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.
In this seotion we study the structure of centralizater of in where is the basic representation.
Let and where is the permutation of the set
Definition 3.1. An algebra of the maps commuting with the action of in these spaces is called the centralizer of in and denoted by
Here we will consider only the algebras for the representations when multiplicity of the irreducible components in the decomposition is equal to one for any IFR
Proposition 3.1. The algebra is simple for general and is generated by the elements if the spectrum of the Casimir operator is simple in the decomposition (3.1).
The proof of this proposition is based on the spectral decomposition of the matrix
The representations of are given by the decomposition of on the components:
The space describes the multiplicity of the h.w. in (3.3):
Proposition 3.2. The representations in (3.3) form the list of all irreducible representations of
Let us use now the graphical representation of the matrices and developed in section 2 to describe the basis in
Proposition 3.3. The elements where and form the orthogonal basis in More precisely where is the basis in and is the unit matrix in
The orthogonality of the elements means that where is the transposition of the matrix The relation (3.5) is proved by the following equalities founded on the orthogonality of CGC: The completeness of this basis in is evident.
Proposition 3.4. The action of the elements in basis have the following form: where and are some constants calculated below.
The relations (3.7) have a simple graphical representation: Using the orthogonality of the basis we obtain a formula for the coefficients
Proposition 3.5. The coefficients in (3.6) are calculated by the following praphical rule: An exact meaning of this formula is given in section 8.
The action of in basis is similar to (3.7): where the coefficient has the following form: From follows the relation
So, we described the algebra and its irreducible representations. It is necessary to note that the basis (3.4) is exactly a of the Young basis in irreducible representations of symmetric group This basis is regular for the embeddings where is formed by the elements and is formed by the elements Indeed, if we consider the restriction the basis in will be formed by the elements where
Let us also note that the decompositions (3.14) determine tho graph of the algebra [KVe1985]. It is evident that really this graph is fixed by decomposition (3.3).
Let us consider now the of Young symmetrizers. For this purpose we introduce the matrices This matrix corresponds to the figure
Proposition 3.3 follows that the set forms an orthogonal set of projectors in
Theorem 3.1. The elements satisfy the following relations: where are the matrix elements of the matrix: is the number of irreducible components in the index numbers the rows of and numbers the columns of
Let us consider the matrix corresponding to the figure
From the theorem 1.5 we have the decomposition or matrix Taking powers of degrees of this equality we obtain a system for the projectors Solving this system we obtain the relation (3.18).
Considering the relation (3.18) as an inductive process for calculating we obtain the explicit expression for Here is a braid (see section 8) where the string with number turned round the strings with numbers (from left to right) times moving counter-clockwise and form the top to the bottom.
Now we can describe the structure of the matrices
Consider a decomposition of into the sum or irreducible components:
The matrix is an element of commuting with the action of Hence it has the following form: where
Let be the module. The decomposition of this module into tho sum of components is: where the spaces are the same as in (3.20).
Let be the matrix acting in of the form:
In accordance with (3.3) this matrix has the following decomposition where is the the permutation operator
Tho decomposition (3.24) and (3.25) is the generalization of the formula (1.37) on matrices with arbitrary and The formula (3.25) gives the method for calculating the matrices
We will write when it will not be ambiguous.