## 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.

## 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 $S\text{-matrix:}$ $S(θ)=1+ Pθ,θ=θ12 =θ1-θ2. (17)$ Here and further we will denote by $1$ the unit matrix in ${M}_{N},$ $1\otimes 1$ and so on. Setting $S{\left(\theta \right)}_{\eta }=1\eta {\left(\theta +\eta \right)}^{-1}+P\theta {\left(\theta +\eta \right)}^{-1}$ 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 ${\theta }_{i}$ $\left(1\le i\le n\right)$ in ${S}_{i}\left(\mathrm{\Theta }\right)={}^{i i+1}S\left({\theta }_{i}-{\theta }_{i+1}\right),$ where $S$ is from (17). We assume $\left\{{Y}_{i}\right\}$ to be pairwise commutative and impose the following conditions $Si(Yi-Yi+1) =Yi+1Si (Yi-Yi+1), (18a) S(Yi-Yi+1) Yi+1=YiS (Yi-Yi+1), (18b) S(Yi-Yi+1) Yj=YjS (Yi-Yi+1), fory≠i,i+1 (18c)$ Formulas (18) are equivalent to the relations $Yi+1si-si Yi=1=si Yi+1-Yisi (19a) Yjsi=siYj fory≠i,i+1 (19b)$ Here and further we will identify ${P}_{{s}_{i}}={}^{i i+1}P$ with ${s}_{i}$ and $1$ with $1\text{.}$ Let us check e.g. the first of these formulas. It results from (18a) that $Yi i+1= (Yi+1si-siYi) Yi i+1,Yij =Yi-Yj. (20)$ Then one can divide (20) by ${Y}_{i i+1}\text{.}$ We see that (19) $⇒$ (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. $\left(a+\left(12\right)\right)·\left(b+\left(23\right)\right)=ab+a\left(23\right)+b\left(12\right)+\left(3,1,2\right),$ $a,b\in ℂ\text{).}$ 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 ${ℋ}_{n}^{\prime }\text{).}$ It is due to Murphy and Drinfeld (see [Dri1986]). Starting with ${S}_{\eta }$ instead of $S$ we get ${ℋ}_{n}\left(\eta \right),$ where $\eta$ should stay for $1$ in (19a). The algebra ${ℋ}_{n}^{\prime }\left(\eta \right)$ isomorphic to ${ℋ}_{n}^{\prime }$ for $\eta \ne 0$ (use the substitution $\left\{{Y}_{i}\to \eta {Y}_{i}\right\}\text{.}$ But this $\eta$ is important to understand ${ℋ}_{n}^{\prime }$ as some quantum object.

To differentiate ${S}_{i}\left(\mathrm{\Theta }\right)$ from ${}^{i i+1}S\left({Y}_{i i+1}\right)$ (see (13b)) let us denote the latter by: $Σi=1+si (Yi-Yi+1), 1≤i The main point is that (16) is equivalent to the following identity in ${ℋ}_{n}^{\prime }\text{:}$ $ΣiΣi+1 Σi=Σi+1 ΣiΣi+1 (1≤i To prove the equivalence we need some kind of Wick (or Poincaré-Birckhof-Witt) theorem for ${ℋ}_{n}^{\prime }\text{.}$ One can deduce directly from (19) that each element $A\in {ℋ}_{n}^{\prime }$ 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 ${Y}_{i i+1},{Y}_{i+1 i+2},{Y}_{i i+1}$ 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 ${ℋ}_{n}^{\prime }\left(\eta \right),$ 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 ${ℋ}_{n}^{\prime }$ extends uniquely to the set $\left\{{\mathrm{\Sigma }}_{w},w\in {S}_{n}\right\}\subset {ℋ}_{n}^{\prime }$ 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 ${ℋ}_{n}^{\prime }\text{.}$ 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 $\left\{{Y}_{1},\cdots ,{Y}_{n}\right\}$ in ${ℋ}_{n}^{\prime }\text{.}$ It is not difficult to prove that this centralizer coincides with the algebra $ℂ\left[{Y}_{1},\cdots ,{Y}_{n}\right]$ 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 ${ℋ}_{n}^{\prime }\text{).}$ 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 ${ℋ}_{n}^{\prime }\text{.}$