## 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.

## $𝔤=𝔰𝔩\left(n\right)$

The basic representation of ${V}_{q}\left(𝔰𝔩\left(n\right)\right)$ have the h.w. ${\omega }_{1}$ and ${V}^{{\omega }_{1}}\simeq {ℂ}^{n}\text{.}$ The matrix elements of the generators do not depend on $q$ in this representation. $π(Xi+)= Eii+1, π(Xi-)= Ei+1i, π(Hi)= Eii-Ei+1,i+1 (4.1)$ where ${\left({E}_{ij}\right)}_{\alpha \beta }={\delta }_{i\alpha }{\delta }_{j\beta }$ are the basic matrices in ${ℂ}^{n}\text{.}$

Tho Tensorial square of ${V}^{{\omega }_{1}}$ is decomposed into the sum of two irreducible components $(Vω1)⊗2= V2ω1⊕Vω2. (4.2)$

The KGC of this decomposition are determined by the relations: $eij2ω1= 11+q-1 (ei⊗ej+q-12ej⊗ei), i where ${e}_{ij}^{2{\omega }_{1}}$ and ${e}_{ij}^{{\omega }_{2}}$ are the bases in ${V}^{2{\omega }_{1}}$ and in ${V}^{{\omega }_{2}}$ respectively.

From the theorem 1.5 we find the basic $R\text{-matrix}$ for ${U}_{q}\left(𝔤𝔩\left(n\right)\right)\text{:}$ $Rω1ω1= P2ω1ω1ω1 q1/2- Pω2ω1ω1 q-1/2 (4.5)$ or using (4.3), (4.4) $Rω1ω1= q1/2∑i=1n Eii⊗Eii+ ∑i≠jn Eij⊗Eji- (q-1/2-q1/2) ∑j>iEjj ⊗Eii. (4.6)$ In the form (4.6) the matrix ${R}^{{\omega }_{1}{\omega }_{1}}$ was found by Jimbo [Jim1985].

For $𝔤=𝔤𝔩\left(n\right)$ the decomposition of ${V}^{\lambda }\otimes {V}^{{\omega }_{1}}$ is well known: $Vλ⊗Vω1= ∑k=1n′ ⊕Vλ(k) (4.7)$ where ${\lambda }^{\left(k\right)}=\left({\lambda }_{1},\dots ,{\lambda }_{k}+1,{\lambda }_{k+1},\dots ,{\lambda }_{n}\right)$ if $\lambda =\left({\lambda }_{1},\dots ,{\lambda }_{n}\right),$ $\left({\lambda }_{i}\in {ℤ}_{+},{\lambda }_{i}\ge {\lambda }_{i+1}\right)$ and the sum is taken only over $k$ satisfying the condition ${\lambda }_{k}>{\lambda }_{k+1}\text{.}$ Using the rule (4.7) and the theorem 3.1 one can calculate the elements ${P}_{\lambda }\left(q,a\right)\text{.}$ The representations $\lambda =k{\omega }_{1}$ and $\lambda ={\omega }_{k}$ are contained in ${\left({V}^{{\omega }_{1}}\right)}^{\oplus k}$ with multiplicity equal to one. It is not difficult to calculate the corresponding projectors ${P}_{\lambda }\text{:}$ $Pkω1 = qk(k-1)4[k]! ∑w∈Sk qℓ(w)2 πw, (4.8) Pωk = q-k(k-1)4[k]! ∑w∈Sk (-q)-ℓ(w)2 πw, (4.9)$ where the sum taken is over the element of symmetric group ${S}_{k},$ ${\pi }_{{s}_{i}}={g}_{i}$ if ${S}_{i}$ is the elementary transposition in ${S}_{k},$ the elements ${g}_{i}$ are defined by (3.2) and ${\pi }_{w}={\pi }_{{s}_{1}}\dots {\pi }_{{s}_{k}}$ if $w={s}_{1}\dots {s}_{k}$ is the representation of $w$ in the nonreducible product of the transpositions, $\left[k\right]!=\left[k\right]\dots \left[1\right],$ $\left[k\right]=\left({q}^{k/2}-{q}^{-k/2}\right)/\left({q}^{1/2}-{q}^{-1/2}\right)\text{.}$ The formulae (4.8), (4.9) were obtained in [Jim1985] as the $q\text{-analogs}$ of the Young sym (antisym) metrisators. The following theorem describes the connection between the algebra ${C}_{N}^{{\omega }_{1}}\left(𝔤𝔩\left(n\right)\right)$ and Hecke algebra ${ℋ}_{N}\left(q\right)\text{.}$ The last one is the assotiative algebra with units generated by the elements $1,{\stackrel{ˆ}{g}}_{i},$ $i=1,\dots ,N-1$ satisfying the following relations: $gˆi gˆi+1 gˆi = gˆi+1 gˆi gˆi+1 gˆi gˆj = gˆj gˆi, |i-j|>1 gˆi2 = (q-1) gˆi+q. (4.10)$

Theorem 4.1. The algebra ${C}_{N}^{{\omega }_{1}}\left(𝔤𝔩\left(n\right)\right)$ is the factor of Hecke algebra ${ℋ}_{N}\left(q\right)$ over the ideal ${J}_{N}$ formed by the elements $∑w∈Sn+1 (-q)-ℓ(w)2 πw=0$ where ${\pi }_{w}$ are the same as in (4.8) and (4.9). If $n+1>N,$ ${C}_{N}^{{\omega }_{1}}\left(𝔤𝔩\left(n\right)\right)\simeq {ℋ}_{N}\left(q\right)\text{.}$ The generators ${g}_{i}$ of ${C}_{N}^{{\omega }_{1}}\left(𝔤𝔩\left(n\right)\right)$ correspond to the generators ${\stackrel{ˆ}{g}}_{i}$ of Hecke algebra ${\stackrel{ˆ}{g}}_{i}={q}^{1/2}{g}_{i}\text{.}$

This theorem in equivalent to the fact that in the space ${\left({V}^{{\omega }_{1}}\right)}^{\otimes N}$ the representation of Hecke algebra $\alpha \left({\stackrel{ˆ}{g}}_{i}\right)={q}^{1/2}{g}_{i}$ (this fact was noted by Jimbo [Jim1985]) is defined. The theorem 4.1 means that the irreducible components of this representation of ${ℋ}_{N}\left(q\right)$ have the Young diagrams $\lambda$ with no more than $n$ rows.

An explicit expression for matrix elements of ${g}_{i}$ in the basics (3.4) of the representation ${W}_{\lambda }$ can be found from (3.9). Omitting the details, we give the answer: $w(λ,λ+ek,λ+ek+eℓ,λ+eℓ) = (qd+12-q-d+12) (qd-12-q-d-12) (q12+q-12) (qd2-q-d2) , (4.11) w(λ,λ+ek,λ+ek+eℓ,λ+ek) = { q12-q-12 qd-1 , k≠ℓ q-12, k=ℓ -q12, ℓ=k+1,λk=λℓ (4.12)$ where $d={\lambda }_{k}-k-{\lambda }_{\ell }+\ell \text{.}$ This expression was given in [Wen1985] and [Ker1987] following a different argumentation.

## Footnotes

${}^{*\text{)}}$ From this point we will use the normalization and the notation of books [Bou1981] for the highest weights and the roots of simple Lie algebras.