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.

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 $$\begin{array}{cc}{A}_x^w=A_{x\left(I\right)}\left(\mathrm{\Theta }\prime \right),\hspace{0.17em}\mathrm{\Theta }\prime =w\left(\mathrm{\Theta }\right),x,w\in S_n\text{.}& \text{(22)}\end{array}$$ 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 $\{x=(x_1,\cdots ,x_n)\in {ℝ}^n,x_i\ne x_j\hspace{0.17em}\text{for}\hspace{0.17em}1\le i\ne j\le n\}\text{.}$ 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 $x,$ ${\oplus }_{x\in S_n}ℂA_x^{\text{id}}$ 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 $\{Y_1,\cdots ,Y_n\}$ in the group algebra $ℂ\left[S_n\right]\text{:}$ $$\begin{array}{cc}{Y}_i\left(S_w\left(\mathrm{\Theta }\right)\right)={\theta }_{w^{-1}\left(i\right)}{S}_w\left(\mathrm{\Theta }\right),\hspace{0.17em}w\in S_n,\hspace{0.17em}1\le i\le n\text{.}& \text{(23')}\end{array}$$ 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 $ℂ\left[S_n\right]$ 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 ${\oplus }_{x\in S_n}ℂA_x^{\text{id}}$ 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 $ℂ\left[S_n\right]$ 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 $$\begin{array}{cc}{Y}_i\left(1\right)={\theta }_i·1,\hspace{0.17em}1=\text{id}\in S_n,\hspace{0.17em}1\le i\le n\text{.}& \text{(24)}\end{array}$$ 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 $ℂ\left[S_n\right]\ni z\to z·S_w\left(\mathrm{\Theta }\right)\in ℂ\left[S_n\right]$ gives an isomorphism $M\left(w\left(\mathrm{\Theta }\right)\right)\to M\left(\mathrm{\Theta }\right),$ which appears to be a ${\mathscr{H}}_n^{\prime }\text{-isomorphism.}$

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 $ℂ\left[S_n\right]$ is equal to $⟨{\mathrm{\Sigma }}_w⟩$ (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 $ℂ\left[S_n\right]$ 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 ${\mathscr{H}}_n^{\prime }$ 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 $𝒴=ℂ[Y_1,\cdots ,Y_n]$ is a maximal commutative in ${\mathscr{H}}_n^{\prime }$ i.e. the commutant (centralizer) of $𝒴$ coincides with $𝒴\text{.}$
(b)	The centre $𝒞$ (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 $𝒞{\mathscr{H}}_{n-m}^{\prime }$ 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 ${\mathscr{H}}_n^{\prime }\left(\eta \right)$ (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 $𝒴=ℂ[Y_1,\cdots ,Y_n]$ 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 $\{y_w,w\in {ℂ}_n\}$ 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 $x\left(\eta \right)\in 𝒴$ 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 $𝒴$ 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.

page history