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 solutions of Yang-Baxter equation connected with the q-deformations of universal enveloping algebras of simple Lie algebras

The q-deformation of the universal enveloping algebras (or quantum universal enveloping algebras QUEA) Uq(𝔤) arose when studying some special class of integrable systems. The definition of Uq(𝔤) for any simple Lie algebra 𝔤 was given in [Dri1985,Dri1986-3,Jim1985]. One can define these algebras by generators and relations.

Definition 1.1 [Dri1985,Dri1986-3,Jim1985]. QUEA Uq(𝔤) is the algebra with generators Xi±, Hi, i=1,,r=rank𝔤 and with the relations: [Hi,Hj]=0, [Hj,Xi±]=± (αjαi)Xi± (1.1) [Xi+,Xj-]= δij sh(h2Hi) sh(h2) ,q=eh (1.2) ij, k=0n (-1)k (nk)qi qi-k(n-k)2 (Xi±)k Xj± (Xi±)n-k =0,qi= q(αiαi)2 (1.3) where Aij is the Cartan matrix of 𝔤, (α,β) is the scalar product of the roots (Aij=(αiαj)(αjαj)), n=1-Aij, ij.

The generators Xi±, Hi play the role of Chevalley basis in U(𝔤). The elements Hi form the commutative subalgebra U(f)Uq(𝔤). This subalgebra is the analog of Cartan subalgebra in U(𝔤). The elements Hi correspond to the simple roots of 𝔤. Let α=i=1rniαi be the root of 𝔤, a corresponding element of f we denote as Hα=i=1rniHi.

Theorem 1.1 [Dri1985,Dri1986-3,Jim1985]. The algebra Uq(𝔤) is the Hopf algebra [Abe1980] (see also Appendix A) with comultiplication and with the antipode γ defined on the generators by the following formulae: Δ(Hi)= Hi1+1Hi, (1.4) Δ(Xi±)= Xi±eh4Hi +e-h4HiXi±, (1.5) γ(Hi)=-Hi, γ(Xi±)=- eh2ρXi± e-h2ρ, (1.6) where ρ=12αΔ+Hα is the element of f and Δ+ is the positive roots of 𝔤.

The parameter q is the deformation parameter. At q=1 the algebra Uq(𝔤) is the universal enveloping algebra of 𝔤. For simple Lie algebras of the algebras Uq(𝔤) are simple for general q. The special values of q are situated in the rational points on the unit circle.

Let 𝔟±𝔤 be the Borel subalgebras in 𝔤. The algebras Uq(𝔟+) and Uq(𝔟-) with generators (Hi,Xi+) and (Hi,Xi-) are the Hopf subalgebras in Uq(𝔤). There is a duality between Uq(𝔟+) and Uq(𝔟-). Let {es} is the basis in Uq(𝔟-) and {es} be the dual basis in Uq(𝔟+).

It is not difficult to prove that the map σΔ:Uq(𝔤)Uq(𝔤)Uq(𝔤), where σ is the permutation in Uq(𝔤)Uq(𝔤) also defines the structure of Hopf algebra on Uq(𝔤) with antipode γ=γ-1.

Theorem 1.2 [Dri1985,Dri1986-3]. The comultiplication Δ and σΔ are connected by the following relation σΔ(a)= RΔ(a)R-1 aUq(𝔤) (1.7) where σ is the permutation operator in Uq(𝔤)Uq(𝔤) and R=sesesUq(𝔤)Uq(𝔤).

The proof of this theorem the following one is based on the construction of double Hopf algebra [Jim1985].

Theorem 1.3. The element R satisfies the relations (Δid)R =R13R23, (1.8a) (idΔ)R= R13R12, (1.8b) R12R13R23= R23R13R12, (1.8c) (idγ)R= R-1. (1.8d) Here R12,R13,R23Uq(𝔤)3, R12=eses1, R13=es1es, R23=1eses.


