{n-1} {\omega }_{n-1} {\omega }_{n-1} , n odd (5.32) {\omega }_{n} {\omega }_{n-1} = { ∑s=1n2-1 (-1) q(ns-s2-(n+1)24+n8) {\omega }_{n} {\omega }_{n-1} {\omega }_{2s-1} {\omega }_{n-1} {\omega }_{n} +qn8-14 {\omega }_{n} {\omega }_{n-1} {\omega }_{n}+{\omega }_{n-1} {\omega }_{n-1} {\omega }_{n} , n even ∑s=0n2-2 (-1)q(ns-s2-n24+n8) {\omega }_{n} {\omega }_{n-1} {\omega }_{2s} {\omega }_{n-1} {\omega }_{n} +qn8-14 {\omega }_{n} {\omega }_{n-1} {\omega }_{n}+{\omega }_{n-1} {\omega }_{n-1} {\omega }_{n} n odd (5.33) {\omega }_{1} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} = { q {\omega }_{1} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{1}+{\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{1} -q-2n-14 {\omega }_{1} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n}{n-1}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{1} , n even q ] {\omega }_{1} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{1}+{\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{1} -q-2n-14 {\omega }_{1} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{\left(\genfrac{}{}{0}{}{n-1}{n}\right)} {\omega }_{1} , n odd. (5.34)$ 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: ${\omega }_{1} {\omega }_{1} \cdots {\omega }_{1} {\omega }_{1} {\omega }_{s} \stackrel{⏞}{\phantom{\rule{8em}{0ex}}} 2n = (qq12)n (1+qn-12)…(1+q12) ∑k1=12n-1 ∑k2≠k1,k∼1 ⋯∑ki≠k1,k∼1,…,ki-1,k∼i-1 ⋯(-q)12(k1+…+kn) {\omega }_{1} {\omega }_{1} \cdots {\omega }_{1} {\omega }_{1} \left[{k}_{1},\dots ,{k}_{n}\right] \stackrel{⏞}{\phantom{\rule{8em}{0ex}}} 2n k∼i = maxj { j∈(1,…,2n)\ (k1,k∼1,…,ki-1,k∼i-1) } . (5.35)$ 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