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
aram@unimelb.edu.au

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 ${ℝ}_{+}=\{x\ge 0\}$ 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 $\mathrm{\Pi }\left(\mathrm{\Theta }\right)=\left({\mathrm{\Pi }}_{i_1}^{j_1}(-{\theta }_1)\right),$ 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 $\hat{S}$ (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 ${\hat{S}}_i\left(\theta \right)$ (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 ${\hat{S}}_i$ 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 $\hat{S}S\hat{S}=\hat{S}S\hat{S},$ $S\hat{S}\hat{S}=\hat{S}\hat{S}S$ (with the same indices and arguments), and the new one: $$\begin{array}{cc}\mathrm{\Pi }\left(u\right){\hat{S}}_1(2u+v)\mathrm{\Pi }(u+v)S_1\left(v\right)=S_1\left(v\right)\mathrm{\Pi }(u+v){\hat{S}}_1(2u+v)\mathrm{\Pi }\left(u\right)\text{.}& \text{(31)}\end{array}$$ 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)) $$\begin{array}{cc}SSS=SSS,\hspace{0.17em}\hat{S}S\hat{S}=\hat{S}S\hat{S},\hspace{0.17em}S\hat{S}\hat{S}=\hat{S}\hat{S}S,\hspace{0.17em}\mathrm{\Pi }\hat{S}\mathrm{\Pi }S=S\mathrm{\Pi }\hat{S}\mathrm{\Pi }& \text{(32a)}\end{array}$$ together with the evident relations (see (12)) $$\begin{array}{cc}[S_i,S_j]=[{\hat{S}}_i,S_j]=[{\hat{S}}_i,{\hat{S}}_j]=[\mathrm{\Pi },S_j]=[\mathrm{\Pi },{\hat{S}}_j]=0& \text{(32b)}\end{array}$$ 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 $k\to \left(\stackrel{\sim }{w}\right){\epsilon }_{k}k\prime ,$ $k\prime \to \left(\stackrel{\sim }{w}\prime \right){\epsilon }_{k\prime }^{\prime }k″$ 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 $${\pi }_k\left(\mathrm{\Theta }\right)=({\theta }_1,\cdots ,{\theta }_{k-1},-{\theta }_k,{\theta }_{k+1},\cdots ,{\theta }_n)(1\le k\le n)\text{.}$$

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 $${\stackrel{\sim }{S}}_i\left(\mathrm{\Theta }\right)=\mathrm{\Pi }(-{\theta }_1)\hspace{0.17em}\text{for}\hspace{0.17em}i=0,{\stackrel{\sim }{S}}_{ik}=S_{ik}\hspace{0.17em}\text{or}\hspace{0.17em}{\hat{S}}_{ik},$$ 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 $$\begin{array}{cc}{S}_i\left(\mathrm{\Theta }\right)={\hat{S}}_i\left(\mathrm{\Theta }\right)=1+s_i({\theta }_i-{\theta }_{i+1}),\hspace{0.17em}\mathrm{\Pi }\left(\mathrm{\Theta }\right)=1-\beta \pi {\theta }_1,\hspace{0.17em}\beta \in ℂ\text{.}& \text{(34)}\end{array}$$ 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 $ℂ\left[{\stackrel{\sim }{S}}_n\right]$ 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 $ℝ$ 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 ${\pi }_k$ $(1\le k\le n)$ 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).

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