Let m,μ and γ be the matrices of multiplication, comultiplication and antipode in algebra Uq(𝔟-): eset=mstτ eτ,Δ(es) =μstτeteτ ,γ(es)= γstet. The matrices μ,m and γ-1 are the matrices of multiplication, comultiplication and antipode in Uq(𝔟+): eset=μτst eτ,Δ(es) =mtτseτet, γ(es)= (γ-1)ts et. From (1.7) we have the commutation relations between the elements es and et in Uq(𝔤): μspqmtpτ eteq=μspq mqtτepet. From this commutation relations and from the definition of R follows (1.8c). To prove (1.8d) we use the definition of antipode (A.): R(idγ)R=es etesγ(et) =mstτeτ γptμqspeq= mstτγptμqsp eτeq=ee. Similarly (idγ)(R)·R= mstτγps μqpteτ eq=ee. The following formulae prove the relations (1.8a) and (1.8b) (Δid)R= Δ(es) es=μsτt eτetes =eτet eτet=R13 R23, (idΔ)R= esΔ(es) =esmτts eteτ=eτ eteteτ =R13R12. The theorem is proved.

The algebras Uq(𝔤) and 𝔤 have a common Cartan subalgebra f generated by the elements Hi. This means that the theories of the finite dimensional representations of Uq(𝔤) and 𝔤 are quite parallel. The following two statements will play the main role in the text.

Proposition 1.1. The finite dimensional representations of Uq(𝔤) are completely irreducible.

Proposition 1.2.

(a) The irreducible finlte-dimensional representations (IFR) of Uq(𝔤) are parametrized by the highest weights λ of the algebra 𝔤.
(b) Any irreducible finite dimensional module Vλ is decomposed into the sum of the weight spaces Vλ=μVλ(μ) and the dimensions of Vλ(μ) are the same as for IFR of 𝔤.

From the complete irreducibility of finite dimensional representations of Uq(𝔤) it follows that any IFR can be considered as some irreducible component of corresponding tensorial power of basic representations. The list of basic representation is given in table 1. 𝔤category An Bn Cn Dn G2 E6 E7 E8 F4 finite dimensional ω1 ω1 ω1 ω1 ω1 ω1 ω7 ω8 ω4 tensorial represent - ωn - ω1,ωn - - - - - Table 1

For algebras Bn and Dn it is natural to separate from the finite dimensional representations the category of tensorial representations (with integers highest weights). In this category the basic representation is vectorial representation with h.w. ω1 in both cases.

It is very important for our purposes that all finite dimensional representations of Uq(𝔤) are contained in tensorial powers of the basic representations.

Below we study the restrictions of universal R-matrices on the IFR's. If ρλ and ρμ are IFR of Uq(𝔤) and Pλμ is the permutation operator on VλVμ (Pλμ:VλVμVμVλ, Pλμ(fg)=gf) then we denote the matrix Pλμ(ρλρμ)(R) as Rλμ and we call it the R-matrix in λ,μ representation (or λ,μ R-matrix). This matrix map VλVμ on VμVλ and satisfies the following relations: (RλμI) (IRνμ) (RνλI)= (IRνλ) (RνμI) (IRλμ). (1.9) The left and the right sides of these relations act from VνVλVμ to VμVλVν.

To find the matrices Rλμ one can use two ways. The first one is the explicit projection of the universal R-matrix by factorising both copies of Uq(𝔤) over corresponding ideals. The second way uses the complete irreducibility of IFR of Uq(𝔤). Because all of IFR are contained in tensorial powers of basic representations of Uq(𝔤) one can construct the matrices Rλμ from a tensorial product of the R-matrices acting in basic representations. The second way gives also the important relations between the matrices Rλμ and we will use this way below.

Consider a Cartan antiinvolution in Uq(𝔤) θ(Xi±)= Xi, θ(Hi)=Hi and automorphism δ corresponding to on authomorphism of Dynkin diagram: δ(Xi±)= Xi*±, δ(Hi)=Hi*.

We define a q-analog C of the element with the maximal length of the Weyl group by the formula γ=δAdCθ.

