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

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 ${ℝ}_{+}=\left\{x\ge 0\right\}$ instead of $ℝ$ with the reflection and its end, i.e. place a mirror at $x=0\text{.}$ 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 $\mathrm{\Pi }\left(\mathrm{\Theta }\right)=\left({\mathrm{\Pi }}_{{i}_{1}}^{{j}_{1}}\left(-{\theta }_{1}\right)\right),$ where $\mathrm{\Theta }=\left({\theta }_{1},\cdots ,{\theta }_{n}\right)$ are the angles in the out-state according to the conventions adopted. Each element of this matrix depends only on the angle ${\theta }_{1}$ of the first particle (after the reflection at $x=0\text{)}$ and on its colours ${i}_{1}$ (before) and ${j}_{1}$ (after) the reflection.

Particles have two phases $\left(\mp \right)\text{.}$ The first is before $\left(\theta <0\right)$ and the second $\left(\theta >0\right)$ 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 $\left(-,-\right),\left(+,+\right),\left(-,+\right),$ (the combination $\left(+,-\right)$ is impossible). For the sake of simplicity we will identify the first two (written $S\text{).}$ Let us denote the $S\text{-matrix}$ of the third type $\left(-,+\right)$ by $\stackrel{ˆ}{S}$ (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 ${S}_{i}\left(\theta \right)$ or ${\stackrel{ˆ}{S}}_{i}\left(\theta \right)$ (see (13b)) for scattering at the intersection point of the $i\text{-th}$ and $\left(i+1\right)\text{-th}$ particles (numbers are from the bottom to the top). We remind that ${S}_{i}$ and ${\stackrel{ˆ}{S}}_{i}$ depend only on ${\theta }_{i}-{\theta }_{i+1}$ and on the corresponding colours of the $i\text{-th}$ and $\left(i-1\right)\text{-th}$ particles.

We should add to (16) its direct analogs $\stackrel{ˆ}{S}S\stackrel{ˆ}{S}=\stackrel{ˆ}{S}S\stackrel{ˆ}{S},$ $S\stackrel{ˆ}{S}\stackrel{ˆ}{S}=\stackrel{ˆ}{S}\stackrel{ˆ}{S}S$ (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) $u=-{\theta }_{1},$ $v={\theta }_{1}-{\theta }_{2},$ $2u+v=-{\theta }_{1}-{\theta }_{2},$ $u+v=-{\theta }_{2}\text{.}$

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 $j\ge 2,$ $|i-j|\ge 2$ 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 $\mathrm{\Theta }=\left({\theta }_{1},\cdots ,{\theta }_{n}\right)$ to some set of angles $\mathrm{\Theta }\prime$ of the in-state can be represented as the sequence $\stackrel{\sim }{w}=\left({\epsilon }_{1}1\prime ,{\epsilon }_{2}2\prime ,\cdots ,{\epsilon }_{n}n\prime \right)$ for ${\epsilon }_{k}=±1,$ $\left(1\prime ,2\prime ,\cdots ,n\prime \right)=w\in {S}_{n}\text{.}$ It means that the angle ${\epsilon }_{k}{\theta }_{k}$ in the set $\mathrm{\Theta }\prime$ is situated at place $k\prime$ (from the bottom to the top at moment $t={t}_{k}\text{),}$ where $1\le k\le n\text{.}$ One has $\stackrel{\sim }{w}=\left(3,-2,-1\right)$ for fig. 6. The composition of two elements $\stackrel{\sim }{w},\stackrel{\sim }{w}\prime$ of this type is quite natural: if $k\to \left(\stackrel{\sim }{w}\right){\epsilon }_{k}k\prime ,$ $k\prime \to \left(\stackrel{\sim }{w}\prime \right){\epsilon }_{k\prime }^{\prime }k″$ then $k\to \left(\stackrel{\sim }{w}\prime \stackrel{\sim }{w}\right)\left({\epsilon }_{k}{\epsilon }_{k\prime }^{\prime }\right)k″\text{.}$

The group of all $\stackrel{\sim }{w}$ can be represented by permutations (acting on $\mathrm{\Theta }\in {ℝ}^{n}$ in the usual manner) multiplied by any number of the reflections $πk(Θ)= ( θ1,⋯,θk-1, -θk,θk+1,⋯, θn ) (1≤k≤n).$

Mathematically, it is the semi-direct product of ${S}_{n}$ and ${\left({ℤ}_{2}\right)}^{n}=\left\{{\pi }_{1}^{{\delta }_{1}}\cdots {\pi }_{n}^{{\delta }_{n}},{\delta }_{k}=0,1\right\}$ with the natural action of ${S}_{n}$ on the latter: $w{\pi }_{k}{w}^{-1}={\pi }_{w\left(k\right)}$ for any $k,w\in {S}_{n}\text{.}$ As an abstract one this group is generated by ${s}_{1},\cdots ,{s}_{n-1}\in {S}_{n}$ and $\pi ={\pi }_{1}$ with the following new relations: ${s}_{1}\pi {s}_{1}\pi =\pi {s}_{1}\pi {s}_{1},$ $\pi {s}_{j}={s}_{j}\pi ,$ $j\ge 2\text{.}$ Each element $\stackrel{\sim }{w}$ of it can be represented in the form $\stackrel{\sim }{w}=\left({\prod }_{k=1}^{n}{\pi }_{k}^{{\delta }_{k}}\right)w,$ $w\in {S}_{n}\text{.}$ We will denote this group by ${\stackrel{\sim }{S}}_{n}\text{.}$ It is called the Weyl group of type ${B}_{n}$ (or ${C}_{n}\text{).}$

