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.

Towards the CFT

There are several possibilities to generalize and extend the above constructions. I shall try to outline only some of them connected with the two-dimensional conformal field theory.

First of all, one can substitute everywhere the $q\text{-analog}$ of Yang's $S\text{.}$ It is written as $$\begin{array}{cc}{S}_i^q\left(\theta \right)=T_i+(q-q^{-1})/(q^{2\theta }-1),1\le i\le n& \text{(37)}\end{array}$$ where $q\in ℂ,$ $\left\{T_i\right\}$ are the generators of the Hecke algebra $H_n^q\text{.}$ They satisfy the following defining relations $$\begin{array}{cc}(T_i-q)(T_i+q^{-1})=0,T_i{T}_{i+1}{T}_i=T_{i+1}{T}_i{T}_{i+1},[T_i,T_j]=0& \text{(38)}\end{array}$$ for $|i-j|\ge 2,$ $1\le i,j\le n\text{.}$ For $q=1$ we arrive at $ℂ\left[S_n\right]\text{.}$

The function $S^q$ was found independently in (one-dimensional) mathematical physics as some solution of (16) and in the theory of representations of $p\text{-adic}$ affine Hecke algebras as an interwiner (see [Che1987] for some details). The latter are defined as ${\mathscr{H}}_n^{\prime }$ but with the term $(q^2-1)Y_{i+1}$ in place of $1$ in formula (19a), where one should substitute $T_i$ for $s_i\text{.}$ In $p\text{-adic}$ papers $q=p^m$ for a prime $p,m\in \mathbb{N}\text{.}$ This way of definition is due to Bernstein, Zelevinsky. In many works $\left\{T_i\right\}$ are considered in some natural representation of $H_n^q$ in ${\left({ℂ}^N\right)}^{\otimes n}$ (Wenzl, Baxter).

Since $T^2\ne 1$ we have two elements $T,T^{-1}$ being on equal grounds. We omit the arguments, but it results in two possible pictures for two-particle $S\text{-matrices}$ instead of the only one above. We can consider intersecting as a passage of a particle over or under the other.

The $S^q$ from (36) is unitary after a proper normalization. But in other non-unitary theories this note can be important.

We have assumed the two-particle intersections to be the only elementary processes. But one can disagree with this assumption. Look at fig. 3a. There is a certain process between the intersections $({\theta }_1,{\theta }_2)$ and $({\theta }_1,{\theta }_2)$ of the corresponding particles. The particle with the angle ${\theta }_1$ should move away from the particle with ${\theta }_2$ after the first intersection and approach particle ${\theta }_3\text{.}$ This transference may be quantum as well. (In fact, any movement can be quantum in some general theory).

Let us consider the arranged symbols ${\hat{A}}_J\left(\mathrm{\Theta }\right),$ which are $A_J\left(\mathrm{\Theta }\right)$ from sec. 1 with some complete set of brackets between some $A_{jk}\left({\theta }_{jk}\right)\text{.}$ For example, $\left(A_{j_1}\left({\theta }_1\right)A_{j_2}\left({\theta }_2\right)\right)A_{j_3}\left({\theta }_3\right),$ $\left(A_{j_1}\left({\theta }_1\right)A_{j_2}\left({\theta }_2\right)\right)\left(A_{j_3}\left({\theta }_3\right)A_{j_4}\left({\theta }_4\right)\right)$ are complete but $\left(A_1{A}_2\right)\left(A_3{A}_4\right)A_5$ is not. The correct arranged symbol should be either $\left(\left(A_1{A}_2\right)\left(A_3{A}_4\right)\right)A_5$ or $\left(A_1{A}_2\right)\left(\left(A_3{A}_4\right)A_5\right)\text{.}$ Here we have omitted $j,\theta \text{.}$ Physically, the last symbol can be interpreted as follows: the particles $A_3,A_4$ are very close one to another, $A_5$ is close to $A_3$ or $A_4$ (it is all the same, since $A_3$ and $A_4$ are very close, more close than $A_5$ to each of them), $A_1$ is close to $A_2,$ the pair $A_1,A_2$ is not close to the triple $A_3,A_4,A_5\text{.}$ In fact, we have the ordered sequence of relations "not close, close, very close, very very close and so on" on the set of $A_J\left(\mathrm{\Theta }\right)\text{.}$

