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.

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 $(N+M)\text{-edged}$ diagram: $$\begin{array}{c} i1 i2 iM μ1 μ2 ⋯ μM λ1 λ2 ⋯ λN j1 j2 jN A \end{array}⟷A_{i_1\dots i_M}^{j_1\dots j_N}$$

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 $V^{\lambda }$ $\text{(}{i}_{\alpha }=1,\dots ,\text{dim}\hspace{0.17em}{V}^{{\lambda }_{\alpha }},$ $j_{\alpha }=1,\dots ,\text{dim}\hspace{0.17em}{V}^{{\lambda }_{\alpha }}\text{).}$

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: $$\begin{array}{cc}\begin{array}{c} i1 iM μ1 ⋯ μM ν1 ⋯ νM λ1 ⋯ λN j1 ⋯ jN A A′ \end{array}{\displaystyle ⟷{\left(AA\prime \right)}_{i_1\cdots i_M}^{j_1\cdots j_N}=\sum _{k_1\cdots k_L}{A}_{i_1\cdots i_M}^{k_1\cdots k_L}{A\prime }_{k_1\cdots k_L}^{j_1\cdots j_N}\text{.}}& \text{(2.2)}\end{array}$$ 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: $$\begin{array}{cc}\begin{array}{c} λ i j \end{array}⟷{\delta }_{ij},i,j=1,\dots ,\text{dim}\hspace{0.17em}{V}^{\lambda }\text{.}& \text{(2.3)}\end{array}$$

