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
aram@unimelb.edu.au

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.

Introduction

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 $q\text{-deformation}$ of a universal enveloping algebra $U_q\left(𝔰𝔩\left(2\right)\right)$ [Dri1985,Dri1986-3,Jim1985]. The algebras $U_q\left(\widehat{𝔤}\right)$ where $\widehat{𝔤}$ 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 $R\text{-matrices}$ found by Jimbo [Jim1986-2] are connected with the Kauffman invariant [Yan1967,Bax1982,Fad1984,KSk1982].

This paper argues that to each algebra $U_q\left(𝔤\right),$ 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 $U_q\left(𝔤\right)\text{.}$ Solutions of the Yang-Baxter equations (they are what the link invariants are built from) will be called $R\text{-matrices}$ following the terminology of QISM.

In the beginning of section 1 some information on the algebras $U_q\left(𝔤\right)$ and on universal $R\text{-matrices}$ connected with them is given. Next the following important theorem is proved.

Theorem (1.5). If $V^{\nu },V^{\lambda }$ and $V^{\mu }$ are irreducible representations of $U_q\left(𝔤\right)$ with highest weights $\nu ,\lambda ,\mu ,$ $V^{\nu }\subset V^{\lambda }\otimes V^{\mu }$ with multiplicity equal to one, $R^{\lambda \mu }$ is $R\text{-matrix,}$ $R^{\lambda \mu }:V^{\lambda }\otimes V^{\mu }\to V^{\mu }\otimes V^{\lambda }$ and $K_{\nu }^{\lambda \mu }:V^{\lambda }\otimes V^{\mu }\to V^{\nu }$ is the matrix of Clebsh-Gordan $q\text{-coefficients}$ (CGC), then $$K_{\nu }^{\mu \lambda }{R}^{\lambda \mu }={(-1)}^{\bar{\nu }}{q}^{\frac{1}{4}(c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right))}{K}_{\nu }^{\lambda \mu }$$ where $\bar{\nu }=0,1,$ $c\left(\nu \right)=⟨\nu ,\nu ⟩+2⟨\rho ,\nu ⟩,$ $\rho =\frac{1}{2}\sum _{\alpha \in {\mathrm{\Delta }}_{+}}\alpha ,$ ${\mathrm{\Delta }}_{+}$ is a set of positive roots of the algebra $𝔤\text{.}$

A fusion procedure [KRS1981] for $R\text{-matrices}$ connected with the algebras $U_q\left(𝔤\right)$ is given by theorem 1.6.

Theorem (1.6). Let $K_{\nu }^{\lambda \mu }\left(a\right)$ be a CGC matrix, $K_{\nu }^{\lambda \mu }\left(a\right):V^{\lambda }\otimes V^{\mu }\to V^{\nu }\subset V^{\lambda }\otimes V^{\mu }$ (the index $a$ numbers $V^{\nu }$ if the multiplicity of $V^{\nu }$ in $V^{\lambda }\otimes V^{\mu }$ is greater than one) then the matrices $R^{\lambda \tau },R^{\mu \tau }$ and $R^{\nu \tau }$ are connected by the following relation $$R^{\nu \tau }{\left(K_{\nu }^{\lambda \mu }\left(a\right)\right)}_{12}={\left(K_{\nu }^{\lambda \mu }\left(a\right)\right)}_{23}{R}_{12}^{\lambda \tau }{R}_{23}^{\mu \tau }\text{.}$$ Their exact meaning is given in section 2.

