Last update: 23 April 2014
This is an excerpt of the paper Notes on Affine Hecke Algebras I. (Degenerated Hecke Algebras and Yangians in Mathematical Physics) by Ivan Cherednik.
Let us describe a sequence of particles at the moment with the (see fig. 2) by the symbol where and are the corresponding sets of the angles and the colours Owing to property (c) the previous or subsequent changes of a set of particles depend only on its symbol (on the colours, angles and on the order of only). Given one has for some describing the set of particles We have expressed considered as an in-state in terms out-states. Every (scalar) coefficient is the element (the amplitude) from to (by definition). Here and can be arbitrary but not the set The is nontrivial only if for an appropriate permutation from the symmeric group
In this formula some misunderstanding is possible. I will comment on it. Any permutation acts on an ordered set of some elements (e.g. coordinates) by the substitution the element at place No. for the element at place No. For example, the transposition interchanges the content of the first and second places. This definition results in the natural formula We see that is necessary in the second equality of (2). In fig. 4 the corresponding is equal to in the one-line notations or in the two-line notation, i.e. takes to
Let us discuss examples. We will omit the indications "in" and "out". One has for and where by definition. Given in the case of fig. 3a we obtain the following relations: where the sum is over all free indices The analogical calculation for fig. 3b should give the same result. We arrive at the identity:
Let us rewrite (4) in a tensor form. We will keep the following notations. Let us consider "multi-matrices" with the multi-indices respectively, of rows and columns for These act on "multi-vectors" by the natural formula If multi-indices are assumed to be (lexicographically) ordered then and are usual vectors and matrices for in place of Given two (from the matrix algebra one can define the tensor product The definition of is quite analogous.
Later on, will be the Kronecker symbol. Put for These matrices are the natural images of in the with respect to the indices and Note that and commutes with for any
Let us introduce as the following matrix (depending on from Now one can represent (4) in the elegant form: If we put for we get the Yang-Baxter equation To check it (to deduce it from (6)) one should carry all the across the other terms in (6) after the substitution Here we have to use the following properties of It is easy either to prove (9) directly or verify them without matrix calculations using the following natural interpretation of