## 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 $q\text{-analog}$ of Brauer-Weyl duality

In this seotion we study the structure of centralizater of ${U}_{q}\left(𝔤\right)$ in ${\left({V}^{\omega }\right)}^{\otimes N},$ where ${V}^{\omega }$ is the basic representation.

Let ${V}^{\left\{\lambda \right\}}={V}^{{\lambda }_{1}}\otimes \dots \otimes {V}^{{\lambda }_{N}}$ and ${V}^{\left\{{\lambda }_{\sigma }\right\}}={V}^{{\lambda }_{{\sigma }_{1}}}\otimes \dots \otimes {V}^{{\lambda }_{{\sigma }_{N}}},$ where $\sigma \in {S}_{N}$ is the permutation of the set $\left(1,\dots ,N\right)\text{.}$

Definition 3.1. An algebra of the maps $𝒴:{V}^{\left\{{\lambda }_{\sigma }\right\}}\to {V}^{\left\{{\lambda }_{\sigma \prime }\right\}}$ commuting with the action of ${V}_{q}\left(𝔤\right)$ in these spaces is called the centralizer of ${U}_{q}\left(𝔤\right)$ in ${V}^{{\lambda }_{1}}\otimes \dots \otimes {V}^{{\lambda }_{N}}$ and denoted by ${C}^{{\lambda }_{1}\dots {\lambda }_{N}}\left(𝔤\right)\text{.}$

Here we will consider only the algebras ${C}^{\omega \dots \omega }\left(𝔤,q\right)\equiv {C}_{n}^{\omega }{\left(𝔤,q\right)}^{*\text{)}}$ for the representations $\omega ,$ when multiplicity of the irreducible components in the decomposition $Vλ⊗Vω=∑μ ⊕Vμ (3.1)$ is equal to one for any IFR ${V}^{\lambda }\text{.}$

Proposition 3.1. The algebra ${C}_{n}^{\omega }\left(𝔤,q\right)$ is simple for general $q$ and is generated by the elements $𝔤i=1⊗…⊗ Rωω(i,i+1) ⊗…⊗1 (3.2)$ if the spectrum of the Casimir operator is simple in the decomposition (3.1).

The proof of this proposition is based on the spectral decomposition of the matrix ${R}^{\omega \omega }\text{.}$

The representations of ${C}_{N}^{\omega }\left(𝔤\right)$ are given by the decomposition of ${\left({V}^{\omega }\right)}^{\otimes N}$ on the ${U}_{q}\left(𝔤\right)\text{-irreducible}$ components: $(Vω)⊗N= ∑λWλ⊗Vλ. (3.3)$

The space ${W}_{\lambda }$ describes the multiplicity of the h.w. $\lambda$ in (3.3): $\text{dim} {W}_{\lambda }=\text{mult}\left({V}^{\lambda }\right)\text{.}$

Proposition 3.2. The representations ${W}_{\lambda }$ in (3.3) form the list of all irreducible representations of ${C}_{N}^{\omega }\left(𝔤\right)\text{.}$

Let us use now the graphical representation of the matrices ${R}^{\lambda \mu }$ and ${K}_{\nu }^{\lambda \mu }$ developed in section 2 to describe the basis in ${W}_{\lambda }\text{.}$

Proposition 3.3. The elements $\cdots \cdots \omega \omega \omega \omega \omega {\lambda }_{2} {\lambda }_{N-1} \lambda ⟷Kλ(q,a), (3.4)$ where $a=\left(\omega ,{\lambda }_{2},\dots ,{\lambda }_{N-1},\lambda \right)$ and ${V}^{{\lambda }_{i}}\subset {V}^{\omega }\otimes {V}^{{\lambda }_{i-1}}$ form the orthogonal basis in ${W}_{\lambda }\text{.}$ More precisely $Kλ(q,a)= Eλ(a)⊗Iλ$ where $\left\{{E}_{\lambda }\left(a\right)\right\}$ is the basis in ${W}_{\lambda }$ and ${I}_{\lambda }$ is the unit matrix in ${V}^{\lambda }\text{.}$

