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

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

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-analog of Brauer-Weyl duality

In this seotion we study the structure of centralizater of Uq(𝔤) in (Vω)N, where Vω is the basic representation.

Let V{λ}=Vλ1VλN and V{λσ}=Vλσ1VλσN, where σSN is the permutation of the set (1,,N).

Definition 3.1. An algebra of the maps 𝒴:V{λσ}V{λσ} commuting with the action of Vq(𝔤) in these spaces is called the centralizer of Uq(𝔤) in Vλ1VλN and denoted by Cλ1λN(𝔤).

Here we will consider only the algebras Cωω(𝔤,q)Cnω(𝔤,q)*) for the representations ω, when multiplicity of the irreducible components in the decomposition VλVω=μ Vμ (3.1) is equal to one for any IFR Vλ.

Proposition 3.1. The algebra Cnω(𝔤,q) 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 Rωω.

The representations of CNω(𝔤) are given by the decomposition of (Vω)N on the Uq(𝔤)-irreducible components: (Vω)N= λWλVλ. (3.3)

The space Wλ describes the multiplicity of the h.w. λ in (3.3): dimWλ=mult(Vλ).

Proposition 3.2. The representations Wλ in (3.3) form the list of all irreducible representations of CNω(𝔤).

Let us use now the graphical representation of the matrices Rλμ and Kνλμ developed in section 2 to describe the basis in Wλ.

Proposition 3.3. The elements ω ω ω ω ω λ2 λN-1 λ Kλ(q,a), (3.4) where a=(ω,λ2,,λN-1,λ) and VλiVωVλi-1 form the orthogonal basis in Wλ. More precisely Kλ(q,a)= Eλ(a)Iλ where {Eλ(a)} is the basis in Wλ and Iλ is the unit matrix in Vλ.

The orthogonality of the elements Kλ(q,a) means that Kλ(q,b) Kλ(q,a)t= δa,b·I, (3.5) where a=(ω,λ2,,λN-1,λ), b=(ω,λ2,,λN-1,λ). Kλt is the transposition of the matrix Kλ. The relation (3.5) is proved by the following equalities founded on the orthogonality of CGC: ω ω ω ω λ2 λ λ2 λ =δλ2λ2 λ2 ω ω λ3 λ λ3 λ ==j=2N-1 δλjλj λ (3.6) The completeness of this basis in Wλ is evident.

Proposition 3.4. The action of the elements gi in basis have the following form: giKλ(q,a) =λiW (λi-1,λi,λi+1,λi) Kλ(q,a) (3.7) where a=(ω,λ2,,λN-1,λ), a=(ω,λ2,,λi,,λN-1,λ) and w(λi-1,λi,λi+1,λi) are some constants calculated below.

The relations (3.7) have a simple graphical representation: ω ω ω ω ω λ2 λi λ =λi ω ω ω ω ω λ2 λi λ w(λi1,λi,λi1,λi). (3.8) Using the orthogonality of the basis Kλ(q,a) we obtain a formula for the coefficients W:

Proposition 3.5. The coefficients W in (3.6) are calculated by the following praphical rule: w(λ1,λ2,λ3,λ2)= λ1 λ2 λ2 λ3* 1χλ2. (3.9) An exact meaning of this formula is given in section 8.

The action of gi-1 in Kλ basis is similar to (3.7): gi-1Kλ(a) =λiw (λi-1,λi,λi+1,λi) Kλ(a) (3.10) where the coefficient w has the following form: w(λ1,λ2,λ3,λ2)= λ1 λ2 λ2 λ3* 1χλ3 (3.11) From gigi-1=1 follows the relation λiw (λi-1,λi,λi+1,λi) w(λi-1λi,λi+1λi) =δλiλi. (3.12)

So, we described the algebra CNω(𝔤) and its irreducible representations. It is necessary to note that the basis (3.4) is exactly a q-analog of the Young basis in irreducible representations of symmetric group SN. This basis is regular for the embeddings C1ω(𝔤) C2ω(𝔤) CNω(𝔤) (3.13) where Cnω(𝔤) is formed by the elements 1,,gN-1 and CN-1ω(𝔤) is formed by the elements 1,,gN-2. Indeed, if we consider the restriction wλ(CNω) |CN-1ω = λwλ (CN-1ω) (3.14) the basis in wλ(CN-1ω) will be formed by the elements Kλ(aλ), where aλ=(ω,,λN-2,λ,λ).

