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 interpret formulas (6), (12), and (13) algebraically. We will now consider symbols as products of generators in the free algebra with generators (they have two indices where can be any number). The quotient algebra of by relations (3) is called Zamolodchikov's algebra. We suppose here and further that is analytical for close to and The latter is quite natural physically (there should be no scattering, when two particles are "parallel" one to another). Here we identify with for being the unit matrix.
Let us impose on relation (6) and the so-called unitaty condition Given in some neighbourhood of we wil denote by the vector subspace in generated by for any Let be its image in
Relations (6), (14) are in fact equivalent to a certain Wick (or Poincaré-Birkhoff-Witt) theorem for every Namely (see e.g. [Zam1978,Che1979]),
|(a)||every is a linear combination of some in|
|(b)||all the are linearly independent in for any multi-indices|
Our other problem is to find the best algebraic language for relations (6), (12), and formula (13). We will start with (see (13b)). Let us denote by the product (13a) for from (10). Then the set of functions satisfies the following conditions (cf. [Che1984]) if (Deduce it from (13)).
Conversally, let be some undetermined functions of with values anywhere and let relation (12) together with be valid for arbitrary
Then we claim that one can uniquely define the set of functions by (15). Moreover, it is possible to omit the condition (15b) in the case of unitary
Summarizing, we see that the tensor mode of rewriting (4) in the form of (6) is very convenient but not the best. The most natural way is to use as the index set (see (15) and (16)).