(Degenerated Hecke Algebras and Yangians in Mathematical Physics)

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 ${\mathbb{R}}_{+}=\{x\ge 0\}$ instead of $\mathbb{R}$ 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 $\mathrm{\Pi}\left(\mathrm{\Theta}\right)=\left({\mathrm{\Pi}}_{{i}_{1}}^{{j}_{1}}(-{\theta}_{1})\right),$ where $\mathrm{\Theta}=({\theta}_{1},\cdots ,{\theta}_{n})$ 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 $(\mp )\text{.}$ The first is before $(\theta <0)$ and the second $(\theta >0)$ 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 $S\text{).}$ Let us denote the $S\text{-matrix}$ of the third type $(-,+)$ by $\stackrel{\u02c6}{S}$ (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 ${\stackrel{\u02c6}{S}}_{i}\left(\theta \right)$ (see (13b)) for scattering at the intersection point of the $i\text{-th}$ and $(i+1)\text{-th}$ particles (numbers are from the bottom to the top). We remind that ${S}_{i}$ and ${\stackrel{\u02c6}{S}}_{i}$ depend only on ${\theta}_{i}-{\theta}_{i+1}$ and on the corresponding colours of the $i\text{-th}$ and $(i-1)\text{-th}$ particles.

We should add to (16) its direct analogs $\stackrel{\u02c6}{S}S\stackrel{\u02c6}{S}=\stackrel{\u02c6}{S}S\stackrel{\u02c6}{S},$ $S\stackrel{\u02c6}{S}\stackrel{\u02c6}{S}=\stackrel{\u02c6}{S}\stackrel{\u02c6}{S}S$ (with the same indices and arguments), and the new one: $$\begin{array}{cc}\mathrm{\Pi}\left(u\right){\stackrel{\u02c6}{S}}_{1}(2u+v)\mathrm{\Pi}(u+v){S}_{1}\left(v\right)={S}_{1}\left(v\right)\mathrm{\Pi}(u+v){\stackrel{\u02c6}{S}}_{1}(2u+v)\mathrm{\Pi}\left(u\right)\text{.}& \text{(31)}\end{array}$$ 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)) $$\begin{array}{cc}SSS=SSS,\hspace{0.17em}\stackrel{\u02c6}{S}S\stackrel{\u02c6}{S}=\stackrel{\u02c6}{S}S\stackrel{\u02c6}{S},\hspace{0.17em}S\stackrel{\u02c6}{S}\stackrel{\u02c6}{S}=\stackrel{\u02c6}{S}\stackrel{\u02c6}{S}S,\hspace{0.17em}\mathrm{\Pi}\stackrel{\u02c6}{S}\mathrm{\Pi}S=S\mathrm{\Pi}\stackrel{\u02c6}{S}\mathrm{\Pi}& \text{(32a)}\end{array}$$ together with the evident relations (see (12)) $$\begin{array}{cc}[{S}_{i},{S}_{j}]=[{\stackrel{\u02c6}{S}}_{i},{S}_{j}]=[{\stackrel{\u02c6}{S}}_{i},{\stackrel{\u02c6}{S}}_{j}]=[\mathrm{\Pi},{S}_{j}]=[\mathrm{\Pi},{\stackrel{\u02c6}{S}}_{j}]=0& \text{(32b)}\end{array}$$ 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 $\mathrm{\Theta}=({\theta}_{1},\cdots ,{\theta}_{n})$ to some set of angles $\mathrm{\Theta}\prime $ of the in-state can be represented as the sequence $\stackrel{\sim}{w}=({\epsilon}_{1}1\prime ,{\epsilon}_{2}2\prime ,\cdots ,{\epsilon}_{n}n\prime )$ for ${\epsilon}_{k}=\pm 1,$ $(1\prime ,2\prime ,\cdots ,n\prime )=w\in {S}_{n}\text{.}$ 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 $\stackrel{\sim}{w}=(3,-2,-1)$ for fig. 6. The composition of two elements $\stackrel{\sim}{w},\stackrel{\sim}{w}\prime $ of this type is quite natural: if $k\to \left(\stackrel{\sim}{w}\right){\epsilon}_{k}k\prime ,$ $k\prime \to \left(\stackrel{\sim}{w}\prime \right){\epsilon}_{k\prime}^{\prime}k\u2033$ then $k\to \left(\stackrel{\sim}{w}\prime \stackrel{\sim}{w}\right)\left({\epsilon}_{k}{\epsilon}_{k\prime}^{\prime}\right)k\u2033\text{.}$

