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.
I shall try to interpret this mathematical trick as some quantization procedure. We will look for observables which correspond to ${\theta}_{1},\cdots ,{\theta}_{n}\text{.}$ (I shall remind that $\text{mtg}{\theta}_{i}$ is the momentum of the $i\text{-th}$ particle. Therefore, a quantization of angles is, in fact, a quantization of impulses).
Yang's $S\text{-matrix}$ (17) is of a very symmetric type. Any in-state ${A}_{{i}_{1}}\left({\theta}_{1}^{\prime}\right)\cdots {A}_{{i}_{n}}\left({\theta}_{n}^{\prime}\right)$ for $\mathrm{\Theta}\prime =w\left(\mathrm{\Theta}\right)$ (see (1,2)) can be expressed in terms of ${A}_{J}\left(\mathrm{\Theta}\right),$ where $J=x\left(I\right)$ for permutations $x\in {S}_{n}\text{.}$ Hence, it is natural to diminish the space of states. We fix $\mathrm{\Theta}$ and some initial set of indices $I\text{.}$ Let $$\begin{array}{cc}{A}_{x}^{w}={A}_{x\left(I\right)}\left(\mathrm{\Theta}\prime \right),\hspace{0.17em}\mathrm{\Theta}\prime =w\left(\mathrm{\Theta}\right),x,w\in {S}_{n}\text{.}& \text{(22)}\end{array}$$ Here $w$ and $x$ play different roles. The orderings of ${\theta}_{1},\cdots ,{\theta}_{n}$ are indexed by $w\text{.}$ They are in one-one correspondence with the "sectors" — the connected components of $\{x=({x}_{1},\cdots ,{x}_{n})\in {\mathbb{R}}^{n},{x}_{i}\ne {x}_{j}\hspace{0.17em}\text{for}\hspace{0.17em}1\le i\ne j\le n\}\text{.}$ Therefore, we will use the visual name "sector" in place of "ordering". The states (for every given sector) are numbered by $x\text{.}$ By definition, ${P}_{x}{A}_{y}^{w}={A}_{xy}^{w},$ $x,y\in {S}_{n}\text{.}$
The natural (but wrong) idea is to introduce the quantum angles ${Y}_{1},\cdots ,{Y}_{n}$ by relations ${Y}_{i}\left({A}_{x}^{w}\right)={\theta}_{{w}^{-1}\left(i\right)}{A}_{x}^{w}\text{.}$ If $\left\{{A}_{x}^{w}\right\}$ were independent it might be possible. But they are linearly dependent. However, one can try the following: $$\begin{array}{cc}{Y}_{i}\left({A}_{\text{id}}^{w}\right)={\theta}_{{w}^{-1}\left(i\right)}{A}_{\text{id}}^{w}\text{.}& \text{(23)}\end{array}$$ Given $I$ we define the action of $\left\{{Y}_{i}\right\}$ only on the "vacuum" states ${A}_{\text{id}}^{w}={A}_{{i}_{1}\cdots {i}_{n}}\left(\mathrm{\Theta}\prime \right)$ for each sector. Let us assume that $N\ge n$ and all ${i}_{1},\cdots ,{i}_{n}$ in are pairwise distinct (fox example $I=(1,2,\cdots ,n)\text{).}$ Then the number of sectors (i.e. orderings of $\left\{{\theta}_{i}\right\}\text{)}$ is equal to the number of states for each of them. Therefore, the definition (23) is, in principle, consistent. If the set $I=({i}_{1},\cdots ,{i}_{n})$ is not "generic" we should be more precise (we will not consider this case here).
All the sectors are glued together by the $S\text{-matrices}$ $\left\{{S}_{w}\left(\mathrm{\Theta}\right)\right\}$ (see (15)). In particular, ${A}_{\text{id}}^{w}={S}_{w}\left(\mathrm{\Theta}\right){A}_{\text{id}}^{\text{id}}\text{.}$ Identifying ${A}_{x}^{\text{id}}$ with $x,$ ${\oplus}_{x\in {S}_{n}}\u2102{A}_{x}^{\text{id}}$ with $\u2102\left[{S}_{n}\right]$ and ${P}_{x}$ with $x$ we obtain the basis $\left\{{S}_{w}\left(\mathrm{\Theta}\right)\right\}$ of eigenvectors for $\{{Y}_{1},\cdots ,{Y}_{n}\}$ in the group algebra $\u2102\left[{S}_{n}\right]\text{:}$ $$\begin{array}{cc}{Y}_{i}\left({S}_{w}\left(\mathrm{\Theta}\right)\right)={\theta}_{{w}^{-1}\left(i\right)}{S}_{w}\left(\mathrm{\Theta}\right),\hspace{0.17em}w\in {S}_{n},\hspace{0.17em}1\le i\le n\text{.}& \text{(23')}\end{array}$$ All these are true for ${\theta}_{1},\cdots ,{\theta}_{n}$ being in a general position only. Simple calculations show that $\left\{{Y}_{i}\right\}$ and $\left\{{s}_{j}\right\}$ satisfy the relations (18-19), where ${S}_{n}$ acts on $\u2102\left[{S}_{n}\right]$ by left multiplications (cf. [Che1986-2]). Deduce this statement from (23').
