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 h.w. of those algebras can be integer or halfinteger vectors. For
on the h.w. vector
The representation with integers h.w. form a category of tensoriol representations of
The basic representation in this category is the representation with h.w.
In the category of finite dimensional representations basic representation of
For there are two basic representations in
this category. These representations are and or
All three possibilities are equivalent. Let us start to study the algebras
and connected with algebras
from the category of a tensorial representation.
Tensorial representations of and
The vectorial representation of
do not depend on
Here are the basic matrices in
is the transposition. The vector representations of
are self-adjoint: The
tensor product of two vectorial representations is decomposed into the sum of three irreducible components
After some calculations one can find the matrices
They are determined by the following formulae
where are normalization constants,
are the bases in
From the theorem 1.5 we obtain the matrix
Substituting the expressions (5.4)-(5.6) in this formula we obtain the representation of
where are the basic matrices in
and the second term is present only for
The matrix (5.8) can be extracted from the work [Jim1986-2]:
where is the solution of Yang-Baxter equation corresponding to the series
The following decompositions are well known:
in (5.9) and
Using the theorem 3.1 and the decompositions (5.9) (5.10) we obtain the formulae for projectors
we have the following representation
is defined by (3.19).
are connected with the Birman-Wenzl algebra. The last one is an associative algebra formed by the generators
with the following relations
Theorem 5.1. The algebras
are factors of Birman-Wenzl algebra with
over the ideals formed by the elements
acting nontrivially only in the multipliers of
with numbers In other multipliers
acts as a unit matrix. Here
is defined by (5.11), (5.12). The generators of B.W. algebra are connected with the generators of
by the identification
the generators are determined by (5.13). For
Consequence. The representation of B.W. algebra with
defined in the space
The structure of algebras
is in agreement with the natural embeddings
Theorem 5.2. The algebras
as a Hopf subalgebra.
This theorem follows from the commutation relations (1.1)-(1.3) and from the formulae (1.4) (1.5) for coproduct in
Let us consider the restriction of the vector representation of
on the subalgebra
are vector and conjugated to vector
in accordance with the notations of the section 2 we represent by the graphs:
From the symmetries of we have
Theorem 5.3. The basic of
have the following block structures at the restriction of
This theorem follows from the formulae (5.8) and (4.6).
Corollary. There exist the following embeddings of the algebras
Using the theorem 5.3 and the action (4.11), (4.12) of
in in one can find the
action of and
Explicit formulae will be given in a separate publication.
The spinor representations
The spinor representations and
each have a dimension For
spinor representation has dimension
The bases in spinor representations are parametrized by the weights
there are auxiliary restrictions on
The spinor representations of
are defined by the action of the generators on the basic vectors
Here r.h.s. if the weight of
does not belong to the set of weights of the space
The tensor product of spinor representations has the following spectral decomposition
Let us describe the KFC of these decompositions. As in the previous part of this section we shall use the graphical representation of the matrices
given in section 2.
It is convenient to consider the space
and the matrix
we shall write
Theorem 5.5. The matrices for
satisfy the following relations:
The proof of this theorem follows from theorem 3.1.
It is interesting that the relations between the matrix elements of
and permit us to introduce
of Clifford algebra. If we write
the relations (5.27) give the relations for the matrices
with normalization condition
Using explicit form of the spinorial and vector representations the matrices
can be calculated exactly. But we will not present here these calculations because they take up too much space and are not very instructive for out purposes.
The matrices are the crossing transformations of
the matrices (see (2.20)). Using the formulae of
the section 5.1 and (5.27) we obtain an expression for the projector
where are some normalisation constants and projector
was given by (5.11).
So, we have the expression for spinor-spinorial through the matrices
are given by (5.30).
Using the commutation relations (5.27) it is easy to prove the following identities:
Here is the joint in which the line
numbered (the numeration being from left to right) is connected with the
in such a way that the goes above