Now we are in a position to calculate the $S\text{-matrix}$ of any picture. To do it one should know $\mathrm{\Theta }$ (in the out-state) and the corresponding transformation $\stackrel{\sim }{w}\in {\stackrel{\sim }{S}}_{n}$ from $\mathrm{\Theta }$ to $\mathrm{\Theta }\prime$ for the in-state (see above). Let $\stackrel{\sim }{w}={s}_{{i}_{\ell }}\cdots {s}_{{i}_{1}}$ be of minimal possible length (written $\ell =\ell \left(\stackrel{\sim }{w}\right)\text{),}$ where $0\le i\le n$ and we denote $\pi$ by ${s}_{0}$ for the sake of uniformity. Then (cf. (13)) $Sw∼(Θ)= S∼iℓ (siℓ-1⋯si1(Θ)) ⋯S∼i2 (si1(Θ)) S∼i2(Θ), (33)$ where $S∼i(Θ)= Π(-θ1) for i=0,S∼ik =Sik or Sˆik,$ if ${i}_{k}\ne 0$ and pair $\left(k,k+1\right)$ of the angles from the set ${s}_{{i}_{k-1}}\cdots {s}_{{i}_{1}}\left(\mathrm{\Theta }\right)$ has the coinciding signs or not. Elements from ${\stackrel{\sim }{S}}_{n}$ act on $\mathrm{\Theta }$ as have been explained. It follows from (32) that ${S}_{\stackrel{\sim }{w}}\left(\mathrm{\Theta }\right)$ does not depend on decomposing of $\stackrel{\sim }{w}\text{.}$

The simplest example of such a theory is as follows. Let $Si(Θ)= Sˆi(Θ) =1+si(θi-θi+1) , Π(Θ)= 1-βπθ1, β∈ℂ. (34)$ Then the only equation we need to verify is (31). To obtain some matrix interpretation of (34) one can use some tensor representation of ${S}_{n}$ (see [Che1984]). Our aim is to quantize the angles. We should substitute some pairwise commuting letters ${Y}_{i}$ for ${\theta }_{i}$ in (34) and (according to the procedure of sec. 5) postulate relations (18) and the natural relations. $(1-βπY1)Y1= -Y1(1-βπY1) , [Yj,Π]= 0 for j>1$ Here $\mathrm{\Pi }\left({Y}_{1}\right)$ corresponds to $\pi$ and therefore should act on $\left({Y}_{1}\cdots {Y}_{n}\right)$ as ${Y}_{1}\to -{Y}_{1}\text{.}$ One obtains the algebra ${\stackrel{\sim }{ℋ}}_{n}^{\prime }$ generated by $ℂ\left[{\stackrel{\sim }{S}}_{n}\right]$ and ${Y}_{1},\cdots ,{Y}_{n}$ with the relations (19) and some new ones $πY1+Y1π=2/β, [Yj,π]=0 for j>1. (35)$ This algebra is a certain degeneration of the affine Hecke algebra of type ${B}_{n}$ or ${C}_{n}$ (see e.g. [Rog1985]). To be more precise it is connected with ${B}_{n},{C}_{n},{D}_{n}$ for $\beta =1,2,0$ (see [Che1984]).

We can use the group ${\stackrel{\sim }{S}}_{n}$ for another problem. Let us consider the usual $ℝ$ 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 $ℂ\left[{\stackrel{\sim }{S}}_{n}\right]\text{.}$ The operator ${\pi }_{k}$ $\left(1\le k\le n\right)$ corresponds to the polarization of $k\text{-th}$ particle in a collection; ${\pi }_{k}{\pi }_{k+1}$ describes the change of polarization from the $k\text{-th}$ to $\left(k+1\right)\text{-th}$ particles. That means that ${\pi }_{k}\left({A}_{j}\left(\mathrm{\Theta }\right)\right)=\text{sgn}\left({\theta }_{k}\right){A}_{J}\left(\mathrm{\Theta }\right),$ ${\pi }_{k}{\pi }_{k+1}\left({A}_{J}\left(\mathrm{\Theta }\right)\right)=\text{sgn}\left({\theta }_{k}{\theta }_{k+1}\right){A}_{J}\left(\mathrm{\Theta }\right)\text{.}$

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 ${S}_{i}={s}_{i}$ for ${\pi }_{i}{\pi }_{i+1}=-1$ (i.e. there is no scattering in this case). At last, ${S}_{i}{\pi }_{i}={\pi }_{i+1}{S}_{i}$ and ${S}_{i}{\pi }_{i+1}={\pi }_{i}{S}_{i}$ (the polarizations are conserved). The quantization of angles give us the following relations (from [Che1991]): $Yi+1si-si Yi=(πiπi+1+1) /2=siYi+1-siYi [si,Yj]= [πk,Yj]= [Yk,Yj]=0 for j≠i,i+1, 1≤k,j≤n.$ in place of (19).