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.

An algebraic interpretation

Let us interpret formulas (6), (12), and (13) algebraically. We will now consider symbols $A_{i_1}\left(\theta \right)\cdots A_{i_n}\left({\theta }_n\right)$ as products of generators $A_{i_1}\left(\theta \right),\cdots ,A_{i_n}\left({\theta }_n\right)$ in the free algebra $TA$ with generators $A_i\left(\theta \right)$ (they have two indices $i,\theta ,$ where $1\le i\le N,\theta$ can be any number). The quotient algebra $𝒜$ of $TA$ by relations (3) is called Zamolodchikov's algebra. We suppose here and further that $S$ is analytical for $\mathrm{\Theta }$ close to $0=(0,\cdots 0)$ and $S\left(0\right)=1\text{.}$ The latter is quite natural physically (there should be no scattering, when two particles are "parallel" one to another). Here we identify $1$ with $1\otimes 1$ for $1$ being the unit matrix.

Let us impose on $S$ relation (6) and the so-called unitaty condition $$\begin{array}{cc}S\left(\theta \right)S(-\theta )={}^{12}1=1\otimes 1=1& \text{(14)}\end{array}$$ Given $\mathrm{\Theta }=({\theta }_1,\cdots ,{\theta }_n)$ in some neighbourhood of $0$ we wil denote by $TA\left\{\mathrm{\Theta }\right\}$ the vector subspace in $TA$ generated by $A_I\left(w\left(\mathrm{\Theta }\right)\right)$ for any $I=(i_1,\cdots ,i_n),$ $w\in S_n\text{.}$ Let $𝒜\left\{\mathrm{\Theta }\right\}$ be its image in $𝒜\text{.}$

Relations (6), (14) are in fact equivalent to a certain Wick (or Poincaré-Birkhoff-Witt) theorem for every $𝒜\left\{\mathrm{\Theta }\right\}\text{.}$ Namely (see e.g. [Zam1978,Che1979]),

(a)	every $A_I\left(w\left(\mathrm{\Theta }\right)\right)$ is a linear combination of some $A_J\left(\mathrm{\Theta }\right)$ in $𝒜\left\{\mathrm{\Theta }\right\}\text{;}$
(b)	all the $A_J\left(\mathrm{\Theta }\right)$ are linearly independent in $𝒜\left\{\mathrm{\Theta }\right\}$ for any multi-indices $J\text{.}$

Here ${\theta }_1,\cdots {\theta }_n$ can be arbitrary complex numbers (possibly not distinct different). The appearance of (14) is quite evident. If one applies (3) twice, he should get the initial binomial because of (b). Physically (14) corresponds to the change of time $t$ from $t_0^{\prime }$ to $t_0>t_0^{\prime }$ then back.

Our other problem is to find the best algebraic language for relations (6), (12), and formula (13). We will start with $S_i$ (see (13b)). Let us denote by $S_w\left(\mathrm{\Theta }\right)$ the product (13a) for $w$ from (10). Then the set of functions $\left\{S_w\left(\mathrm{\Theta }\right)\right\}$ satisfies the following conditions (cf. [Che1984]) $$\begin{array}{cc}{S}_x\left(y\left(\mathrm{\Theta }\right)\right)S_y\left(\mathrm{\Theta }\right)=S_{xy}\left(\mathrm{\Theta }\right),& \text{(15a)}\end{array}$$ if $$\begin{array}{cc}l\left(xy\right)=l\left(x\right)+l\left(y\right),x,y\in S_n\text{.}& \text{(15b)}\end{array}$$ (Deduce it from (13)).

Conversally, let $S_i=S_{s_i}$ be some undetermined functions of $\theta$ with values anywhere and let relation (12) together with $$\begin{array}{cc}{S}_{i+1}\left(u\right)S_i(u+v)S_{i+1}\left(v\right)=S_i\left(v\right)S_{i+1}(u+v)S_i\left(v\right)& \text{(16)}\end{array}$$ be valid for arbitrary $u={\theta }_i-{\theta }_{i+1},$ $v={\theta }_{i+1}-{\theta }_{i+2}\text{.}$

Then we claim that one can uniquely define the set of functions $\left\{S_w\left(\mathrm{\Theta }\right)\right\}$ by (15). Moreover, it is possible to omit the condition (15b) in the case of unitary $S(S\left(\theta \right)S(-\theta )=1)\text{.}$

Summarizing, we see that the tensor mode of rewriting (4) in the form of (6) is very convenient but not the best. The most natural way is to use $S_n$ as the index set (see (15) and (16)).

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