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 complicate our space. The idea is to consider the half-line instead of with the reflection and its end, i.e. place a mirror at Some typical picture of interactions is in fig. 6. As before is the number of colours, axioms (a), (b), (c), (d) (see sec. 1) are valid. But now we have the reflection. We connect with it the scattering matrix where are the angles in the out-state according to the conventions adopted. Each element of this matrix depends only on the angle of the first particle (after the reflection at and on its colours (before) and (after) the reflection.
Particles have two phases The first is before and the second is after the reflection. Respectively, one should consider 3 types of two-particle amplitudes, when the phases (the signs of the angles) of particles in the out-state are (the combination is impossible). For the sake of simplicity we will identify the first two (written Let us denote the of the third type by (cf. [Che1984]). Look at fig. 7. Here the out-state is the state after the intersection.
We omit here the symbolic and multi-index language of sec. 1, 2 and we will use at once the notations or (see (13b)) for scattering at the intersection point of the and particles (numbers are from the bottom to the top). We remind that and depend only on and on the corresponding colours of the and particles.
We should add to (16) its direct analogs (with the same indices and arguments), and the new one: Here (see fig. 8)
We claim that the identities (with the indices and arguments from (16), (31)) together with the evident relations (see (12)) for provide the independence of any scattering matrix of the internal picture of intersections. In a word (32) is equivalent to axiom (c) from sec. 1.
Let us describe the corresponding group of symmetries. Now the transformation of a given out-state with the angles to some set of angles of the in-state can be represented as the sequence for It means that the angle in the set is situated at place (from the bottom to the top at moment where One has for fig. 6. The composition of two elements of this type is quite natural: if then
The group of all can be represented by permutations (acting on in the usual manner) multiplied by any number of the reflections
Mathematically, it is the semi-direct product of and with the natural action of on the latter: for any As an abstract one this group is generated by and with the following new relations: Each element of it can be represented in the form We will denote this group by It is called the Weyl group of type (or
Now we are in a position to calculate the of any picture. To do it one should know (in the out-state) and the corresponding transformation from to for the in-state (see above). Let be of minimal possible length (written where and we denote by for the sake of uniformity. Then (cf. (13)) where if and pair of the angles from the set has the coinciding signs or not. Elements from act on as have been explained. It follows from (32) that does not depend on decomposing of
The simplest example of such a theory is as follows. Let Then the only equation we need to verify is (31). To obtain some matrix interpretation of (34) one can use some tensor representation of (see [Che1984]). Our aim is to quantize the angles. We should substitute some pairwise commuting letters for in (34) and (according to the procedure of sec. 5) postulate relations (18) and the natural relations. Here corresponds to and therefore should act on as One obtains the algebra generated by and with the relations (19) and some new ones This algebra is a certain degeneration of the affine Hecke algebra of type or (see e.g. [Rog1985]). To be more precise it is connected with for (see [Che1984]).
We can use the group for another problem. Let us consider the usual as a space (without any reflections). However, assume that there are two different non-changing types of particles (two "polarizations"). We assume that the scattering process is described by Yang's two-particle whenever they are of the same type (polarization). Otherwise the scattering is trivial. The simplest algebra of observables for collections of polarised particles is The operator corresponds to the polarization of particle in a collection; describes the change of polarization from the to particles. That means that
Let us demonstrate that is exactly what we need (see [Che1991]). Firstly, it is a solution of the Yang-Baxter equation (in the form of (16)). Then for (i.e. there is no scattering in this case). At last, and (the polarizations are conserved). The quantization of angles give us the following relations (from [Che1991]): in place of (19).