Quantized universal enveloping algebras, the Yang-Baxter equation and invariants of links, I
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.
All formulae in thin caae are very similar to those in part one of the previous section.
The h.w. of finite dimensional representation of are parametrized by the numbers
is the h.w. vector,
The basic representation of
have the h.w.
The tensor square of the basic representation is the sum of three irreducible components:
and we have the following spectral decomposition of
As in the cases of
and one can calculate the matrices
Substituting these matrices into (6.3) we obtain the matrix
This matrix can also be extracted from [Jim1986-2].
The matrices and
for any h.w. can be found from the theorems 2. - 3. and from the ramification rule:
As in the case
we have the following propositions.
Proposition 6.1. The algebra
is the factor of B.-W. algebra with
over the ideal formed by the elements
acting nontrivially only in the multipliers of
with numbers The elements
are defined by (5.11) with the matrix
If we consider the block basis in connected with the embedding
we obtain the representation of
in the block form with the blocks constructed from
Proposition 6.3. The matrices
have the following block structure in the basis (6.9):
where we use the notations of sections 2 and 5.