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.
The irreducible finite dimensional representation of
are parametrized by h.w.
where and are the fundamental weights (see [Bou1981]) and
the h.w. vector,
The basic representation of have h.w.
Let us consider the basics in
formed by weight vectors,
The roots are shown on the figure 1.
In this basis the elements
are represented by the matrices:
In this normalization the commutation relations between the generators
have the following form:
The representation is selfdual:
A tensorial product of the two basic representations of
has the following decomposition:
In this decomposition
It is not difficult to calculate the matrix
So the matrix
is given explicitly.
Theorem 7.1. The matrix
satisfies the following relations:
From the theorem 1.5 we have the spectral decomposition of
One-dimensional projector is defined in section 1 (1.47):
Substituting (7.8), (7.9) in the l.h.s. of 7.6 and using the decomposition of units in
we obtain (7.6).
The equality (7.7) is obtained by rotating the pictures (7.6) by
Considering the equalities (7.6) and (7.7) as a system of linear equations for
we obtain an expression for these matrices in terms of the matrix
Let us consider now the restriction of to the
triple formed by
The representation is the sum of three
are formed by the vectors
For CGC we have:
where are the bases in
are the bases of a one.
In accordance with the graphical representation of CGC given in section 2 we can associate tho following picture with the formulae (7.12) and (7.13)
An index corresponds to a representation. An index
corresponds to a representation.
From (7.4) and (7.12)-(7.14) we obtain the
form of the matrix
From (7.15) we obtain the following expression for projector
and the similar expression for crossing-conjugate matrix. For one dimensional projector there is a similar representation:
Substituting (7.15)-(7.17) in (7.10) and (7.11) we obtain an explicit expression for