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.

Yang's $S\text{-matrix}$

Let us discuss the basic example of a factorized $S\text{-matrix}$ (a solution of (6)). The following one is the so-called Yang's $S\text{-matrix:}$ $$\begin{array}{cc}S\left(\theta \right)=1+P\theta ,\theta ={\theta }_{12}={\theta }_1-{\theta }_2\text{.}& \text{(17)}\end{array}$$ Here and further we will denote by $1$ the unit matrix in $M_N,$ $1\otimes 1$ and so on. Setting $S{\left(\theta \right)}_{\eta }=1\eta {(\theta +\eta )}^{-1}+P\theta {(\theta +\eta )}^{-1}$ for any $\eta$ we get an unitary $S\text{-matrix,}$ satisfying (6) and (14). In particular, $S{\left(\theta \right)}_0=P$ corresponds to a world without any scattering. To verify (6) for $S$ (or $S_{\eta }\text{)}$ is an easy exercise. Nevertheless, one can wish to prove (6) without any calculations like it was made for $P$ (see above). The best way is to find an interpretation of $S$ as a transposition of something $\text{(}P$ interchanges the tensor components). I know four ways to do this. Two of them are based, respectively, on some algebraic geometry (see [Che1979]) and on the theory of the so-called Knizhnik-Zamolodchikov equation from the two-dimensional conformal field theory (see [KZa1984,Che1991]). I will explain here and below only other two making use of (degenerated) affine Hecke algebras and the so-called Yangians [Dri1986].

Mathematically, the idea is simple enough. Let us substitute some operators $Y_i$ for ${\theta }_i$ $(1\le i\le n)$ in $S_i\left(\mathrm{\Theta }\right)={}^{i\hspace{0.17em}i+1}S({\theta }_i-{\theta }_{i+1}),$ where $S$ is from (17). We assume $\left\{Y_i\right\}$ to be pairwise commutative and impose the following conditions $$\begin{array}{cc}{S}_i(Y_i-Y_{i+1})=Y_{i+1}{S}_i(Y_i-Y_{i+1}),& \text{(18a)}\\ S(Y_i-Y_{i+1})Y_{i+1}=Y_{i}S(Y_i-Y_{i+1}),& \text{(18b)}\\ S(Y_i-Y_{i+1})Y_j=Y_{j}S(Y_i-Y_{i+1}),\text{for}y\ne i,i+1& \text{(18c)}\end{array}$$ Formulas (18) are equivalent to the relations $$\begin{array}{cc}{Y}_{i+1}{s}_i-s_i{Y}_i=1=s_i{Y}_{i+1}-Y_i{s}_i& \text{(19a)}\\ Y_j{s}_i=s_i{Y}_j\text{for}y\ne i,i+1& \text{(19b)}\end{array}$$ Here and further we will identify $P_{s_i}={}^{i\hspace{0.17em}i+1}P$ with $s_i$ and $1$ with $1\text{.}$ Let us check e.g. the first of these formulas. It results from (18a) that $$\begin{array}{cc}{Y}_{i\hspace{0.17em}i+1}=(Y_{i+1}{s}_i-s_i{Y}_i)Y_{i\hspace{0.17em}i+1},Y_{ij}=Y_i-Y_j\text{.}& \text{(20)}\end{array}$$ Then one can divide (20) by $Y_{i\hspace{0.17em}i+1}\text{.}$ We see that (19) $\Rightarrow$ (18), but the converse holds true only for $\left\{Y_i\right\}$ in a "general position".

We have arrived at the following object. Let $ℂ\left[S_n\right]{=}_w\oplus ℂw$ be the group algebra of $S_n\ni w$ with the natural multiplication law: $w·w\prime =ww\prime \in S_n$ (e.g. $(a+\left(12\right))·(b+\left(23\right))=ab+a\left(23\right)+b\left(12\right)+(3,1,2),$ $a,b\in ℂ\text{).}$ It is nothing else but the algebra of formal linear combinations of permutations. Its extension by pairwise commutative symbols $Y_1,\cdots ,Y_n$ with relations (19) is called the degenerated affine Hecke algebra (written ${\mathscr{H}}_n^{\prime }\text{).}$ It is due to Murphy and Drinfeld (see [Dri1986]). Starting with $S_{\eta }$ instead of $S$ we get ${\mathscr{H}}_n\left(\eta \right),$ where $\eta$ should stay for $1$ in (19a). The algebra ${\mathscr{H}}_n^{\prime }\left(\eta \right)$ isomorphic to ${\mathscr{H}}_n^{\prime }$ for $\eta \ne 0$ (use the substitution $\{Y_i\to \eta Y_i\}\text{.}$ But this $\eta$ is important to understand ${\mathscr{H}}_n^{\prime }$ as some quantum object.

