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.

Introduction

The main aim of these notes, based on my lectures at Bonn University (May-June, 1990), is to demonstrate that degenerated affine Hecke algebras (and Yangians) are quite natural from mathematical and physical points of view, and are necessary to understand the very classic object — the symmetric group $S_n\text{.}$ There is no need to discuss much its important role in mathematics and natural sciences, the representation theory and the theory of combinations. But $S_n$ is a key thing not only for science.

The great philosopher Immanuel Kant found that our consciousness is based on two a priori notions of space and time. I like his books very much and consider him as the best philosopher in Europe. Nevertheless, I think he made a mistake. One should add a third notion to these two. I mean the notion of combination.

I do not attempt to discuss this point here. One can find easily many arguments in favor of the thesis that calculus of permutations (transpositions) interchanges is the main work our brain can do and likes to do. Gambles give good models. They are among the most exciting things for human beings, Moreover, the primitive ones are more exciting, because here the fundamental concept of combination is as pure as possible. However, gambling is nothing else but the applied representation theory of $S_n\text{.}$

It is strange enough for me that the symmetric group is not a necessary thing to study for students in physical departments. The space-time is, but not $S_n\text{.}$ I can understand to a certain extent why it is so for mathematicians (they do not try to understand the universe and for them consistent notions are on equal grounds). As for physics (old and new) its most significant parts (e.g. the diagrammatic method, the string theory, the two-dimensional conforma! theory) are very closely connected with combinatorics. Moreover, the modern representation theory was created as a base for quantization and is now such a base. But its classic ground is undoubtedly in Young's works on $S_n\text{.}$

I shall begin now the detailed exposition of the simplest example of a physical theory based on $S_n$ only. This one is the most natural way to quantum groups. We will discuss A. Zamolodchikov's world of two-dimensional elementary factorized particles. Let the space be $ℝ$ with the only coordinate $x,$ and let $t$ be the usual time. The life of a free particle can be represented as a line in ${ℝ}^2\text{.}$ This line (the graph of the movement of the particle) can be determined (see fig. 1) by some (initial) point $(t_0,x\left(t_0\right))$ and by the angle $\theta$ from the $t\text{-axis}$ to it $(-\pi /2<\theta <\pi /2)\text{.}$ The momentum of the particle is $p=mtg\left(\theta \right)$ (in the proper units), where $m$ is its mass.

Let us suppose that the masses of all the particles are the same and (a) there is no mechanical interaction.

It means that individual momenta of two or more particles are conserved at the points of intersections of corresponding lines. In some evident sense the particles are transparent to each other (see fig. 2). It is not a billiard.

This world is too boring. We should add some quantum scattering to make the life of these particles more interesting. Let us consider "coloured" particles of types (colours) $1,2,\dots ,N\text{.}$ The particles are permitted to (b) change the colours (types) only at the points of intersections.

It is the second postulate. Moreover, (c) the $S\text{-matrix}$ of any collision (the set of all the amplitudes) does not depend on
concrete initial positions of the particles.

I should comment on axiom (c). One can always slightly deform the initial positions (the $x\text{-coordinates}$ $x_1,\dots ,x_n$ at the time $t=t_0\text{)}$ to have only two-particle intersections in the future and in the past like in fig. 2. By the standard principles of quantum mechanics any element of the total $S\text{-matrix}$ can be obtained as a certain sum of products of two-particle amplitudes over the intersection points. Thus, it results from (c) that if we know two-particle $S\text{-matrices}$ we can calculate any amplitude, depending in fact only on the angles and on the order of the points of the in-state and out-state.

The last things we have to declare to obtain Zamolodchikov's world are some locality (or causality) properties and some invariances: (d) every two-particle $S\text{-matrix}$ depends only on the colours of these two particles and on the difference of their angles. Other particles do not affect it.

A few words about the origin of all these properties. It was conjectured in Zamolodchikov brothers' paper [Zam1978] that the quantum scattering processes should be of this type for $O\left(N\right)\text{-symmetric}$ chiral field models in two-dimensions like Pohlmeyer's $\sigma \text{-model.}$ Besides the asymptotic freedom and the existence of isovector $N\text{-plets}$ of massive particles ("coloured particles") the following property is of great importance. This quantum model possesses an infinite set of conservation laws extending their classical counterparts. It gives us the conservation of individual momenta.

Zamolodchikov's conjecture was proved (at some physical level of strictness). I am not able to discuss the details here. It is worth mentioning that particles in [Zam1978] and in other papers are relativistic. Therefore, the authors use rapidities $\theta$ $\text{(}{p}^0=\text{mch}\left(\theta \right),$ $p^1=\text{msh}\left(\theta \right)$ for the relativistic momentum $p=(p^0,p^1)\text{)}$ instead of angles. The dependence of $S\text{-matrices}$ on differences of angles (see (d)) is nothing else but the relativistic invariance. Some unitary conditions and crossing-symmetry relations are of physical importance as well. We will omit the latter here and pay little attention to the first.

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