## 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.

## Graphical representation of the $R\text{-matrices}$

The graphical representation given below is very useful for showing the relations between the $R\text{-matrices}$ and KGC. This representation shows the combersome formulae in terms of simple figures and makes these relations intuitively understandable.

i) The matrix $A$ mapping the space ${V}^{{\lambda }_{1}}\otimes \cdots \otimes {V}^{{\lambda }_{N}}$  in ${V}^{{\mu }_{1}}\otimes \cdots \otimes {V}^{{\mu }_{M}}$ is represented by $\left(N+M\right)\text{-edged}$ diagram: ${i}_{1} {i}_{2} {i}_{M} {\mu }_{1} {\mu }_{2} \cdots {\mu }_{M} {\lambda }_{1} {\lambda }_{2} \cdots {\lambda }_{N} {j}_{1} {j}_{2} {j}_{N} A ⟷ Ai1…iMj1…jN$

The indices $\left\{i\right\},\left\{j\right\}$ and $\left\{\lambda \right\},\left\{\mu \right\}$ are called states on the edges and the colours of the edges correspondingly. The states numerate the bases in the spaces ${V}^{\lambda }$ $\text{(}{i}_{\alpha }=1,\dots ,\text{dim} {V}^{{\lambda }_{\alpha }},$ ${j}_{\alpha }=1,\dots ,\text{dim} {V}^{{\lambda }_{\alpha }}\text{).}$

ii) The product of the matrices $A$ and $A\prime$ is represented by an ordered from top to bottom combination of the diagrams $A$ and $A\prime \text{.}$ If $A\prime :{V}^{{\lambda }_{1}}\otimes \cdots \otimes {V}^{{\lambda }_{N}}\to {V}^{{\nu }_{1}}\otimes \cdots \otimes {V}^{{\nu }_{L}}$ and $A\prime :{V}^{{\nu }_{1}}\otimes \cdots \otimes {V}^{{\nu }_{L}}\to {V}^{{\mu }_{1}}\otimes \cdots \otimes {V}^{{\mu }_{M}}$ the diagram $AA\prime$ is: ${i}_{1} {i}_{M} {\mu }_{1} \cdots {\mu }_{M} {\nu }_{1} \cdots {\nu }_{M} {\lambda }_{1} \cdots {\lambda }_{N} {j}_{1} \cdots {j}_{N} A A\prime ⟷ (AA′)i1⋯iMj1⋯jN =∑k1⋯kL Ai1⋯iMk1⋯kL A′k1⋯kLj1⋯jN. (2.2)$ The junction of the edges corresponds to summation over the states on these edges.

iii) The unit matrix acting in ${V}^{\nu }$ represented by vertical line: $\lambda i j ⟷δij, i,j=1,…,dim Vλ. (2.3)$

Let us represent the matrices ${R}^{\lambda \mu },$ ${\left({R}^{\lambda \mu }\right)}^{-1},$ ${K}^{\lambda \mu },$ $C$ and ${C}^{-1}$ by the following diagrams: $i j \lambda \mu k \ell ⟷(Rλμ)ijkℓ =(eit⊗ejt) Rλμ(ek⊗eℓ) (2.4) i j \lambda \mu k \ell ⟷((Rμλ)-1)ijkℓ =(eit⊗ejt) (Rμλ)-1 (ek⊗eℓ) (2.5) i j \lambda \mu k a ⟷ (Kνλμ(q,a))kij ⟷ i j \lambda \mu k \nu a (2.6) i j \lambda {\lambda }^{*} ⟷(c-1)ij, i j {\lambda }^{*} \lambda ⟷(c-1)ji (2.7) {\lambda }^{*} \lambda i j ⟷cij, \lambda {\lambda }^{*} i j ⟷cji. (2.8)$ Further on we assume that each line acquires the colour $\lambda$ which defines the states on this line.

Using the rules (2.1)-(2.8) we obtain the graphical representation of the formulae from section 1:

