Notes on Affine Hecke Algebras I.
(Degenerated Hecke Algebras and Yangians in Mathematical Physics)

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 23 April 2014

Notes and References

This is an excerpt of the paper Notes on Affine Hecke Algebras I. (Degenerated Hecke Algebras and Yangians in Mathematical Physics) by Ivan Cherednik.

Mirrors and polarizations

Let us complicate our space. The idea is to consider the half-line +={x0} instead of with the reflection and its end, i.e. place a mirror at x=0. Some typical picture of interactions is in fig. 6. As before N 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 Π(Θ)=(Πi1j1(-θ1)), where Θ=(θ1,,θn) are the angles in the out-state according to the conventions adopted. Each element of this matrix depends only on the angle θ1 of the first particle (after the reflection at x=0) and on its colours i1 (before) and j1 (after) the reflection.

Particles have two phases (). The first is before (θ<0) and the second (θ>0) 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 S). Let us denote the S-matrix of the third type (-,+) by Sˆ (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 Si(θ) or Sˆi(θ) (see (13b)) for scattering at the intersection point of the i-th and (i+1)-th particles (numbers are from the bottom to the top). We remind that Si and Sˆi depend only on θi-θi+1 and on the corresponding colours of the i-th and (i-1)-th particles.

We should add to (16) its direct analogs SˆSSˆ=SˆSSˆ, SSˆSˆ=SˆSˆS (with the same indices and arguments), and the new one: Π(u)Sˆ1 (2u+v)Π (u+v)S1(v)= S1(v)Π(u+v) Sˆ1(2u+v) Π(u). (31) Here (see fig. 8) u=-θ1, v=θ1-θ2, 2u+v=-θ1-θ2, u+v=-θ2.

We claim that the identities (with the indices and arguments from (16), (31)) SSS=SSS, SˆSSˆ=SˆSSˆ, SSˆSˆ=SˆSˆS, ΠSˆΠS=SΠSˆΠ (32a) together with the evident relations (see (12)) [Si,Sj]= [Sˆi,Sj]= [Sˆi,Sˆj]= [Π,Sj]= [Π,Sˆj]=0 (32b) for j2, |i-j|2 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 Θ=(θ1,,θn) to some set of angles Θ of the in-state can be represented as the sequence w=(ε11,ε22,,εnn) for εk=±1, (1,2,,n)=wSn. It means that the angle εkθk in the set Θ is situated at place k (from the bottom to the top at moment t=tk), where 1kn. One has w=(3,-2,-1) for fig. 6. The composition of two elements w,w of this type is quite natural: if k(w)εkk, k(w)εkk then k(ww)(εkεk)k.

The group of all w can be represented by permutations (acting on Θn in the usual manner) multiplied by any number of the reflections πk(Θ)= ( θ1,,θk-1, -θk,θk+1,, θn ) (1kn).

Mathematically, it is the semi-direct product of Sn and (2)n={π1δ1πnδn,δk=0,1} with the natural action of Sn on the latter: wπkw-1=πw(k) for any k,wSn. As an abstract one this group is generated by s1,,sn-1Sn and π=π1 with the following new relations: s1πs1π=πs1πs1, πsj=sjπ, j2. Each element w of it can be represented in the form w=(k=1nπkδk)w, wSn. We will denote this group by Sn. It is called the Weyl group of type Bn (or Cn).

Now we are in a position to calculate the S-matrix of any picture. To do it one should know Θ (in the out-state) and the corresponding transformation wSn from Θ to Θ for the in-state (see above). Let w=sisi1 be of minimal possible length (written =(w)), where 0in and we denote π by s0 for the sake of uniformity. Then (cf. (13)) Sw(Θ)= Si (si-1si1(Θ)) Si2 (si1(Θ)) Si2(Θ), (33) where Si(Θ)= Π(-θ1)for i=0,Sik =SikorSˆik, if ik0 and pair (k,k+1) of the angles from the set sik-1si1(Θ) has the coinciding signs or not. Elements from Sn act on Θ as have been explained. It follows from (32) that Sw(Θ) does not depend on decomposing of w.

The simplest example of such a theory is as follows. Let Si(Θ)= Sˆi(Θ) =1+si(θi-θi+1) ,Π(Θ)= 1-βπθ1,β. (34) Then the only equation we need to verify is (31). To obtain some matrix interpretation of (34) one can use some tensor representation of Sn (see [Che1984]). Our aim is to quantize the angles. We should substitute some pairwise commuting letters Yi for θi in (34) and (according to the procedure of sec. 5) postulate relations (18) and the natural relations. (1-βπY1)Y1= -Y1(1-βπY1) ,[Yj,Π]= 0forj>1 Here Π(Y1) corresponds to π and therefore should act on (Y1Yn) as Y1-Y1. One obtains the algebra n generated by [Sn] and Y1,,Yn with the relations (19) and some new ones πY1+Y1π=2/β, [Yj,π]=0forj>1. (35) This algebra is a certain degeneration of the affine Hecke algebra of type Bn or Cn (see e.g. [Rog1985]). To be more precise it is connected with Bn,Cn,Dn for β=1,2,0 (see [Che1984]).

We can use the group Sn 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 S whenever they are of the same type (polarization). Otherwise the scattering is trivial. The simplest algebra of observables for collections of n polarised particles is [Sn]. The operator πk (1kn) corresponds to the polarization of k-th particle in a collection; πkπk+1 describes the change of polarization from the k-th to (k+1)-th particles. That means that πk(Aj(Θ))=sgn(θk)AJ(Θ), πkπk+1(AJ(Θ))=sgn(θkθk+1)AJ(Θ).

Let us demonstrate that Si(Θ)= (πiπi+1+1) (θi-θi+1)-1 /2+si (36) is exactly what we need (see [Che1991]). Firstly, it is a solution of the Yang-Baxter equation (in the form of (16)). Then Si=si for πiπi+1=-1 (i.e. there is no scattering in this case). At last, Siπi=πi+1Si and Siπi+1=πiSi (the polarizations are conserved). The quantization of angles give us the following relations (from [Che1991]): Yi+1si-si Yi=(πiπi+1+1) /2=siYi+1-siYi [si,Yj]= [πk,Yj]= [Yk,Yj]=0 forji,i+1, 1k,jn. in place of (19).

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