The orthogonality of the elements ${K}_{\lambda }\left(q,a\right)$ means that $Kλ(q,b) Kλ(q,a)t= δa,b·I, (3.5)$ where $a=\left(\omega ,{\lambda }_{2},\dots ,{\lambda }_{N-1},\lambda \right),$ $b=\left(\omega ,{\lambda }_{2}^{\prime },\dots ,{\lambda }_{N-1}^{\prime },\lambda \right)\text{.}$ ${K}_{\lambda }^{t}$ is the transposition of the matrix ${K}_{\lambda }\text{.}$ The relation (3.5) is proved by the following equalities founded on the orthogonality of CGC: $\omega \omega \omega \cdots \omega {\lambda }_{2} \lambda {\lambda }_{2}^{\prime } \lambda \cdots \cdots =δλ2λ2′ {\lambda }_{2} \omega \omega {\lambda }_{3} \lambda {\lambda }_{3}^{\prime } \lambda =…=∏j=2N-1 δλjλj′ \lambda (3.6)$ The completeness of this basis in ${W}_{\lambda }$ is evident.

Proposition 3.4. The action of the elements ${g}_{i}$ in basis have the following form: $giKλ(q,a) =∑λi′W (λi-1,λi,λi+1,λi′) Kλ(q,a′) (3.7)$ where $a=\left(\omega ,{\lambda }_{2},\dots ,{\lambda }_{N-1},\lambda \right),$ $a\prime =\left(\omega ,{\lambda }_{2},\dots ,{\lambda }_{i}^{\prime },\dots ,{\lambda }_{N-1},\lambda \right)$ and $w\left({\lambda }_{i-1},{\lambda }_{i},{\lambda }_{i+1},{\lambda }_{i}^{\prime }\right)$ are some constants calculated below.

The relations (3.7) have a simple graphical representation: $\cdots \cdots \omega \omega \omega \omega \omega {\lambda }_{2} {\lambda }_{i} \lambda =∑λi′ \cdots \cdots \omega \omega \omega \omega \omega {\lambda }_{2} {\lambda }_{i} \lambda w(λi1,λi,λi1,λi′). (3.8)$ Using the orthogonality of the basis ${K}_{\lambda }\left(q,a\right)$ we obtain a formula for the coefficients $W\text{:}$

Proposition 3.5. The coefficients $W$ in (3.6) are calculated by the following praphical rule: $w(λ1,λ2,λ3,λ2′)= {\lambda }_{1} {\lambda }_{2} {\lambda }_{2}^{\prime } {\lambda }_{3}^{*} 1χλ2. (3.9)$ An exact meaning of this formula is given in section 8.

The action of ${g}_{i}^{-1}$ in ${K}_{\lambda }$ basis is similar to (3.7): $gi-1Kλ(a) =∑λi′w‾ (λi-1,λi,λi+1,λi′) Kλ(a′) (3.10)$ where the coefficient $\stackrel{‾}{w}$ has the following form: $w‾(λ1,λ2,λ3,λ2′)= {\lambda }_{1} {\lambda }_{2} {\lambda }_{2}^{\prime } {\lambda }_{3}^{*} 1χλ3 (3.11)$ From ${g}_{i}{g}_{i}^{-1}=1$ follows the relation $∑λi′w (λi-1,λi,λi+1,λi′) w‾(λi-1λi′,λi+1λi″) =δλiλi″. (3.12)$

