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 interest in the quantum inverse scattering method (QISM) and in the solutions of Yang-Baxter equations 1 has significantly increased recently, for it happened that interesting mathematical constructions lie at the basis of QISM - these are noncommutative and noncocommutative Hopf algebras. The first example of such an algebra was the q-deformation of a universal enveloping algebra Uq(𝔰𝔩(2)) [Dri1985,Dri1986-3,Jim1985]. The algebras Uq(𝔤ˆ) where 𝔤ˆ is a simple or a Kac-Moody algebra, have been constructed (un works) by Drinfeld and Jimbo [Dri1985,Dri1986-3,Jim1985]. All these Hopf algebras are connected with an entire class of solutions of Yang-Baxter equation.

V. Jones [Jon1986] has recently shown that under certain additional conditions solutions of the Yang-Baxter equations can be used for constructing invariants of links. He also showed that two-parameters invariant of links constructed by Freyd-Yetter, Lickorish-Millet, Ocneanu and Hoste (FILMOH) [FYL1985] is connected with the solution of the Yang-Baxter equation found by Jimbo and Cherednik [Skl1982,SklNONE,Skl1985,Che1982]. Following this idea Turaev [Tur1988] showed that the R-matrices found by Jimbo [Jim1986-2] are connected with the Kauffman invariant [Yan1967,Bax1982,Fad1984,KSk1982].

This paper argues that to each algebra Uq(𝔤), where 𝔤 is a simple Lie algebra, we can associate a countable set of invariants of links. If a link contains k components then an invariant depending on one continuous parameter and k discrete parameters can be defined for it. These discrete parameters correspond to a highest weight of the algebra Uq(𝔤). Solutions of the Yang-Baxter equations (they are what the link invariants are built from) will be called R-matrices following the terminology of QISM.

In the beginning of section 1 some information on the algebras Uq(𝔤) and on universal R-matrices connected with them is given. Next the following important theorem is proved.

Theorem (1.5). If Vν,Vλ and Vμ are irreducible representations of Uq(𝔤) with highest weights ν,λ,μ, VνVλVμ with multiplicity equal to one, Rλμ is R-matrix, Rλμ:VλVμVμVλ and Kνλμ:VλVμVν is the matrix of Clebsh-Gordan q-coefficients (CGC), then KνμλRλμ= (-1)ν q14(c(ν)-c(λ)-c(μ)) Kνλμ where ν=0,1, c(ν)=ν,ν+2ρ,ν, ρ=12αΔ+α, Δ+ is a set of positive roots of the algebra 𝔤.

A fusion procedure [KRS1981] for R-matrices connected with the algebras Uq(𝔤) is given by theorem 1.6.

Theorem (1.6). Let Kνλμ(a) be a CGC matrix, Kνλμ(a):VλVμVνVλVμ (the index a numbers Vν if the multiplicity of Vν in VλVμ is greater than one) then the matrices Rλτ,Rμτ and Rντ are connected by the following relation Rντ (Kνλμ(a))12= (Kνλμ(a))23 R12λτR23μτ. Their exact meaning is given in section 2.

Section 2 introduces a graphical interpretation of R-matrices and of CGC in order to clarify visually their relations. All of these relations are expressed graphically. The last theorem 1.8 of section 1 is proven with the help of this graphical interpretation.

This graphical interpretation is actively exploited in subsequent sections.

Section 3 explores the structure of the centraliser of the algebra Uq(𝔤) in the space (Vω)N, where ω is the basic representation of Uq(𝔤) (see section 1). These results are applied for calculating the matrices Rλμ through the matrix Rωω. It is shown that the matrices gi=1Rωω 1 form the centraliser CNω(𝔤)=C(Uq(𝔤),(Vω)N). From the complete irreducibility of finite dimensional representation of Uq(𝔤) follows the analog of Weyl's duality: (Vω)N= λVλWλ where Vλ is an irreducible finite dimensional Uq(𝔤)-module and Wλ is an irreducible CNω(𝔤)-module. In the spaces Wλ orthogonal basis analogous to the Young basis for Sn built. It is shown how to calculate the matrix elements of gi in this basis, when CGC Kμλω are known. A q-analog of the Young basis is built out of gi. (A q-analog of the Young projectors). Using the expression for the Young projectors through the matrices Rωω it is shown how the matrices Rλμ are expressed through Rωω.

