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 basic representation of have the h.w. and The matrix elements of the generators do not depend on in this representation. where are the basic matrices in
Tho Tensorial square of is decomposed into the sum of two irreducible components
The KGC of this decomposition are determined by the relations: where and are the bases in and in respectively.
From the theorem 1.5 we find the basic for or using (4.3), (4.4) In the form (4.6) the matrix was found by Jimbo [Jim1985].
For the decomposition of is well known: where if and the sum is taken only over satisfying the condition Using the rule (4.7) and the theorem 3.1 one can calculate the elements The representations and are contained in with multiplicity equal to one. It is not difficult to calculate the corresponding projectors where the sum taken is over the element of symmetric group if is the elementary transposition in the elements are defined by (3.2) and if is the representation of in the nonreducible product of the transpositions, The formulae (4.8), (4.9) were obtained in [Jim1985] as the of the Young sym (antisym) metrisators. The following theorem describes the connection between the algebra and Hecke algebra The last one is the assotiative algebra with units generated by the elements satisfying the following relations:
Theorem 4.1. The algebra is the factor of Hecke algebra over the ideal formed by the elements where are the same as in (4.8) and (4.9). If The generators of correspond to the generators of Hecke algebra
This theorem in equivalent to the fact that in the space the representation of Hecke algebra (this fact was noted by Jimbo [Jim1985]) is defined. The theorem 4.1 means that the irreducible components of this representation of have the Young diagrams with no more than rows.
An explicit expression for matrix elements of in the basics (3.4) of the representation can be found from (3.9). Omitting the details, we give the answer: where This expression was given in [Wen1985] and [Ker1987] following a different argumentation.
From this point we will use the normalization and the notation of books [Bou1981] for the highest weights and the roots of simple Lie algebras.