(Degenerated Hecke Algebras and Yangians in Mathematical Physics)

Last update: 23 April 2014

This is an excerpt of the paper *Notes on Affine Hecke Algebras I. (Degenerated Hecke Algebras and Yangians in Mathematical Physics)* by Ivan Cherednik.

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 $$\begin{array}{cc}{S}_{i}({Y}_{i}-{Y}_{i+1})={Y}_{i+1}{S}_{i}({Y}_{i}-{Y}_{i+1}),& \text{(18a)}\\ S({Y}_{i}-{Y}_{i+1}){Y}_{i+1}={Y}_{i}S({Y}_{i}-{Y}_{i+1}),& \text{(18b)}\\ S({Y}_{i}-{Y}_{i+1}){Y}_{j}={Y}_{j}S({Y}_{i}-{Y}_{i+1}),\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}y\ne i,i+1& \text{(18c)}\end{array}$$ Formulas (18) are equivalent to the relations $$\begin{array}{cc}{Y}_{i+1}{s}_{i}-{s}_{i}{Y}_{i}=1={s}_{i}{Y}_{i+1}-{Y}_{i}{s}_{i}& \text{(19a)}\\ {Y}_{j}{s}_{i}={s}_{i}{Y}_{j}\phantom{\rule{1em}{0ex}}\text{for}\phantom{\rule{1em}{0ex}}y\ne i,i+1& \text{(19b)}\end{array}$$ 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 $\u2102\left[{S}_{n}\right]{=}_{w}\oplus \u2102w$
be the *group algebra* of ${S}_{n}\ni w$ with the natural multiplication law:
$w\xb7w\prime =ww\prime \in {S}_{n}$ (e.g.
$(a+\left(12\right))\xb7(b+\left(23\right))=ab+a\left(23\right)+b\left(12\right)+(3,1,2),$
$a,b\in \u2102\text{).}$ 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:
$${\mathrm{\Sigma}}_{i}=1+{s}_{i}({Y}_{i}-{Y}_{i+1}),\phantom{\rule{2em}{0ex}}1\le i<n$$
The main point is that (16) is equivalent to the following identity in ${\mathscr{H}}_{n}^{\prime}\text{:}$
$$\begin{array}{cc}{\mathrm{\Sigma}}_{i}{\mathrm{\Sigma}}_{i+1}{\mathrm{\Sigma}}_{i}={\mathrm{\Sigma}}_{i+1}{\mathrm{\Sigma}}_{i}{\mathrm{\Sigma}}_{i+1}\phantom{\rule{2em}{0ex}}(1\le i<n)\text{.}& \text{(21)}\end{array}$$
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 $\u27e8A\u27e9\text{.}$ The only thing we need is to show
that $\u27e8{\mathrm{\Sigma}}_{i}{\mathrm{\Sigma}}_{i+1}{\mathrm{\Sigma}}_{i}\u27e9$
and $\u27e8{\mathrm{\Sigma}}_{i+1}{\mathrm{\Sigma}}_{i}{\mathrm{\Sigma}}_{i+1}\u27e9$
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 $\u2102[{Y}_{1},\cdots ,{Y}_{n}]$ 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 ${\mathscr{H}}_{n}^{\prime}\text{.}$