Formally speaking, a system of brackets is not complete if $\hat{A}$ contains a segment of type $\left({\hat{A}}^1\right)\left({\hat{A}}^2\right)\left({\hat{A}}^3\right)$ for some arranged symbols ${\hat{A}}^1,{\hat{A}}^2,{\hat{A}}^3\text{.}$ In this case ${\hat{A}}^1$ and ${\hat{A}}^3$ are at the same level (close, very close, ...) with respect to ${\hat{A}}^2\text{.}$ We forbid it, that resembles very much the Pauli principle in quantum mechanics.

Given some arranged ${\hat{A}}_J\left(\mathrm{\Theta }\right)$ for any $\mathrm{\Theta }=({\theta }_1,\cdots ,{\theta }_n),$ we can define to following two elementary operations. One can interchange two adjacent terms $A_{j_k}\left({\theta }_k\right)$ and $A_{j_{k+1}}\left({\theta }_{k+1}\right),$ but only if they are in brackets. The corresponding quantum $S_k$ will be introduced like in sec. 1-3. Another operation (written ${\mathrm{\Phi }}_k\text{)}$ is the passage $$\begin{array}{cc}{\hat{A}}_J=\cdots \left({\hat{A}}^1\left(A_{j_k}\left({\theta }_k\right){\hat{A}}^2\right)\right)\cdots \to \cdots \left(\left({\hat{A}}^1{A}_{j_k}\left({\theta }_k\right)\right){\hat{A}}^2\right)\cdots & \text{(39)}\end{array}$$ for some arranged ${\hat{A}}^1,{\hat{A}}^2$ or the analogous transformation from $\left({\hat{A}}^1{A}_k\right){\hat{A}}^2$ to ${\hat{A}}^1\left(A_k{\hat{A}}^2\right)\text{.}$ Given $k$ the corresponding ${\hat{A}}^1,{\hat{A}}^2$ (if any) can be found uniquely. For example, $$A_{i_1}\left({\theta }_1\right)\left(A_{i_2}\left({\theta }_2\right)A_{i_3}\left({\theta }_3\right)\right)={\mathrm{\Sigma }}_J{\mathrm{\Phi }}_I^J({\theta }_1,{\theta }_2,{\theta }_3)\left(A_{j_1}\left({\theta }_1\right)A_{j_2}\left({\theta }_2\right)\right)A_{j_3}\left({\theta }_3\right),$$ where ${\mathrm{\Phi }}_{i_1{i}_2{i}_3}^{j_1{j}_2{j}_3}\left(\mathrm{\Theta }\right)$ are the amplitudes from the in-state, where $A_2$ is more close to $A_3$ than to $A_1,$ to the out-state, where $A_2$ is more close to $A_1\text{;}$ $\mathrm{\Phi }=\left({\mathrm{\Phi }}_I^J\right)\text{.}$

