Last update: 2 June 2014
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 graphical representation given below is very useful for showing the relations between the and KGC. This representation shows the combersome formulae in terms of simple figures and makes these relations intuitively understandable.
i) The matrix mapping the space in is represented by diagram:
The indices and are called states on the edges and the colours of the edges correspondingly. The states numerate the bases in the spaces
ii) The product of the matrices and is represented by an ordered from top to bottom combination of the diagrams and If and the diagram is: The junction of the edges corresponds to summation over the states on these edges.
iii) The unit matrix acting in represented by vertical line:
Let us represent the matrices and by the following diagrams: Further on we assume that each line acquires the colour 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 is represented by the following graphical equality ii) the relation (1.12) has the following representation iii) the representation of from (1.39) is iv) the theorem 1.5 is represented by the equalities v) the representations of the decompositions (1.37) and (1.38) are: vi) the theorem 1.5 is represented by the following equalities vii) crossing-symmetry means that the diagrams representing and can be rotated by viii) the representation of the crossing symmetries of is where ix) the theorem 1.8 is represented by the following graphical equalities: x) Taking the crossing conjugation of the decompositions (1.37) and (1.38) we obtain the equalities xi) the orthogonality relations (1.14) have the following representation
So, the 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 and we will use it many times.
|Proof of Theorem 1.8.|
Using the crossing-symmetry of (1.43) and the relation we obtain that the relations (1.48) and (1.49) are equivalent. For the matrices 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 is the basic representation. Let us prove that it holds for any if it holds for To prove this statement it is sufficient to consider a set of equalities following from (2.4)-(2.7) The equality (1.49) is proved. Using the crossing-symmetry of we obtain (1.48) from (1.49).