It is not difficult to check that in basic representations the element C has the form C=Sqρ/2 where S is the corresponding element of the Weyl group for U(𝔤).

(b) The element R satisfies the relation (idδ) (R-1)= (1C) (idθ) (R)(1C-1).

Definition 1.2. If VνVλVμ, the projection matrix Kνλμ(q,a):VλVμVν is called the matrix of Klebsch-Gordan coefficient (KGC) (here index a numbers the components Vν in VλVμ if the multiplicity of Vν is more than one).

The matrices Kνλμ and Kνμλ are orthogonal if νν. They are normalized when mult(Vν)=mult(Vν)=1 Kνλμ (Kνμλ) =δννIν. (1.14)

Theorem 1.4. The matrices Rλμ and Kνλμ for mult(Vν)=1 satisfy the following relations Pμλ R(q)λμ Pμλ = SS R(q)μλ s-1s-1 =Rλμ (q-1)-1, (1.15) C-1Kνλμ (q)(CC) = (-1)ν Kν*λ*μ* (q-1), (1.16) Kνλμ(q) Pμλ = (-1)ν Kνμλ(q-1), (1.17) Rλμ(q) = Rλμ(q). (1.18)

Here ν=0,1 is the parity of Vν in VλVμ and is the transposition.

Following theorems are the keys ones to study the matrices Rλμ.

Theorem 1.5. Let VνVλVμ and mult(Vν)=1 then Rλμ(q) (Kνλμ(q)) = (-1)ν qc(ν)-c(λ)-c(μ)4 Kνμλ(q) (1.19) Kνμλ(q) Rλμ(q) = (-1)ν qc(ν)-c(λ)-c(μ)4 Kνλμ(q) (1.20) where ν is the parity of Vν in VλVμ and c(ν) is the value of Casimir operator of 𝔤 on the irreducible representation with highest weight ν: c=i=1r Hi2+ αΔXα X-α,Hi =(H,ωi) (1.21) c(ν)=ν2+ 2ρν. (1.22)


From the definition of universal R-matrix follows the equality RλμΔλμ (a)=Δμλ (a)Rλμ (1.23) where Δλμ(a)=(ρλρμ)Δ(a). The representation Δλμ is reducible and Kνλμ are the projectors on the irreducible components: Δλμ(a) Kνλμ = Kνλμ ρν(a) (1.24) Δμλ(a) Kνμλ = Kνμλ ρν(a). (1.25) From (1.23) and (1.24) we have Rλμ Kνλμ ρν(a)= Δμλ(a) Rλμ Kνλμ. (1.26) Comparing with (1.25) we obtain Rλμ Kνλμ= Rν(q) Kνμλ (1.27) where Rν(q) is some function of q,λ,μ,ν. To find Rν(q) we consider the limit q1. Let φνλμ(q) be the highest weight vector in VνVλVμ. From (1.27) follows the equality Rλμ(q) φνλμ(q) =Rν(q) φνμλ(q). (1.28) At q1 we have Rλμ = Pλμ (1+(q-1)τ+O((q-1)2)), (1.29) τ = 12iHi Hi+αΔ+ XαX-α (1.30) φνλμ(q) = φλμ+(q-1) ψνλμ+ O((q-1)2), (1.31) Rν(q) = (-1)ν (1+(q-1)sν+O((q-1)2)). (1.32) Here τ𝔤𝔤 (𝔤U(𝔤)) is the so-called classical τ-matrix [Bax1982,Fad1984]. φνλμ is the h.w.v. of 𝔤, and ψνλμ and sν some unknown vectors and constants.

Theorem 1.4 implies symmetry relations for φνλμ and ψνλμ: Pλμφνλμ =(-1)ν φνμλ, Pλμψνλμ =(-1)ν+1 ψνμλ. (1.33) Comparing the coefficients at (q-1) in (1.28) we obtain the equation for sν: Pλμτφνλμ +Pλμψνλμ =(-1)ν ψνμλ+ (-1)ν sνφνμλ. Multiplying this equality by (φνμλ) from the left and using the symmetry relations (1.33) we obtain the following expression for sν sν=(φνλμ) τφνλμ.