In a contrast with sec. 1 these two operations (processes) exist only for some $k\text{.}$ E.g. let us consider $$\hat{A}=\left(\left(A_1{A}_2\right)\left(A_3{A}_4\right)\right)\left(A_5\left(A_6\left(A_7{A}_8\right)\right)\right)\text{.}$$ One can apply only the operations $S_1,S_3,S_7,{\mathrm{\Phi }}_2,{\mathrm{\Phi }}_3,{\mathrm{\Phi }}_5,{\mathrm{\Phi }}_6,{\mathrm{\Phi }}_7$ to this $\hat{A}\text{.}$ It is quite natural to postulate the identities $$[{\mathrm{\Phi }}_i,{\mathrm{\Phi }}_j]=[S_i,S_j]=[S_i,{\mathrm{\Phi }}_k]=0,\hspace{0.17em}|i-j|>1,\hspace{0.17em}k\ne i,i+1\text{.}$$ The reason is that ${\mathrm{\Phi }}_i{\mathrm{\Phi }}_j$ and ${\mathrm{\Phi }}_j{\mathrm{\Phi }}_i$ induce for $|i-j|>1$ the same changes of brackets. It holds true for the permutations and changes of brackets in the case $[S,S]$ or $[S,\mathrm{\Phi }]$ as well. The other relations are of the following type (see fig. 3). We begin with ${\hat{A}}_I=A_1\left(A_2{A}_3\right)$ and use here the abbreviations ${\theta }_{12}=({\theta }_1,{\theta }_2),$ ${\theta }_{123}=({\theta }_1,{\theta }_2,{\theta }_3)$ and so on. One has $$\begin{array}{ccc}& & S_2\left({\theta }_{12}\right){\mathrm{\Phi }}_2\left({\theta }_{312}\right)S_1\left({\theta }_{13}\right){\mathrm{\Phi }}_2\left({\theta }_{132}\right)S_2\left({\theta }_{23}\right){\mathrm{\Phi }}_2\left({\theta }_{123}\right)\\ & =& {\mathrm{\Phi }}_2\left({\theta }_{321}\right)S_2\left({\theta }_{23}\right){\mathrm{\Phi }}_2\left({\theta }_{231}\right)S_2\left({\theta }_{13}\right){\mathrm{\Phi }}_2\left({\theta }_{213}\right)S_2\left({\theta }_{12}\right)\text{.}\end{array}$$ This equality is quite analogous to identity (2.6) from [MSe1989] (see also [Dri1989-2]). It is small wonder since our symbolic language and appropriate pictures are very close to these of [MSe1989,Dri1989-2].

Here we assume that $S,$ $\mathrm{\Phi }$ depend on the corresponding parameters in the natural order. In general, $\mathrm{\Phi }$ can depend on many indices and parameters. E.g. ${\mathrm{\Phi }}_6$ for (39) may have 4 matrix indices and be a function of ${\theta }_{5\hspace{0.17em}6\hspace{0.17em}7\hspace{0.17em}8}=({\theta }_5,{\theta }_6,{\theta }_7,{\theta }_8)\text{.}$ In some sense the order of $A_7$ and $A_8$ is not important for ${\mathrm{\Phi }}_6$ since they both are at the same level with respect to $A_6\text{.}$ In particular, the dependence of ${\mathrm{\Phi }}_6$ on the indices of $A_7,A_8$ should be symmetric. The development of this point can give some version of the axiom system from [MSe1989], where any $\mathrm{\Phi }$ are defined by means of the comultiplication in terms of the least possible $\mathrm{\Phi }$ (with 3 matrix indices). The penthagone relation arises in this way. We note that our angles are, in fact, parallel to the conformal dimensions (see e.g. [Dri1989-2]).

I'd like to give another example of connections between the two-dimensional conformal theory and the affine Hecke algebras. The so-called Knizhnik-Zamolodchikov equation for the n-point function of the Wess-Zumino-Witten model can be written in terms of $ℂ\left[S_n\right]$ only. It has the following natural "affine" generalization $$\begin{array}{cc}\kappa dG/d{z}_i=(\sum _j\left(ij\right){(z_i-z_j)}^{-1}+x_i{z}_i^{-1})G,& \text{(40)}\end{array}$$ where $1\le i\ne j\le n,$ $G(z_1,\cdots ,z_n)$ takes its values in the algebra $𝒜$ generated by $ℂ\left[S_n\right]$ and some operators $\left\{x_i\right\}$ with the relations $w{\kappa }_i{w}^{-1}=x_{w\left(i\right)},$ $w\in S_n\text{.}$ Here $\left(ij\right)$ are the usual transposition, $\kappa \in ℂ\text{.}$ It is easy to show (see [Che1991]), that the cross-derivative integrability conditions for (40) are equivalent to the relations $$\begin{array}{cc}[x_i,x_j+\left(ij\right)]=0=[\left(ij\right),x_i+x_j]\text{.}& \text{(41)}\end{array}$$ The latter (together with the conditions $w{x}_i{w}^{-1}=x_{w\left(i\right)}\text{)}$ coincide with the defining relations for $Y_i$ (sec. 5) if $$Y_i=-x_i-\sum _{n\ge j>i}\left(ij\right),\hspace{0.17em}1\le i\le n\text{.}$$ Hence, this $𝒜$ should be 6ome quotient of the degenerated affine Hecke algebra ${\mathscr{H}}_n^{\prime }\text{.}$

To get the usual Knizhnik-Zamolodchikov equation one should put $x_i=0$ for any $i\text{.}$ It gives us the so-called Murphy surjection $Y_i\to -\sum _{n\ge j>i}\left(ij\right)$ of ${\mathscr{H}}_n^{\prime }$ onto $ℂ\left[S_n\right]$ (see [Dri1986]). This homorphism of algebras is important in the theory of $S_n\text{.}$ For example, the centre of $ℂ\left[S_n\right]$ is generated by symmetric polynomials in the images of $\left\{Y_i\right\}$ (cf. theorem 3 and [Che1987]).