So, we described the algebra ${C}_{N}^{\omega }\left(𝔤\right)$ and its irreducible representations. It is necessary to note that the basis (3.4) is exactly a $q\text{-analog}$ of the Young basis in irreducible representations of symmetric group ${S}_{N}\text{.}$ This basis is regular for the embeddings $C1ω(𝔤)⊂ C2ω(𝔤)⊂…⊂ CNω(𝔤) (3.13)$ where ${C}_{n}^{\omega }\left(𝔤\right)$ is formed by the elements $1,\dots ,{g}_{N-1}$ and ${C}_{N-1}^{\omega }\left(𝔤\right)$ is formed by the elements $1,\dots ,{g}_{N-2}\text{.}$ Indeed, if we consider the restriction $wλ(CNω) |CN-1ω = ∑λ′⊕wλ (CN-1ω) (3.14)$ the basis in ${w}_{\lambda \prime }\left({C}_{N-1}^{\omega }\right)$ will be formed by the elements ${K}_{\lambda }\left({a}_{\lambda \prime }\right),$ where ${a}_{\lambda \prime }=\left(\omega ,\dots ,{\lambda }_{N-2},\lambda \prime ,\lambda \right)\text{.}$

Let us also note that the decompositions (3.14) determine tho graph of the algebra ${C}_{N}^{\omega }\left(𝔤\right)$ [KVe1985]. It is evident that really this graph is fixed by decomposition (3.3).

Let us consider now the $q\text{-analog}$ of Young symmetrizers. For this purpose we introduce the matrices ${P}_{\lambda }\left(q,a\right):{\left({V}^{\omega }\right)}^{\otimes N}\to {\left({V}^{\omega }\right)}^{\otimes N},$ $\text{Im}\left({P}_{\lambda }\left(q,a\right)\right)\simeq {V}^{\lambda }\text{:}$ $Pλ(q,a)= Kλt(q,a) Kλ(q,a)= (Eλ(a)⊗Eλt(a)) ⊗Iλ. (3.15)$ This matrix corresponds to the figure $\lambda a a \cdots \cdots \omega \omega \omega \omega \omega \omega = \cdots \cdots \cdots \cdots \lambda {\lambda }_{N-1} {\lambda }_{N-1} {\lambda }_{2} {\lambda }_{2} \omega \omega \omega \omega \omega \omega \omega \omega (3.16)$

Proposition 3.3 follows that the set ${P}_{\lambda }\left(q,a\right)$ forms an orthogonal set of projectors in ${\left({V}^{a}\right)}^{\otimes N}\text{:}$ $Pλ(q,a) Pμ(q,b)= δλμδa,b. (3.17)$

Theorem 3.1. The elements ${P}_{\lambda }$ satisfy the following relations: $\lambda a a \cdots \cdots \omega \omega \omega \omega \omega \omega =∑n=0r-1 qn,λ a a \omega \cdots \omega \omega \omega \cdots \omega \omega \phantom{\rule{0ex}{1.9em}}\right\} 2n \text{crosses} (3.18)$ where $a=\left(\omega ,{\lambda }_{2},\dots ,{\lambda }_{N-1},\lambda \right),$ $a\prime =\left(\omega ,{\lambda }_{2},\dots ,{\lambda }_{N-1}\right),$ ${q}_{n,\lambda }$ are the matrix elements of the matrix: $A= (qnc(λ)-c(λN-1)-c(ω)2) λ∈λN-1⊗ω, n=0,…,r-1 (3.19)$ $r$ is the number of irreducible components in ${\lambda }_{N-1}\otimes \omega ,$ the index $n$ numbers the rows of $A$ and $\lambda$ numbers the columns of $A\text{.}$