The group of all $\stackrel{\sim}{w}$ can be represented by permutations (acting on $\mathrm{\Theta}\in {\mathbb{R}}^{n}$ in the usual manner) multiplied by any number of the reflections $${\pi}_{k}\left(\mathrm{\Theta}\right)=({\theta}_{1},\cdots ,{\theta}_{k-1},-{\theta}_{k},{\theta}_{k+1},\cdots ,{\theta}_{n})\phantom{\rule{1em}{0ex}}(1\le k\le n)\text{.}$$

Mathematically, it is the semi-direct product of ${S}_{n}$ and
${\left({\mathbb{Z}}_{2}\right)}^{n}=\{{\pi}_{1}^{{\delta}_{1}}\cdots {\pi}_{n}^{{\delta}_{n}},{\delta}_{k}=0,1\}$
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)) $$\begin{array}{cc}{S}_{\stackrel{\sim}{w}}\left(\mathrm{\Theta}\right)={\stackrel{\sim}{S}}_{{i}_{\ell}}\left({s}_{{i}_{\ell -1}}\cdots {s}_{{i}_{1}}\left(\mathrm{\Theta}\right)\right)\cdots {\stackrel{\sim}{S}}_{{i}_{2}}\left({s}_{{i}_{1}}\left(\mathrm{\Theta}\right)\right){\stackrel{\sim}{S}}_{{i}_{2}}\left(\mathrm{\Theta}\right),& \text{(33)}\end{array}$$ where $${\stackrel{\sim}{S}}_{i}\left(\mathrm{\Theta}\right)=\mathrm{\Pi}(-{\theta}_{1})\hspace{0.17em}\text{for}\hspace{0.17em}i=0,\phantom{\rule{1em}{0ex}}{\stackrel{\sim}{S}}_{ik}={S}_{ik}\hspace{0.17em}\text{or}\hspace{0.17em}{\stackrel{\u02c6}{S}}_{ik},$$ if ${i}_{k}\ne 0$ and pair $(k,k+1)$ 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
$$\begin{array}{cc}{S}_{i}\left(\mathrm{\Theta}\right)={\stackrel{\u02c6}{S}}_{i}\left(\mathrm{\Theta}\right)=1+{s}_{i}({\theta}_{i}-{\theta}_{i+1}),\hspace{0.17em}\mathrm{\Pi}\left(\mathrm{\Theta}\right)=1-\beta \pi {\theta}_{1},\hspace{0.17em}\beta \in \u2102\text{.}& \text{(34)}\end{array}$$
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.
$$(1-\beta \pi {Y}_{1}){Y}_{1}=-{Y}_{1}(1-\beta \pi {Y}_{1}),\hspace{0.17em}[{Y}_{j},\mathrm{\Pi}]=0\hspace{0.17em}\text{for}\hspace{0.17em}j>1$$
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
${\stackrel{\sim}{\mathscr{H}}}_{n}^{\prime}$ generated by
$\u2102\left[{\stackrel{\sim}{S}}_{n}\right]$ and
${Y}_{1},\cdots ,{Y}_{n}$ with the relations (19) and some new ones
$$\begin{array}{cc}\pi {Y}_{1}+{Y}_{1}\pi =2/\beta ,\hspace{0.17em}[{Y}_{j},\pi ]=0\hspace{0.17em}\text{for}\hspace{0.17em}j>1\text{.}& \text{(35)}\end{array}$$
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 $\mathbb{R}$ 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 $\u2102\left[{\stackrel{\sim}{S}}_{n}\right]\text{.}$ The operator ${\pi}_{k}$ $(1\le k\le n)$ 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 $(k+1)\text{-th}$ 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 $$\begin{array}{cc}{S}_{i}\left(\mathrm{\Theta}\right)=({\pi}_{i}{\pi}_{i+1}+1){({\theta}_{i}-{\theta}_{i+1})}^{-1}/2+{s}_{i}& \text{(36)}\end{array}$$ 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]): $$\begin{array}{c}{Y}_{i+1}{s}_{i}-{s}_{i}{Y}_{i}=({\pi}_{i}{\pi}_{i+1}+1)/2={s}_{i}{Y}_{i+1}-{s}_{i}{Y}_{i}\\ [{s}_{i},{Y}_{j}]=[{\pi}_{k},{Y}_{j}]=[{Y}_{k},{Y}_{j}]=0\hspace{0.17em}\text{for}\hspace{0.17em}j\ne i,i+1,1\le k,j\le n\text{.}\end{array}$$ in place of (19).