Using the symmetry relations (1.33) and the property of highest weight vectors the value of sν one can express through the Casimir operators sν=14 (c(ν)-c(λ)-c(μ)). (1.34)

From the symmetry relations (1.15) and from the definition of Rλμ we have Rν(q) Rν(q-1) =1 (1.35) Rν(q) (-1)ν qα,q. (1.36)

Moreover, the function Rν(q) has no poles and zeros for finite values of q. From Liouville theorem follows that there is only one function Rν(q) satisfying the relations (1.29), (1.34)-(1.36) Rν(q)= (-1)ν qc(ν)-c(λ)-c(μ)4.

The equality (1.20) is the transposition of (1.19). The Theorem is proved.

Consequence. When all irreducible components in VλVμ have the multiplicity equal to one the matrices Rλμ and (Rλμ)-1 have the following decompositions: Rλμ(q) = νλμ (-1)ν qc(ν)-c(λ)-c(μ)4 Pνλμ(q) (1.37) (Rλμ(q))-1 = νλμ (-1)ν q-c(ν)-c(λ)-c(μ)4 Pνμλ(q) (1.38) where Pνλμ(q)= Kνμλ(q) Kνλμ(q). (1.39) These R-matrices will be called R-matrices with simple spectrum.

Let us consider the space VτVλVμ and the matrices R12τλ=Rτλ1:VτVλVμVλVτVμ, R23τμ=1Rτμ:VλVτVμVλVμVτ, (Kνλμ)23=1Kνλμ:VτVλVμVτVν, (Kνλμ)12=Kνλμ1:VλVμVτVνVτ, Rτν:VτVνVνVτ and similar matrices R23μτ, R12λτ, (Kνλμ)23, (Kνλμ)23, (Kνλμ)12, Rντ.

Theorem 1.6. The matrices introduced above satisfy the following relations Rτν (Kνλμ)23 = (Kνλμ)12 R23τμR12τλ, (1.40) Rντ (Kνλμ)12 = (Kνλμ)23 R12λτR23μτ. (1.41)


The restriction or the relation (1.6) on the irreducible representation gives: s ( ρν(es) ρτ(es) ) (KνλμI) =(KνλμI) s,tρλ(et) ρμ(es) ρτ(etes). (1.42) Multiplying this formula by the permutation operator Pντ:VνVτVτVν from the left and using the equality Pντ (KνλμI) P13P23=Iτ Kνλμ we obtain (1.41). The relation (1.40) is proved similarly.

Crossing-symmetry of the universal R-matrices (1.8d) implies crossing-symmetry of the matrices Rλμ and Kνλμ:

Theorem 1.7. The matrices Rλμ and Kνλμ satisfy the following crossing-symmetry relations: (PμλRλμ)? = (C1) (PμλRλμ)t1 (C-1I) (1.43) j (Kνλμ)kij (C)j = χλχν (Kλνμ*)ik (1.44) i(Kνλμ)kij (C)i = χμχν (Kμλ*ν)jk (1.45) where t1 is transposition over the first space, λ* is the highest weight (h.w.) vector conjugated with λ by Carton automorphism, C=Sq-ρ2, χλ=trVλ (q-ρ). (1.46)

Using the explicit formula for coproduct in it is not difficult to prove the first statement of the following theorem.

Theorem 1.8.

(a) The matrix (Kλλ*)ij =1χλCji (1.47) is the projector on ω the one-dimensional subrepresentation in VλVλ*.
(b) The matrices Rλλ* and Rλλ satisfy the relations Rλλ* (Kλλ*) = q-c(λ)2 (-1)[λ] (Kλ*λ) (1.48) tr2 ( (Iq-ρ) (Rλλ)±1 ) = q±c(λ)2 ·I. (1.49) Here [λ]=0,1, tr2 is the matrix trace over the second space in VλVλ.

The proof of the second statement of this theorem is given in the next section, where we introduce graphical representation for the R-matrices and for KGC.

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