 Proof. Let us consider the matrix ${R}^{{\lambda }_{N-1}\omega }\left(q,a\prime \right)$ corresponding to the figure $a\prime a\prime \omega \cdots \omega \omega \omega \cdots \omega {\lambda }_{N-1}$ From the theorem 1.5 we have the decomposition or matrix ${R}^{{\lambda }_{N-1}\omega }\left(q,a\prime \right){R}^{\omega {\lambda }_{N-1}}\left(q,a\prime \right)$ $a\prime a\prime \omega \cdots \omega \omega \omega \cdots \omega {\lambda }_{N-1} =∑λ∈λN-1⊗ω qc(λ)-c(λN-1)-c(ω)2 a a \omega \cdots \omega \omega \cdots \omega \lambda .$ Taking powers of degrees $n=0,1,\dots ,r-1$ of this equality we obtain a system for the projectors ${P}_{\lambda }\left(q,a\right)\text{.}$ Solving this system we obtain the relation (3.18). Considering the relation (3.18) as an inductive process for calculating ${P}_{\lambda }\left(q,a\right)$ we obtain the explicit expression for ${P}_{\lambda }\left(q,a\right)\text{:}$ $Pλ(q,a)= ∑n1=0r1-1… ∑nN-1=0rN-1-1 qn1λ qn2λN-1… qnN-1,ω× \omega \cdots \omega \omega \cdots \omega {\alpha }_{\left({n}_{1},\dots ,{n}_{N-1}\right)} . (3.19)$ Here ${\alpha }_{\left({n}_{1},\dots ,{n}_{N-1}\right)}$ is a braid (see section 8) where the string with number $i$ turned round the strings with numbers $1,\dots ,i-1$ (from left to right) $2{n}_{i}$ times moving counter-clockwise and form the top to the bottom. $\square$

Now we can describe the structure of the matrices ${R}^{\lambda \mu }\text{.}$

Consider a decomposition of ${V}^{\lambda }\otimes {V}^{\mu }$ into the sum or ${U}_{q}\left(𝔤\right)$ irreducible components: $Vλ⊗Vμ= ∑νWνλμ ⊗Vν (3.20)$

The matrix ${R}^{\lambda \mu }$ is an element of $\text{End}\left({V}^{\lambda }\otimes {V}^{\mu }\right)\to \text{End}\left({V}^{\lambda }\otimes {V}^{\mu }\right)$ commuting with the action of ${U}_{q}\left(𝔤\right)\text{.}$ Hence it has the following form: $Rλμ=∑ν Rνλμ⊗Iν (3.21)$ where ${R}_{\nu }^{\lambda \mu }:{W}_{\nu }^{\lambda \mu }\to {W}_{\nu }^{\lambda \mu }\text{.}$

Let ${W}_{\nu }$ be the ${C}_{N+M}^{\omega }\text{-irreducible}$ module. The decomposition of this module into tho sum of ${C}_{N}^{\omega }×{C}_{M}^{\omega }\text{-irreducible}$ components is: $Wν|CNω×CMω =∑λμWνλμ ⊗Wλ⊗Wμ (3.22)$ where the spaces ${W}_{\nu }^{\lambda \mu }$ are the same as in (3.20).

Let ${R}^{\left(N,M\right)}$ be the matrix acting in ${\left({V}^{\omega }\right)}^{\otimes \left(N+M\right)}$ of the form: $R(N,M)⟷ \cdots \cdots \cdots \cdots (3.23)$

In accordance with (3.3) this matrix has the following decomposition $R(N,M)=∑ν Rν(N,M)⊗Iν$ $Rν(N,M)= ∑λ,μ Rνλμ⊗ σλμ (3.25)$ where ${\sigma }^{\lambda \mu }:{W}_{\lambda }\otimes {W}_{\mu }\to {W}_{\mu }\otimes {W}_{\lambda }$ is the the permutation operator ${\sigma }^{\lambda \mu }\left({f}_{\lambda }\otimes {g}_{\mu }\right)={g}_{\mu }\otimes {f}_{\lambda }\text{.}$

Tho decomposition (3.24) and (3.25) is the generalization of the formula (1.37) on matrices ${R}^{\lambda \mu }$ with arbitrary $\lambda$ and $\mu \text{.}$ The formula (3.25) gives the method for calculating the matrices ${R}^{\lambda \mu }\text{.}$

## Footnotes

${}^{*\text{)}}$ We will write ${C}_{N}^{\omega }\left(𝔤\right)$ when it will not be ambiguous.