Section 2 introduces a graphical interpretation of $R\text{-matrices}$ 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 $U_q\left(𝔤\right)$ in the space ${\left(V^{\omega }\right)}^{\otimes N},$ where $\omega$ is the basic representation of $U_q\left(𝔤\right)$ (see section 1). These results are applied for calculating the matrices $R^{\lambda \mu }$ through the matrix $R^{\omega \omega }\text{.}$ It is shown that the matrices $$g_i=1\otimes \dots \otimes R^{\omega \omega }\otimes \dots \otimes 1$$ form the centraliser $C_N^{\omega }\left(𝔤\right)=C(U_q\left(𝔤\right),{\left(V^{\omega }\right)}^{\otimes N})\text{.}$ From the complete irreducibility of finite dimensional representation of $U_q\left(𝔤\right)$ follows the analog of Weyl's duality: $${\left(V^{\omega }\right)}^{\otimes N}=\sum _{\lambda }{V}^{\lambda }\otimes W_{\lambda }$$ where $V^{\lambda }$ is an irreducible finite dimensional $U_q\left(𝔤\right)\text{-module}$ and $W_{\lambda }$ is an irreducible $C_N^{\omega }\left(𝔤\right)\text{-module.}$ In the spaces $W_{\lambda }$ orthogonal basis analogous to the Young basis for $S_n$ built. It is shown how to calculate the matrix elements of $g_i$ in this basis, when CGC $K_{\mu }^{\lambda \omega }$ are known. A $q\text{-analog}$ of the Young basis is built out of $g_i\text{.}$ (A $q\text{-analog}$ of the Young projectors). Using the expression for the Young projectors through the matrices $R^{\omega \omega }$ it is shown how the matrices $R^{\lambda \mu }$ are expressed through $R^{\omega \omega }\text{.}$

The results of sections 1-3 are applied to the $R\text{-matrices}$ connected with the algebra $U_q\left(𝔤𝔩\left(n\right)\right)$ 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 $C_N^{\omega }\left(𝔤𝔩\left(n\right)\right)$ is new. Other results (obtained in [Jim1985,Jon1986]) are given merely for the purpose of drawing a fuller picture.

Section 5 focuses on $R\text{-matrices}$ connected with the algebras $U_q\left(𝔰𝔬(2n+1)\right)$ and $U_q\left(𝔰𝔬\left(2n\right)\right)\text{.}$ $R\text{-matrices}$ corresponding to tensorial representation of these algebras are analysed. The matrices $R^{{\omega }_1{\omega }_1},$ found earlier by Jimbo [Jim1986-2], are constructed using the results of sections 1-3. It is shown that $C_N^{{\omega }_1}\left(𝔰𝔬(2n+1)\right)$ and $C_N^{{\omega }_1}\left(𝔰𝔬\left(2n\right)\right)$ 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 $R^{{\omega }_1{\omega }_1}$corresponding to the embeddings $𝔰𝔬(2n+1)\supset 𝔤𝔩\left(n\right),$ $𝔰𝔬\left(2n\right)\supset 𝔤𝔩\left(n\right)$ is given. The second half of section 5 deals with spinorial representations of $U_q\left(𝔰𝔬(2n+1)\right)$ and $U_q\left(𝔰𝔬\left(2n\right)\right)\text{.}$ It is shown that the matrices $K_s^{{\omega }_{1}s},$ where $s$ are spinorial representations, form a $q\text{-analog}$ of the Clifford algebra, finally, it is shown how spinorial $R\text{-matrices}$ are expressed through the matrices $K_s^{{\omega }_{1}s}$ and $R^{{\omega }_1{\omega }_1}\text{.}$

Section 6 is concerned with the solutions of Yang-Baxter equation connected with $U_q\left(𝔰𝔭\left(2n\right)\right)\text{.}$ All of the results are quite similar to tensorial representations of the algebras $U_q\left(𝔰𝔬(2n+1)\right)$ and $U_q\left(𝔰𝔬\left(2n\right)\right)\text{.}$

A matrix $R^{{\omega }_1{\omega }_1}$ for the algebra $U_q\left(G_2\right)$ is found in section 7, based on the results of sections 1-3. A block structure of $R^{{\omega }_1{\omega }_1}$ corresponding to the embedding $G_2\supset 𝔰𝔩\left(2\right)$ 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 $R^{\lambda \mu }$ connected with irreducible representations of the algebras $U_q\left(𝔤\right)$ 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 $R^{\lambda \mu }$ to the calculation of the basic invariant (built on the matrix $R^{{\omega }_1{\omega }_1}\text{).}$

Section 9 discusses the invariants connected with the $q\text{-deformations}$ 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 $R\text{-matrices}$ is reduced to the calculation of Kauffman invariants for reconstructed links.

In Section 10 a rule by which a basic invariant connected with $U_q\left(G_2\right)$ 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.

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