To finish this part of my notes I will describe without going into detail some quantum counterpart of (41).

Let us consider the space $ℝ$ with the glass at the point $x=0\text{.}$ It is transparent for particles from sec. 1, but passing through this glass is assumed to be quantum. One can connect with this process two one-index matrices $X(+\theta ),$ $\hat{X}(-\theta )$ respectively for $\theta <0$ and $\theta >0$ (see fig. 9). E.g. for $\theta <0$ $$A_i{\left(\theta \right)}_{\text{in}}\stackrel{\text{glass}}{=}\sum _j{X}_i^j\left(\theta \right)A_j{\left(\theta \right)}_{\text{out}}$$ The factorization relations are close to (6). We will write them down in term $s$ of $R=PS\text{:}$ $$\begin{array}{ccc}{}^{1}X\left({\theta }_1\right){}^{12}R\left({\theta }_{12}\right){}^2\stackrel{}{ˆ}(-{\theta }_2)& =& {}^2\stackrel{}{ˆ}(-{\theta }_2){}^{12}R\left({\theta }_{12}\right){}^{1}X\left({\theta }_1\right),\\ {}^{12}R\left({\theta }_{12}\right){}^{1}X\left({\theta }_1\right){}^{2}X\left({\theta }_2\right)& =& {}^{2}X\left({\theta }_2\right){}^{1}X\left({\theta }_1\right){}^{12}R\left({\theta }_{12}\right),\\ {}^{12}R\left({\theta }_{12}\right){}^2\stackrel{}{ˆ}(-{\theta }_2){}^1\stackrel{}{ˆ}(-{\theta }_1)& =& {}^1\stackrel{}{ˆ}(-{\theta }_1){}^2\stackrel{}{ˆ}(-{\theta }_2){}^{12}R\left({\theta }_{12}\right)\text{.}& \text{(42)}\end{array}$$ Of course, $R$ should be a solution of (8) as well.

These equalities hold true (follow from (8)) if one formally substitute $X={}^{10}R\left({\theta }_{10}\right),$ $\hat{X}={}^{01}R\left({\theta }_{01}\right),$ ${\theta }_0=0,$ where $0$ is some other tensor index. Really, the transmission through the glass can be interpreted as intersecting with the particle of angle ${\theta }_0=0$ and colour $=0,$ where the latter does not change its colour in any quantum interactions.

The natural problem is to combine $S,\mathrm{\Phi },$ mirrors (not more than 2), polarizations (any number) and glasses (any number) in one picture. Then to consider more complicated spaces (circumferences, elliptic curves) and find interesting examples. Only some fragments of this heavy construction are clear (see [Che1984,Che1991,MSe1989]).

Let $R=R_{\eta }=\theta {(\theta +\eta )}^{-1}+\eta {(\theta +\eta )}^{-1}P$ (see sec. 5), $X=\hat{X}\text{.}$ We can consider (42) over $ℂ\left[S_n\right]$ in a natural manner. One identifies permutations with the corresponding matrices and supposes ${}^{i}X$ to be some undeterminate functions with the following action of $S_n:w{}^{i}X{w}^{-1}={}^{w\left(i\right)}X\text{.}$ We have $R_i={}^{i\hspace{0.17em}i+1}R=1+\eta s_i/{\theta }_i+\theta \left(\eta \right)$ as $\eta \to 0\text{.}$ Let us impose the analogical restrictions $X_i=1+\eta x_i/\theta +o\left(\eta \right)$ on $X_i={}^{i}X$ (in particular, $x_{w\left(i\right)}=w{x}_i{w}^{-1}\text{).}$ Then (42) results in (41).

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