To differentiate $S_i\left(\mathrm{\Theta }\right)$ from ${}^{i\hspace{0.17em}i+1}S\left(Y_{i\hspace{0.17em}i+1}\right)$ (see (13b)) let us denote the latter by: $${\mathrm{\Sigma }}_i=1+s_i(Y_i-Y_{i+1}),1\le i<n$$ The main point is that (16) is equivalent to the following identity in ${\mathscr{H}}_n^{\prime }\text{:}$ $$\begin{array}{cc}{\mathrm{\Sigma }}_i{\mathrm{\Sigma }}_{i+1}{\mathrm{\Sigma }}_i={\mathrm{\Sigma }}_{i+1}{\mathrm{\Sigma }}_i{\mathrm{\Sigma }}_{i+1}(1\le i<n)\text{.}& \text{(21)}\end{array}$$ To prove the equivalence we need some kind of Wick (or Poincaré-Birckhof-Witt) theorem for ${\mathscr{H}}_n^{\prime }\text{.}$ One can deduce directly from (19) that each element $A\in {\mathscr{H}}_n^{\prime }$ has the unique representation of the following type: $A=\sum _{w}w{y}_w,$ where $w\in S_n,$ $y_w$ are some polynomials in $Y_1,\cdots ,Y_n\text{.}$ Let us denote this sum for $A$ after the converse substitution $Y_i\to {\theta }_i$ by $⟨A⟩\text{.}$ The only thing we need is to show that $⟨{\mathrm{\Sigma }}_i{\mathrm{\Sigma }}_{i+1}{\mathrm{\Sigma }}_i⟩$ and $⟨{\mathrm{\Sigma }}_{i+1}{\mathrm{\Sigma }}_i{\mathrm{\Sigma }}_{i+1}⟩$ coincide, respectively, with the l.h.s and r.h.s of (16). Let us carry all the $Y_i,Y_{i+1},Y_{i+2}$ in (21) over ${\mathrm{\Sigma }}_i,{\mathrm{\Sigma }}_{i+1}$ by means of (18) from the left to the right. Then one obtains $Y_i-Y_{i+1},$ $Y_i-Y_{i+2}$ and $Y_{i+1}-Y_{i+2}$ instead of $Y_{i\hspace{0.17em}i+1},Y_{i+1\hspace{0.17em}i+2},Y_{i\hspace{0.17em}i+1}$ in the l.h.s of (21) and the same elements but in the opposite order in the r.h.s. These differences are exactly what we need. By the way, ${\mathrm{\Sigma }}_{\eta }$ in the natural notations is involutive in ${\mathscr{H}}_n^{\prime }\left(\eta \right),$ i.e. $\left({\mathrm{\Sigma }}_{\eta }\right)\left({\mathrm{\Sigma }}_{\eta }\right)=1$ (cf. (14)). The next theorem (see [Che1987], proposition 3.1 and [Rog1985]) results directly from (21) and (18).

Theorem 1. The collection of elements $\left\{{\mathrm{\Sigma }}_i\right\}$ from ${\mathscr{H}}_n^{\prime }$ extends uniquely to the set $\{{\mathrm{\Sigma }}_w,w\in S_n\}\subset {\mathscr{H}}_n^{\prime }$ with the following properties:

(a)	${\mathrm{\Sigma }}_x{\mathrm{\Sigma }}_y={\mathrm{\Sigma }}_{xy}$ if $\ell \left(xy\right)=\ell \left(x\right)+\ell \left(y\right),$ ${\mathrm{\Sigma }}_{\text{id}}=1$
(b)	${\mathrm{\Sigma }}_w{Y}_i{\mathrm{\Sigma }}_w^{-1}=Y_{w\left(i\right)},$ $w\in S_n,$ $1\le i\le n\text{.}$

Here (a) is in fact (15). This property can be deduced from (b) (or (18)). Formulas (21), (6) are particular cases of this property. Therefore, the Yang-Baxter relation for Yang's $S$ is a direct consequence of the definition of ${\mathscr{H}}_n^{\prime }\text{.}$ Let us discuss this point.

We see that the l.h.s and r.h.s of (a) induce (operating by conjugations) the same permutation of $\left\{Y_i\right\}\text{.}$ Hence, the product ${\mathrm{\Sigma }}_x{\mathrm{\Sigma }}_y$ should be equal to ${\mathrm{\Sigma }}_{xy}$ modulo multiplications by some elements from the centralizer (commutant) of $\{Y_1,\cdots ,Y_n\}$ in ${\mathscr{H}}_n^{\prime }\text{.}$ It is not difficult to prove that this centralizer coincides with the algebra $ℂ[Y_1,\cdots ,Y_n]$ of polynomials of $\left\{Y_i\right\}$ (see theorem 3). In particular, (6) for Yang's $S$ has to be true up to a multiplication by a scalar function in ${\theta }_1,{\theta }_2,{\theta }_3$ (use the Wick theorem for ${\mathscr{H}}_n^{\prime }\text{).}$ Then it is easy to get (6) from this weaker statement. Thus, we have verified, in principle, the Yang-Baxter identity without any calculations. Only by means of formula (b), which is the definition of ${\mathscr{H}}_n^{\prime }\text{.}$

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