## Notes on Affine Hecke Algebras I.(Degenerated Hecke Algebras and Yangians in Mathematical Physics)

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.

## Quantization of angles

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 $Axw=Ax(I) (Θ′), Θ′=w(Θ), x,w∈Sn. (22)$ 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 $\left\{x=\left({x}_{1},\cdots ,{x}_{n}\right)\in {ℝ}^{n},{x}_{i}\ne {x}_{j} \text{for} 1\le i\ne j\le n\right\}\text{.}$ 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: $Yi(Aidw) =θw-1(i) Aidw. (23)$ 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 $I=\left(1,2,\cdots ,n\right)\text{).}$ 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 $I=\left({i}_{1},\cdots ,{i}_{n}\right)$ 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 $x,$ ${\oplus }_{x\in {S}_{n}}ℂ{A}_{x}^{\text{id}}$ with $ℂ\left[{S}_{n}\right]$ and ${P}_{x}$ with $x$ we obtain the basis $\left\{{S}_{w}\left(\mathrm{\Theta }\right)\right\}$ of eigenvectors for $\left\{{Y}_{1},\cdots ,{Y}_{n}\right\}$ in the group algebra $ℂ\left[{S}_{n}\right]\text{:}$ $Yi(Sw(Θ)) =θw-1(i) Sw(Θ), w∈ Sn, 1≤i≤n. (23')$ 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 $ℂ\left[{S}_{n}\right]$ 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 ${\oplus }_{x\in {S}_{n}}ℂ{A}_{x}^{\text{id}}$ for the initial sector $\mathrm{\Theta }\prime =\mathrm{\Theta }\text{.}$ Starting with other sectors we will obtain some isomorphic representations of ${ℋ}_{n}^{\prime }\text{.}$ 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 $\mathrm{\Theta }=\left({\theta }_{1},\cdots ,{\theta }_{n}\right)$ let us denote by $M\left(\mathrm{\Theta }\right)$ the space $ℂ\left[{S}_{n}\right]$ with the natural left (regular) action of ${S}_{n}$ and the action of ${Y}_{1},\cdots ,{Y}_{n}\in {ℋ}_{n}^{\prime },$ which can be uniquely determined by means of the following relations $Yi(1)=θi ·1, 1=id∈Sn, 1≤i≤n. (24)$ Then $M\left(\mathrm{\Theta }\right)$ is an irreducible ${ℋ}_{n}^{\prime }\text{-module}$ for $\mathrm{\Theta }$ being in a general position $\text{(}{\theta }_{i}-{\theta }_{j}\ne 1$ for any $i,j\text{).}$ The operator $ℂ\left[{S}_{n}\right]\ni z\to z·{S}_{w}\left(\mathrm{\Theta }\right)\in ℂ\left[{S}_{n}\right]$ gives an isomorphism $M\left(w\left(\mathrm{\Theta }\right)\right)\to M\left(\mathrm{\Theta }\right),$ which appears to be a ${ℋ}_{n}^{\prime }\text{-isomorphism.}$

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 $ℂ\left[{S}_{n}\right]$ is equal to $⟨{\mathrm{\Sigma }}_{w}⟩$ (coincides with ${\mathrm{\Sigma }}_{w}$ after the substitution $\left\{{Y}_{i}\to {\theta }_{i},1\le i\le n\right\},$ 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 $\left\{{S}_{w}\left(\mathrm{\Theta }\right),w\in {S}_{n}\right\}$ form a basis in $ℂ\left[{S}_{n}\right]$ 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 ${ℋ}_{n}^{\prime }$ is more natural than $ℂ\left[{S}_{n}\right]$ from some other physical point of view. Let us summarize its "quantum" properties.

Theorem 3

 (a) The subalgebras $𝒴=ℂ\left[{Y}_{1},\cdots ,{Y}_{n}\right]$ is a maximal commutative in ${ℋ}_{n}^{\prime }$ i.e. the commutant (centralizer) of $𝒴$ coincides with $𝒴\text{.}$ (b) The centre $𝒞$ (commutant) of ${ℋ}_{n}^{\prime }$ consists of all symmetric polynomials in ${Y}_{1},\cdots ,{Y}_{n}$ (due to I. Bernstein). (c) Haag-duality. The commutant of ${ℋ}_{m}^{\prime }\subset {ℋ}_{n}^{\prime },$ where ${ℋ}_{n}^{\prime }$ is generated by ${s}_{1},\cdots ,{s}_{m-1}$ and ${Y}_{1},\cdots ,{Y}_{m},$ is equal to $𝒞{ℋ}_{n-m}^{\prime }$ generated by $𝒞,{s}_{m+1},\cdots ,{s}_{n-1},{Y}_{m+1},\cdots ,{Y}_{n}$ (see e.g. [Che1987]).

Compare (c) with the corresponding axiom from [FRS1989]. As for ${S}_{n}$ the commutant of $ℂ\left[{S}_{m}\right]\subset ℂ\left[{S}_{n}\right]$ modulo the centre is more than complimentary $ℂ\left[{S}_{n-m}\right]\text{.}$

Let us consider ${ℋ}_{n}^{\prime }\left(\eta \right)$ (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 ${ℋ}_{n}^{\prime }\left(0\right),$ which is the "quasi-classical" limit of ${ℋ}_{n}^{\prime }\left(\eta \right)$ and especially simple. For example, it is evident that any element $x\in {ℋ}_{n}^{\prime }\left(0\right)$ can be represented in the form $x=\sum w{y}_{w},$ $w\in {S}_{n}$ for appropriate polynomials ${y}_{w}={y}_{w}\left({Y}_{1},\cdots ,{Y}_{n}\right)\text{.}$ 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, $x{Y}_{k}={Y}_{k}x⇒{y}_{w}\left({Y}_{k}-{Y}_{{w}^{-1}\left(k\right)}\right)\equiv 0\text{.}$ The latter is impossible. In particular, the subalgebra $𝒴=ℂ\left[{Y}_{1},\cdots ,{Y}_{n}\right]$ coincides with its commutant in ${ℋ}_{n}^{\prime }\left(0\right)\text{.}$

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 $\left\{{y}_{w},w\in {ℂ}_{n}\right\}$ 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 $k\in {ℤ}_{+}$ 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 $x\left(\eta \right)\in 𝒴$ in this neighbourhood. We have proved the required statement for small $|\eta |\text{.}$ But any ${ℋ}_{n}^{\prime }\left(\eta \right)$ for $\eta \ne 0$ is isomorphic to ${ℋ}_{n}^{\prime }\left(1\right)={ℋ}_{n}^{\prime }$ (see above). Hence, the coincidence of $𝒴$ 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 $A\in {ℋ}_{n}^{\prime }$ 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.