The results of sections 1-3 are applied to the R-matrices connected with the algebra Uq(𝔤𝔩(n)) in section 4. This section has a methodological character. Most probably an explicit description of the ideal by which the Hecke algebra should be factorized in order to obtain CNω(𝔤𝔩(n)) is new. Other results (obtained in [Jim1985,Jon1986]) are given merely for the purpose of drawing a fuller picture.

Section 5 focuses on R-matrices connected with the algebras Uq(𝔰𝔬(2n+1)) and Uq(𝔰𝔬(2n)). R-matrices corresponding to tensorial representation of these algebras are analysed. The matrices Rω1ω1, found earlier by Jimbo [Jim1986-2], are constructed using the results of sections 1-3. It is shown that CNω1(𝔰𝔬(2n+1)) and CNω1(𝔰𝔬(2n)) are both equal to a factor of Birman-Wenzl algebra [J. Birman, H. Wenzl] over an ideal given explicitly. A block structure of the matrices Rω1ω1corresponding to the embeddings 𝔰𝔬(2n+1)𝔤𝔩(n), 𝔰𝔬(2n)𝔤𝔩(n) is given. The second half of section 5 deals with spinorial representations of Uq(𝔰𝔬(2n+1)) and Uq(𝔰𝔬(2n)). It is shown that the matrices Ksω1s, where s are spinorial representations, form a q-analog of the Clifford algebra, finally, it is shown how spinorial R-matrices are expressed through the matrices Ksω1s and Rω1ω1.

Section 6 is concerned with the solutions of Yang-Baxter equation connected with Uq(𝔰𝔭(2n)). All of the results are quite similar to tensorial representations of the algebras Uq(𝔰𝔬(2n+1)) and Uq(𝔰𝔬(2n)).

A matrix Rω1ω1 for the algebra Uq(G2) is found in section 7, based on the results of sections 1-3. A block structure of Rω1ω1 corresponding to the embedding G2𝔰𝔩(2) is obtained.

Section 8 applies the result obtained in the previous sections to knot theory. It is shown how to build with the help of the solutions of Yang-Baxter equation satisfying certain auxiliary condition representations of the group of partially coloured braids and invariants of links as characters of these representations. This result is a natural generalisation of Jones construction [Jon1986]. Next, it is shown that the matrices Rλμ connected with irreducible representations of the algebras Uq(𝔤) satisfy the necessary conditions and generate a class of invariants of links. Two ways of description are offered for these invariants. In the first way the invariants are interpreted as characters of the braid group representations. In the second way the invariants are looked upon as statistical sums (state functionals) on the diagram of the link. At the end of the section a combinatorial rule, which helps to reduce the calculation of invariants built on the matrices Rλμ to the calculation of the basic invariant (built on the matrix Rω1ω1).

Section 9 discusses the invariants connected with the q-deformations of classical Lie algebras. The basic invariants connected with these algebras give invariant and the Kauffman invariant. Theorem 9.1 is a new result which allows us to reduce the calculation of the Kauffman invariant to the calculation of the FILMOH invariant for some reconstructed links. At the end of the section it is shown how the calculation of invariants built on spinorial R-matrices is reduced to the calculation of Kauffman invariants for reconstructed links.

In Section 10 a rule by which a basic invariant connected with Uq(G2) can be calculated by means of Jones's invariant [Abe1980] is given.

Finally, in conclusion some hypotheses are formulated.

I thank L.D. Faddeev, I.M. Gelfand, V.G. Drinfeld, A.N. Krillov, S.V. Kerov, M.A. Semenov-Tian-Shansky, E.K. Sklyanin, V.G. Turaev and O.Ya. Viro for useful and stimulating discussions.

page history