## 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.

## $𝔤=𝔰𝔭\left(2n\right)$

All formulae in thin caae are very similar to those in part one of the previous section.

The h.w. of finite dimensional representation of are parametrized by the numbers $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{n}\right),$ ${\lambda }_{1}\ge \dots \ge {\lambda }_{n},$ ${\lambda }_{i}\in {ℤ}_{+}\text{.}$ If ${e}_{\lambda }$ is the h.w. vector, ${H}_{i}{e}_{\lambda }=\left({\lambda }_{i}-{\lambda }_{i+1}\right){e}_{\lambda },$ $i=1,\dots ,n-1,$ ${H}_{n}{e}_{\lambda }={\lambda }_{n}{e}_{\lambda },$ ${X}_{i}^{+}{e}_{\lambda }=0\text{.}$

The basic representation of ${U}_{q}\left(𝔰𝔭\left(2n\right)\right)$ have the h.w. ${\omega }_{1}=\left(1,0,\dots ,0\right),$ ${V}^{{\omega }_{1}}\simeq {ℂ}^{2n}$ $π(Xi+) = Eii+1- EN-i,N-i+1, i=1,…,n-1 π(Xn-) = Enn+1, π(Xi-)= π(Xi+)t π(Hi) = Eii- Ei+1,i+1- EN-i+1,N-i+1+ EN-i,N-i π(Hn) = 2Enn-2 En+1,n+1. (6.1)$ The tensor square of the basic representation is the sum of three irreducible components: $Vω1⊗Vω1= V2ω1⊕ Vω2⊕ V(0) (6.2)$ and we have the following spectral decomposition of ${R}^{{\omega }_{1}{\omega }_{1}}\text{:}$ $Rω1ω1= q12P2ω1ω1ω1 -q-12 Pω2ω1ω1- q-2n+12 P0ω1ω1. (6.3)$

As in the cases of $𝔤=𝔰𝔬\left(2n+1\right)$ and $𝔰𝔬\left(2n\right)$ one can calculate the matrices ${K}_{2{\omega }_{1}}^{{\omega }_{1}{\omega }_{1}},$ ${K}_{{\omega }_{2}}^{{\omega }_{1}{\omega }_{1}},$ and ${K}_{0}^{{\omega }_{1}{\omega }_{1}}\text{.}$ Substituting these matrices into (6.3) we obtain the matrix ${R}^{{\omega }_{1}{\omega }_{1}}\text{:}$ $Rω1ω1 = q12∑iEii ⊗Eii+∑i≠j,j′ Eji⊗Eij +q-12∑i≠i′ Eii′⊗Ei′i+ (q12-q-12) ∑i>jEii⊗ Ejj -(q12-q-12) ∑i>jEiEj qj‾-i‾2 Ei′j⊗Eij′ (6.4)$ where $i\prime =2n+1-i\text{;}$ ${ϵ}_{i}=1,$ $i=1,\dots ,n,$ ${ϵ}_{j}=-1,$ $i=n+1,\dots ,2n\text{;}$ $\stackrel{‾}{i}=i-\frac{1}{2},$ $i\le n,$ $\stackrel{‾}{i}=i+\frac{1}{2},$ $i\ge n+1\text{.}$ This matrix can also be extracted from [Jim1986-2].

The matrices ${R}^{\lambda \mu }$ and ${P}_{\lambda }$ for any h.w. $\lambda ,\mu$ can be found from the theorems 2. - 3. and from the ramification rule: $Vλ⊗Vω1= ∑k=12n⊕ Vλ(k) (6.5)$ where ${\lambda }^{\left(k\right)}=\left({\lambda }_{1},\dots ,{\lambda }_{k}+1,\dots ,{\lambda }_{n}\right),$ ${\lambda }^{\left(n+k\right)}=\left({\lambda }_{1},\dots ,{\lambda }_{k}-1,\dots ,{\lambda }_{k}\right),$ $k=1,\dots ,n\text{.}$

As in the case $𝔤=𝔰𝔬\left(2n+1\right),𝔰𝔬\left(2n\right)$ we have the following propositions.

Proposition 6.1. The algebra ${C}_{N}^{{\omega }_{1}}\left(𝔰𝔭\left(2n\right)\right)$ is the factor of B.-W. algebra with $m=\left(i\left({q}^{1/2}-{q}^{-1/2}\right)\right),$ $\ell =i{q}^{-\frac{2n+1}{2}}$ over the ideal formed by the elements $J=Pωn+1 (q)1…n+1 (6.6)$ acting nontrivially only in the multipliers of ${\left({V}^{{\omega }_{1}}\right)}^{\otimes N}$ with numbers $i\le n+1\text{.}$ The elements ${P}_{k}\left(q\right)$ are defined by (5.11) with the matrix $Q= ( 111 q q-k qk-2-2n q2 q-2k q2(k-2-2n) ) (6.7)$ and ${C}_{M}^{{\omega }_{1}}\left(𝔰𝔭\left(2n\right)\right)\simeq B{W}_{M}$ for $n+1>M\text{.}$

If we consider the block basis in ${V}^{{\omega }_{1}}$ connected with the embedding ${U}_{q}\left(𝔰𝔭\left(2n\right)\right)\supset {U}_{q}\left(𝔰𝔩\left(n\right)\right),$ $Vω1=V∼ω1⊕V∼ω1*, dim V∼ω1=dim V∼ω1*=n (6.8)$ we obtain the representation of ${R}^{{\omega }_{1}{\omega }_{1}}$ in the block form with the blocks constructed from ${U}_{q}\left(𝔰𝔩\left(n\right)\right)$ $R\text{-matrices.}$

Proposition 6.3. The matrices ${\left({R}^{{\omega }_{1}{\omega }_{1}}\right)}^{±1}$ have the following block structure in the basis (6.9): $= 0 0 0 0 a ( +q-n+12 ) 0 0 0 0 0 0 0 (6.9) = 0 0 0 0 0 0 0 -a ( +qn+12 ) 0 0 0 0 (6.10)$ $a=q12-q-12$ where we use the notations of sections 2 and 5.