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: $$\begin{array}{cc}\begin{array}{c} i j λ μ k ℓ \end{array}⟷{\left(R^{\lambda \mu }\right)}_{ij}^{k\ell }=(e_i^t\otimes e_j^t)R^{\lambda \mu }(e_k\otimes e_{\ell })& \text{(2.4)}\\ \begin{array}{c} i j λ μ k ℓ \end{array}⟷{\left({\left(R^{\mu \lambda }\right)}^{-1}\right)}_{ij}^{k\ell }=(e_i^t\otimes e_j^t){\left(R^{\mu \lambda }\right)}^{-1}(e_k\otimes e_{\ell })& \text{(2.5)}\\ \begin{array}{c} i j λ μ k a \end{array}⟷{\left(K_{\nu }^{\lambda \mu }(q,a)\right)}_k^{ij}⟷\begin{array}{c} i j λ μ k ν a \end{array}& \text{(2.6)}\\ \begin{array}{c} i j λ λ* \end{array}⟷{\left(c^{-1}\right)}^{ij},\begin{array}{c} i j λ* λ \end{array}⟷{\left(c^{-1}\right)}^{ji}& \text{(2.7)}\\ \begin{array}{c} λ* λ i j \end{array}⟷c_{ij},\begin{array}{c} λ λ* i j \end{array}⟷c_{ji}\text{.}& \text{(2.8)}\end{array}$$ 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 $$\begin{array}{cc}\begin{array}{c} λ μ \end{array}=\begin{array}{c} λ μ \end{array}=\begin{array}{c} λ μ \end{array}& \text{(2.9)}\end{array}$$ ii) the relation (1.12) has the following representation $$\begin{array}{cc}\begin{array}{c} λ μ ν \end{array}=\begin{array}{c} λ μ ν \end{array}& \text{(2.10)}\end{array}$$ iii) the representation of $P_{\nu }^{\lambda \mu }$ from (1.39) is $$\begin{array}{cc}\begin{array}{c} i j λ μ ν μ λ k ℓ \end{array}⟷P_{\nu }^{\lambda \mu }& \text{(2.11)}\end{array}$$ iv) the theorem 1.5 is represented by the equalities $$\begin{array}{cc}\begin{array}{c} λ μ ν \end{array}=\begin{array}{c} λ μ ν \end{array}{(-1)}^{\nu }{q}^{\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}},& \text{(2.12)}\\ \begin{array}{c} λ μ ν \end{array}=\begin{array}{c} λ μ ν \end{array}{(-1)}^{\nu }{q}^{\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}}\text{.}& \text{(2.13)}\end{array}$$ v) the representations of the decompositions (1.37) and (1.38) are: $$\begin{array}{cc}{\displaystyle \begin{array}{c} λ μ \end{array}=\sum _{\nu }\begin{array}{c} λ μ ν μ λ \end{array}{(-1)}^{\bar{\nu }}{q}^{\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}},}& \text{(2.14)}\\ {\displaystyle \begin{array}{c} λ μ \end{array}=\sum _{\nu }\begin{array}{c} λ μ ν μ λ \end{array}{(-1)}^{\bar{\nu }}{q}^{-\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}}\text{.}}& \text{(2.15)}\end{array}$$ vi) the theorem 1.5 is represented by the following equalities $$\begin{array}{cc}\begin{array}{c} τ λ μ ν a \end{array}=\begin{array}{c} τ λ μ ν a \end{array}& \text{(2.16)}\\ \begin{array}{c} τ λ μ ν a \end{array}=\begin{array}{c} τ λ μ ν a \end{array}& \text{(2.17)}\end{array}$$ vii) crossing-symmetry means that the diagrams representing $R^{\lambda \mu }$ and ${\left(R^{\lambda \mu }\right)}^{-1}$ can be rotated by ${90}^{\circ }\text{:}$ $$\begin{array}{cc}\begin{array}{c} λ μ \end{array}=\begin{array}{c} λ μ \end{array}=\begin{array}{c} μ λ \end{array}& \text{(2.18)}\\ \begin{array}{c} λ μ \end{array}=\begin{array}{c} λ μ \end{array}=\begin{array}{c} μ λ \end{array}& \text{(2.19)}\end{array}$$ viii) the representation of the crossing symmetries of $K_{\nu }^{\lambda \mu }$ is $$\begin{array}{cc}\begin{array}{c} λ μ ν λ* \end{array}=\begin{array}{c} λ μ ν \end{array},\begin{array}{c} λ ν μ \end{array}=\begin{array}{c} ν μ λ \end{array}& \text{(2.20)}\end{array}$$ where $\begin{array}{c} λ μ ν \end{array}=\frac{1}{\sqrt{{\chi }_{\nu }}}\begin{array}{c} λ μ ν \end{array}\text{.}$ ix) the theorem 1.8 is represented by the following graphical equalities: $$\begin{array}{cc}\begin{array}{c} λ λ* \end{array}⟷\sqrt{{\chi }_{\lambda }}·K^{\lambda {\lambda }^{*}}& \text{(2.21)}\\ \begin{array}{c} λ* λ \end{array}=\begin{array}{c} λ* λ \end{array}{q}^{-\frac{c\left(\lambda \right)}{2}}& \text{(2.22)}\\ \begin{array}{c} λ λ \end{array}=\begin{array}{c} λ \end{array}·q^{-\frac{c\left(\lambda \right)}{2}}& \text{(2.23)}\\ \begin{array}{c} λ \end{array}=\begin{array}{c} λ \end{array}·q^{\frac{c\left(\lambda \right)}{2}}\text{.}& \text{(2.24)}\end{array}$$ x) Taking the crossing conjugation of the decompositions (1.37) and (1.38) we obtain the equalities $$\begin{array}{cc}{\displaystyle \begin{array}{c} λ μ \end{array}=\sum _{\nu \in {\mu }^{*}\otimes \lambda }{(-1)}^{\bar{\nu }}{q}^{-\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}}··\begin{array}{c} μ λ ν λ μ \end{array}}& \text{(2.25)}\\ {\displaystyle \begin{array}{c} λ μ \end{array}=\sum _{\nu \in {\mu }^{*}\otimes \lambda }{(-1)}^{\bar{\nu }}{q}^{\frac{c\left(\nu \right)-c\left(\lambda \right)-c\left(\mu \right)}{4}}··\begin{array}{c} μ λ ν λ μ \end{array}\text{.}}& \text{(2.26)}\end{array}$$ xi) the orthogonality relations (1.14) have the following representation $$\begin{array}{cc}\begin{array}{c} λ μ ν1 ν2 \end{array}=\begin{array}{c} ν1 \end{array}\delta {\nu }_1{\nu }_2\text{.}& \text{(2.27)}\end{array}$$

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 $$\begin{array}{cc}{\left(c^{-1}\right)}^{t}c={(-1)}^{\left[\lambda \right]}{q}^{-\rho }& \text{(2.28)}\end{array}$$ 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) $$\begin{array}{cc}\begin{array}{ccc}\begin{array}{c} μ \end{array}& =& \begin{array}{c} λ ω μ \end{array}=\begin{array}{c} λ μ ω \end{array}=\begin{array}{c} λ μ ω \end{array}\\ & =& q^{\frac{c\left(\omega \right)}{2}}\begin{array}{c} λ ω μ \end{array}=q^{\frac{c\left(\omega \right)}{2}}\begin{array}{c} λ ω μ \end{array}=q^{\frac{c\left(\omega \right)+c\left(\lambda \right)}{2}}\begin{array}{c} λ ω μ \end{array}\\ & =& q^{\frac{c\left(\omega \right)+c\left(\lambda \right)}{2}}{\left(q^{\frac{c\left(\mu \right)-c\left(\lambda \right)-c\left(w\right)}{4}}\right)}^2\begin{array}{c} λ ω μ μ \end{array}=q^{\frac{c\left(\mu \right)}{2}}\begin{array}{c} μ \end{array}\text{.}\end{array}& \text{(2.29)}\end{array}$$ The equality (1.49) is proved. Using the crossing-symmetry of $R^{\lambda \mu }$ we obtain (1.48) from (1.49). $\square$

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