i) the relation ${R}^{\lambda \mu }{\left({R}^{\mu \lambda }\right)}^{-1}={\left({R}^{\mu \lambda }\right)}^{-1}{R}^{\lambda \mu }$ is represented by the following graphical equality $\lambda \mu = \lambda \mu = \lambda \mu (2.9)$ ii) the relation (1.12) has the following representation $\lambda \mu \nu = \lambda \mu \nu (2.10)$ iii) the representation of ${P}_{\nu }^{\lambda \mu }$ from (1.39) is $i j \lambda \mu ν \mu \lambda k \ell ⟷Pνλμ (2.11)$ iv) the theorem 1.5 is represented by the equalities $\lambda \mu \nu = \lambda \mu \nu (-1)ν qc(ν)-c(λ)-c(μ)4, (2.12) \lambda \mu \nu = \lambda \mu \nu (-1)ν qc(ν)-c(λ)-c(μ)4. (2.13)$ v) the representations of the decompositions (1.37) and (1.38) are: $\lambda \mu =∑ν \lambda \mu ν \mu \lambda (-1)ν‾ qc(ν)-c(λ)-c(μ)4, (2.14) \lambda \mu =∑ν \lambda \mu ν \mu \lambda (-1)ν‾ q-c(ν)-c(λ)-c(μ)4. (2.15)$ vi) the theorem 1.5 is represented by the following equalities $\tau \lambda \mu \nu a = \tau \lambda \mu \nu a (2.16) \tau \lambda \mu \nu a = \tau \lambda \mu \nu a (2.17)$ vii) crossing-symmetry means that the diagrams representing ${R}^{\lambda \mu }$ and ${\left({R}^{\lambda \mu }\right)}^{-1}$ can be rotated by ${90}^{\circ }\text{:}$ $\lambda \mu = \lambda \mu = \mu \lambda (2.18) \lambda \mu = \lambda \mu = \mu \lambda (2.19)$ viii) the representation of the crossing symmetries of ${K}_{\nu }^{\lambda \mu }$ is $\lambda \mu \nu {\lambda }^{*} = \lambda \mu \nu , \lambda \nu \mu = \nu \mu \lambda (2.20)$ where $\begin{array}{c} \lambda \mu \nu \end{array}=\frac{1}{\sqrt{{\chi }_{\nu }}}\begin{array}{c} \lambda \mu \nu \end{array}\text{.}$ ix) the theorem 1.8 is represented by the following graphical equalities: $\lambda {\lambda }^{*} ⟷χλ·Kλλ* (2.21) {\lambda }^{*} \lambda = {\lambda }^{*} \lambda q-c(λ)2 (2.22) \lambda \lambda = \lambda ·q-c(λ)2 (2.23) \lambda = \lambda ·qc(λ)2. (2.24)$ x) Taking the crossing conjugation of the decompositions (1.37) and (1.38) we obtain the equalities $\lambda \mu =∑ν∈μ*⊗λ (-1)ν‾ q-c(ν)-c(λ)-c(μ)4 ·· \mu \lambda \nu \lambda \mu (2.25) \lambda \mu =∑ν∈μ*⊗λ (-1)ν‾ qc(ν)-c(λ)-c(μ)4 ·· \mu \lambda \nu \lambda \mu . (2.26)$ xi) the orthogonality relations (1.14) have the following representation $\lambda \mu {\nu }_{1} {\nu }_{2} = {\nu }_{1} δν1ν2. (2.27)$

So, the $R\text{-matrices,}$ Clebsch-Gordan coefficients and the relations between them can be represented by flexible lines on the surface. Moreover, these lines can be glued into a graph with three edged vertices. These graphs can be deformed according to the rules i)-xi).

This representation of the matrices is very convenient for manipulations with the matrices ${R}^{\lambda \mu },{K}_{\nu }^{\lambda \mu }$ and we will use it many times.

 Proof of Theorem 1.8. Using the crossing-symmetry of $R\text{-matrices}$ (1.43) and the relation $(c-1)tc= (-1)[λ] q-ρ (2.28)$ we obtain that the relations (1.48) and (1.49) are equivalent. For the matrices ${R}^{\lambda {\lambda }^{*}}$ with simple spectrum the relation (1.48) follows from the theorem 1.5. In the general case we use an induction procedure. The relation (1.49) takes place if $\lambda$ is the basic representation. Let us prove that it holds for any $\mu \subset \lambda \otimes \omega$ if it holds for $\lambda \text{.}$ To prove this statement it is sufficient to consider a set of equalities following from (2.4)-(2.7) $\mu = \lambda \omega \mu = \lambda \mu \omega = \lambda \mu \omega = qc(ω)2 \lambda \omega \mu =qc(ω)2 \lambda \omega \mu =qc(ω)+c(λ)2 \lambda \omega \mu = qc(ω)+c(λ)2 (qc(μ)-c(λ)-c(w)4)2 \lambda \omega \mu \mu =qc(μ)2 \mu . (2.29)$ The equality (1.49) is proved. Using the crossing-symmetry of ${R}^{\lambda \mu }$ we obtain (1.48) from (1.49). $\square$