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.

Graphical representation of the R-matrices

The graphical representation given below is very useful for showing the relations between the R-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 Vλ1VλN
 in Vμ1VμM is represented by (N+M)-edged diagram: i1 i2 iM μ1 μ2 μM λ1 λ2 λN j1 j2 jN A Ai1iMj1jN

The indices {i},{j} and {λ},{μ} are called states on the edges and the colours of the edges correspondingly. The states numerate the bases in the spaces Vλ (iα=1,,dimVλα, jα=1,,dimVλα).

ii) The product of the matrices A and A is represented by an ordered from top to bottom combination of the diagrams A and A. If A:Vλ1VλNVν1VνL and A:Vν1VνLVμ1VμM the diagram AA is: i1 iM μ1 μM ν1 νM λ1 λN j1 jN A A (AA)i1iMj1jN =k1kL Ai1iMk1kL Ak1kLj1jN. (2.2) The junction of the edges corresponds to summation over the states on these edges.

iii) The unit matrix acting in Vν represented by vertical line: λ i j δij, i,j=1,,dim Vλ. (2.3)

Let us represent the matrices Rλμ, (Rλμ)-1, Kλμ, C and C-1 by the following diagrams: i j λ μ k (Rλμ)ijk =(eitejt) Rλμ(eke) (2.4) i j λ μ k ((Rμλ)-1)ijk =(eitejt) (Rμλ)-1 (eke) (2.5) i j λ μ k a (Kνλμ(q,a))kij i j λ μ k ν a (2.6) i j λ λ* (c-1)ij, i j λ* λ (c-1)ji (2.7) λ* λ i j cij, λ λ* i j cji. (2.8) Further on we assume that each line acquires the colour λ 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 Rλμ(Rμλ)-1=(Rμλ)-1Rλμ is represented by the following graphical equality λ μ = λ μ = λ μ (2.9) ii) the relation (1.12) has the following representation λ μ ν = λ μ ν (2.10) iii) the representation of Pνλμ from (1.39) is i j λ μ ν μ λ k Pνλμ (2.11) iv) the theorem 1.5 is represented by the equalities λ μ ν = λ μ ν (-1)ν qc(ν)-c(λ)-c(μ)4, (2.12) λ μ ν = λ μ ν (-1)ν qc(ν)-c(λ)-c(μ)4. (2.13) v) the representations of the decompositions (1.37) and (1.38) are: λ μ =ν λ μ ν μ λ (-1)ν qc(ν)-c(λ)-c(μ)4, (2.14) λ μ =ν λ μ ν μ λ (-1)ν q-c(ν)-c(λ)-c(μ)4. (2.15) vi) the theorem 1.5 is represented by the following equalities τ λ μ ν a = τ λ μ ν a (2.16) τ λ μ ν a = τ λ μ ν a (2.17) vii) crossing-symmetry means that the diagrams representing Rλμ and (Rλμ)-1 can be rotated by 90: λ μ = λ μ = μ λ (2.18) λ μ = λ μ = μ λ (2.19) viii) the representation of the crossing symmetries of Kνλμ is λ μ ν λ* = λ μ ν , λ ν μ = ν μ λ (2.20) where λ μ ν = 1χν λ μ ν . ix) the theorem 1.8 is represented by the following graphical equalities: λ λ* χλ·Kλλ* (2.21) λ* λ = λ* λ q-c(λ)2 (2.22) λ λ = λ ·q-c(λ)2 (2.23) λ = λ ·qc(λ)2. (2.24) x) Taking the crossing conjugation of the decompositions (1.37) and (1.38) we obtain the equalities λ μ =νμ*λ (-1)ν q-c(ν)-c(λ)-c(μ)4 ·· μ λ ν λ μ (2.25) λ μ =νμ*λ (-1)ν qc(ν)-c(λ)-c(μ)4 ·· μ λ ν λ μ . (2.26) xi) the orthogonality relations (1.14) have the following representation λ μ ν1 ν2 = ν1 δν1ν2. (2.27)

So, the R-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 Rλμ,Kνλμ and we will use it many times.

Proof of Theorem 1.8.

Using the crossing-symmetry of R-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 Rλλ* 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 λ is the basic representation. Let us prove that it holds for any μλω if it holds for λ. To prove this statement it is sufficient to consider a set of equalities following from (2.4)-(2.7) μ = λ ω μ = λ μ ω = λ μ ω = qc(ω)2 λ ω μ =qc(ω)2 λ ω μ =qc(ω)+c(λ)2 λ ω μ = qc(ω)+c(λ)2 (qc(μ)-c(λ)-c(w)4)2 λ ω μ μ =qc(μ)2 μ . (2.29) The equality (1.49) is proved. Using the crossing-symmetry of Rλμ we obtain (1.48) from (1.49).

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