We have collected the vacuum states $\left\{{A}_{\text{id}}^{w}\right\}$ (they linearly generate all the states) together in the space of states ${\oplus}_{x\in {S}_{n}}\u2102{A}_{x}^{\text{id}}$ for the initial sector $\mathrm{\Theta}\prime =\mathrm{\Theta}\text{.}$ Starting with other sectors we will obtain some isomorphic representations of ${\mathscr{H}}_{n}^{\prime}\text{.}$ The $S\text{-matrices}$ will be interwiners between these "sector" representations. In fact, this interpretation of $S$ is very close to the ideology of superselection sectors (see [FRS1989,MSc1990]). We will not discuss here the latter, but formulate the corresponding mathematical theorem. As a matter of fact it has been partially proven.
Theorem 2 (see [Rog1985,Che1986-2]). Given $\mathrm{\Theta}=({\theta}_{1},\cdots ,{\theta}_{n})$ let us denote by $M\left(\mathrm{\Theta}\right)$ the space $\u2102\left[{S}_{n}\right]$ with the natural left (regular) action of ${S}_{n}$ and the action of ${Y}_{1},\cdots ,{Y}_{n}\in {\mathscr{H}}_{n}^{\prime},$ which can be uniquely determined by means of the following relations $$\begin{array}{cc}{Y}_{i}\left(1\right)={\theta}_{i}\xb71,\hspace{0.17em}1=\text{id}\in {S}_{n},\hspace{0.17em}1\le i\le n\text{.}& \text{(24)}\end{array}$$ Then $M\left(\mathrm{\Theta}\right)$ is an irreducible ${\mathscr{H}}_{n}^{\prime}\text{-module}$ for $\mathrm{\Theta}$ being in a general position $\text{(}{\theta}_{i}-{\theta}_{j}\ne 1$ for any $i,j\text{).}$ The operator $\u2102\left[{S}_{n}\right]\ni z\to z\xb7{S}_{w}\left(\mathrm{\Theta}\right)\in \u2102\left[{S}_{n}\right]$ gives an isomorphism $M\left(w\left(\mathrm{\Theta}\right)\right)\to M\left(\mathrm{\Theta}\right),$ which appears to be a ${\mathscr{H}}_{n}^{\prime}\text{-isomorphism.}$
This theorem is in fact equivalent to theorem 1. Indeed, any ${S}_{w}\left(\mathrm{\Theta}\right)$ considered as a function of ${\theta}_{1},\cdots ,{\theta}_{n}$ with its values in $\u2102\left[{S}_{n}\right]$ is equal to $\u27e8{\mathrm{\Sigma}}_{w}\u27e9$ (coincides with ${\mathrm{\Sigma}}_{w}$ after the substitution $\{{Y}_{i}\to {\theta}_{i},1\le i\le n\},$ where all the $\left\{Y\right\}$ should be collected on the right). Therefore, the isomorphism above is a direct corollary of statement (b) of theorem 1. The irreducibility of $M\left(\mathrm{\Theta}\right)$ is clear, because $\{{S}_{w}\left(\mathrm{\Theta}\right),w\in {S}_{n}\}$ form a basis in $\u2102\left[{S}_{n}\right]$ of eigenvectors with respect to $\left\{{Y}_{i}\right\}$ with pairwise distinct eigenvalues (for $\mathrm{\Theta}$ in a general position).
It is worth mentioning that ${\mathscr{H}}_{n}^{\prime}$ is more natural than $\u2102\left[{S}_{n}\right]$ from some other physical point of view. Let us summarize its "quantum" properties.