Let us also note that the decompositions (3.14) determine tho graph of the algebra CNω(𝔤) [KVe1985]. It is evident that really this graph is fixed by decomposition (3.3).

Let us consider now the q-analog of Young symmetrizers. For this purpose we introduce the matrices Pλ(q,a):(Vω)N(Vω)N, Im(Pλ(q,a))Vλ: Pλ(q,a)= Kλt(q,a) Kλ(q,a)= (Eλ(a)Eλt(a)) Iλ. (3.15) This matrix corresponds to the figure λ a a ω ω ω ω ω ω = λ λN-1 λN-1 λ2 λ2 ω ω ω ω ω ω ω ω (3.16)

Proposition 3.3 follows that the set Pλ(q,a) forms an orthogonal set of projectors in (Va)N: Pλ(q,a) Pμ(q,b)= δλμδa,b. (3.17)

Theorem 3.1. The elements Pλ satisfy the following relations: λ a a ω ω ω ω ω ω =n=0r-1 qn,λ a a ω ω ω ω ω ω } 2ncrosses (3.18) where a=(ω,λ2,,λN-1,λ), a=(ω,λ2,,λN-1), qn,λ are the matrix elements of the matrix: A= (qnc(λ)-c(λN-1)-c(ω)2) λλN-1ω,n=0,,r-1 (3.19) r is the number of irreducible components in λN-1ω, the index n numbers the rows of A and λ numbers the columns of A.


Let us consider the matrix RλN-1ω(q,a) corresponding to the figure a a ω ω ω ω ω λN-1

From the theorem 1.5 we have the decomposition or matrix RλN-1ω(q,a)RωλN-1(q,a) a a ω ω ω ω ω λN-1 =λλN-1ω qc(λ)-c(λN-1)-c(ω)2 a a ω ω ω ω λ . Taking powers of degrees n=0,1,,r-1 of this equality we obtain a system for the projectors Pλ(q,a). Solving this system we obtain the relation (3.18).

Considering the relation (3.18) as an inductive process for calculating Pλ(q,a) we obtain the explicit expression for Pλ(q,a): Pλ(q,a)= n1=0r1-1 nN-1=0rN-1-1 qn1λ qn2λN-1 qnN-1,ω× ω ω ω ω α(n1,,nN-1) . (3.19) Here α(n1,,nN-1) is a braid (see section 8) where the string with number i turned round the strings with numbers 1,,i-1 (from left to right) 2ni times moving counter-clockwise and form the top to the bottom.

Now we can describe the structure of the matrices Rλμ.

Consider a decomposition of VλVμ into the sum or Uq(𝔤) irreducible components: VλVμ= νWνλμ Vν (3.20)

The matrix Rλμ is an element of End(VλVμ)End(VλVμ) commuting with the action of Uq(𝔤). Hence it has the following form: Rλμ=ν RνλμIν (3.21) where Rνλμ:WνλμWνλμ.

Let Wν be the CN+Mω-irreducible module. The decomposition of this module into tho sum of CNω×CMω-irreducible components is: Wν|CNω×CMω =λμWνλμ WλWμ (3.22) where the spaces Wνλμ are the same as in (3.20).

Let R(N,M) be the matrix acting in (Vω)(N+M) of the form: R(N,M) (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 σλμ:WλWμWμWλ is the the permutation operator σλμ(fλgμ)=gμfλ.

Tho decomposition (3.24) and (3.25) is the generalization of the formula (1.37) on matrices Rλμ with arbitrary λ and μ. The formula (3.25) gives the method for calculating the matrices Rλμ.


*) We will write CNω(𝔤) when it will not be ambiguous.

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