Theorem 3
(a) | The subalgebras $\mathcal{Y}=\u2102[{Y}_{1},\cdots ,{Y}_{n}]$ is a maximal commutative in ${\mathscr{H}}_{n}^{\prime}$ i.e. the commutant (centralizer) of $\mathcal{Y}$ coincides with $\mathcal{Y}\text{.}$ |
(b) | The centre $\mathcal{C}$ (commutant) of ${\mathscr{H}}_{n}^{\prime}$ consists of all symmetric polynomials in ${Y}_{1},\cdots ,{Y}_{n}$ (due to I. Bernstein). |
(c) | Haag-duality. The commutant of ${\mathscr{H}}_{m}^{\prime}\subset {\mathscr{H}}_{n}^{\prime},$ where ${\mathscr{H}}_{n}^{\prime}$ is generated by ${s}_{1},\cdots ,{s}_{m-1}$ and ${Y}_{1},\cdots ,{Y}_{m},$ is equal to $\mathcal{C}{\mathscr{H}}_{n-m}^{\prime}$ generated by $\mathcal{C},{s}_{m+1},\cdots ,{s}_{n-1},{Y}_{m+1},\cdots ,{Y}_{n}$ (see e.g. [Che1987]). |
Compare (c) with the corresponding axiom from [FRS1989]. As for ${S}_{n}$ the commutant of $\u2102\left[{S}_{m}\right]\subset \u2102\left[{S}_{n}\right]$ modulo the centre is more than complimentary $\u2102\left[{S}_{n-m}\right]\text{.}$
Let us consider ${\mathscr{H}}_{n}^{\prime}\left(\eta \right)$ (see above) with the relations ${Y}_{i+1}{s}_{i}-{s}_{i}{Y}_{i}=\eta ={s}_{i}{Y}_{i+1}-{Y}_{i}{s}_{i}$ in place of (19a). Here $\eta $ plays the role of the Planck constant $\hslash \text{.}$ For $\eta =0$ we get the algebra ${\mathscr{H}}_{n}^{\prime}\left(0\right),$ which is the "quasi-classical" limit of ${\mathscr{H}}_{n}^{\prime}\left(\eta \right)$ and especially simple. For example, it is evident that any element $x\in {\mathscr{H}}_{n}^{\prime}\left(0\right)$ can be represented in the form $x=\sum w{y}_{w},$ $w\in {S}_{n}$ for appropriate polynomials ${y}_{w}={y}_{w}({Y}_{1},\cdots ,{Y}_{n})\text{.}$ Moreover, if ${y}_{w}\ne 0$ for some $w\ne \text{id}$ then $x{Y}_{k}\ne {Y}_{k}x$ for any $k$ such that $w\left(k\right)\ne k\text{.}$ Indeed, $x{Y}_{k}={Y}_{k}x\Rightarrow {y}_{w}({Y}_{k}-{Y}_{{w}^{-1}\left(k\right)})\equiv 0\text{.}$ The latter is impossible. In particular, the subalgebra $\mathcal{Y}=\u2102[{Y}_{1},\cdots ,{Y}_{n}]$ coincides with its commutant in ${\mathscr{H}}_{n}^{\prime}\left(0\right)\text{.}$
There is a nice mathematical trick to extend the above statement to any $\eta $ sufficiently close to $0\text{.}$ To calculate this commutant for any $\eta $ one should solve linear equations for coefficients of the polynomials $\{{y}_{w},w\in {\u2102}_{n}\}$ in the decomposition $x=\mathrm{\Sigma}w{y}_{w}\text{.}$ If the commutant contains $x\left(\eta \right)$ with ${y}_{w}\ne 0$ for some $w\ne \text{id}$ then certain determinants of minors are to be equal to zero (and vice versa). To be more precise, given $k\in {\mathbb{Z}}_{+}$ the rank of the above system for polynomials ${y}_{w}$ of degree $\le k$ for such $\eta $ is less than the corresponding rank for $\eta =0\text{.}$ The determinants are scalar polynomial functions in $\eta \text{.}$ Some of them do not equal zero at $\eta =0$ $\text{(}x={y}_{\text{id}}$ for $\eta =0\text{).}$ Hence, they have no common zeroes not only at $0$ but in a neighbourhood of $\eta =0\text{.}$ Therefore, $\text{rank}\left(\eta \right)=\text{rank}\left(0\right)$ and $x\left(\eta \right)\in \mathcal{Y}$ in this neighbourhood. We have proved the required statement for small $\left|\eta \right|\text{.}$ But any ${\mathscr{H}}_{n}^{\prime}\left(\eta \right)$ for $\eta \ne 0$ is isomorphic to ${\mathscr{H}}_{n}^{\prime}\left(1\right)={\mathscr{H}}_{n}^{\prime}$ (see above). Hence, the coincidence of $\mathcal{Y}$ and its commutant holds true for arbitrary $\eta $ as well.
The best way to prove (a, b, c) is to use the following statement. each element $A\in {\mathscr{H}}_{n}^{\prime}$ has the unique representation: $A={\mathrm{\Sigma}}_{w}{\mathrm{\Sigma}}_{w}{y}_{w},$ where $w\in {S}_{n},$ ${Y}_{w}$ are some rational function in ${y}_{1},\cdots ,{Y}_{n},$ $\left\{{\mathrm{\Sigma}}_{w}\right\}$ are from theorem 1.