## 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-Baxter identities

Let us describe a sequence of $n$ particles at the moment $t={t}_{0}$ with the $x\text{-coordinates}$ ${x}_{1}<{x}_{2}<\dots <{x}_{n}$ (see fig. 2) by the symbol ${A}_{J}\left(\mathrm{\Theta }\right)={A}_{{j}_{1}}\left({\theta }_{1}\right)\cdots {A}_{{j}_{n}}\left({\theta }_{n}\right),$ where $\mathrm{\Theta }=\left({\theta }_{1},\dots ,{\theta }_{n}\right)$ and $J=\left({j}_{1},\dots ,{j}_{n}\right)$ are the corresponding sets of the angles and the colours $\left(-\pi /2<{\theta }_{k}<\pi /2,1\le {j}_{k}\le N\right)\text{.}$ Owing to property (c) the previous or subsequent changes of a set of particles depend only on its symbol ${A}_{J}\left(\mathrm{\Theta }\right)$ (on the colours, angles and on the order of $x\text{-coordinates}$ only). Given ${A}_{I}\left(\mathrm{\Theta }\prime \right)$ one has $AI(Θ′)in= ∑JSIJ(Θ,Θ′) AJ(Θ)out (1)$ for some $I=\left({i}_{1},\cdots ,{i}_{n}\right),$ $\mathrm{\Theta }\prime =\left({\theta }_{1}^{\prime },\cdots ,{\theta }_{n}^{\prime }\right)$ describing the set of particles $t={t}_{0}^{\prime }<{t}_{0}\text{.}$ We have expressed ${A}_{I}\left(\mathrm{\Theta }\prime \right)$ considered as an in-state in terms out-states. Every (scalar) coefficient ${S}_{I}^{J}$ is the $S\text{-matrix}$ element (the amplitude) from ${A}_{I}\left(\mathrm{\Theta }\prime \right)$ to ${A}_{J}\left(\mathrm{\Theta }\right)$ (by definition). Here $I$ and $J$ can be arbitrary $\left(1\le {i}_{k},{j}_{k}\le N\right)$ but not the set $\mathrm{\Theta }\prime \text{.}$ The $S\text{-matrix}$ is nontrivial only if $Θ′=w(Θ)= ( θw-1(1), θw-1(2),⋯ θw-1(n) ) (2)$ for an appropriate permutation $w$ from the symmeric group ${S}_{n}\text{.}$

In this formula some misunderstanding is possible. I will comment on it. Any permutation $w:\left(1,2,\dots ,n\right)\to \left(1\prime ,2\prime ,\cdots ,n\prime \right)$ acts on an ordered set $s=\left(x,y,z,\cdots ,\right)$ of some elements (e.g. coordinates) by the substitution the element at place No. $i$ for the element at place No. $i\prime \text{.}$ For example, the transposition $w=\left(12\right):\left(1,2,3,\cdots ,n\right)\to 21$ interchanges the content of the first and second places. This definition results in the natural formula $v\left(w\left(s\right)\right)=\left(v·w\right)\left(s\right),$ $w,v\in {S}_{n}\text{.}$ We see that ${w}^{-1}$ is necessary in the second equality of (2). In fig. 4 the corresponding $w$ is equal to $\left(4,5,1,3,2\right)$ in the one-line notations or $\genfrac{}{}{0}{}{1 2 3 4 5}{4 5 1 3 2}$ in the two-line notation, i.e. takes $\left(1 2 3 4 5\right)$ to $\left(3 5 4 1 2\right)\text{;}$ ${w}^{-1}=\left(\genfrac{}{}{0}{}{1 2 3 4 5}{3 5 4 1 2}\right)=\left(3,5,4,1,2\right)\text{.}$

Let us discuss examples. We will omit the indications "in" and "out". One has for $n=2$ and ${\theta }_{1}<{\theta }_{2},$ ${\theta }_{1}^{\prime }={\theta }_{2},$ ${\theta }_{2}^{\prime }={\theta }_{1}\text{:}$ $Ai1(θ2) Ai2(θ1)= ∑i1,i2 Si1i2j1j2 (θ12)Aj1 (θ1)Aj2 (θ2), (3)$ where ${\theta }_{ij}={\theta }_{i}-{\theta }_{j}$ by definition. Given $I=\left({i}_{1},{i}_{2},{i}_{3}\right),$ $J=\left({j}_{1},{j}_{2},{j}_{3}\right)$ in the case of fig. 3a we obtain the following relations: $Ai1(θ3) Ai2(θ2) Ai3(θ1) = ∑Si2i3k2k3 (θ12)Ai1(θ3) Ak2(θ1)Ak3 (θ2) = ∑Si2i3k2k3 (θ12) Si1k2j1ℓ2 (θ13)Aj1 (θ1)Aℓ2 (θ3)Ak3 (θ2) = ∑Si2i3k2k3 (θ12) Si1k2j1l2 (θ13) Sl2k3j2j3 (θ23) AJ(Θ),$ where the sum is over all free indices $\left({j}_{1},{j}_{2},{j}_{3},{k}_{2},{k}_{3},{l}_{2}\right)\text{.}$ The analogical calculation for fig. 3b should give the same result. We arrive at the identity: $∑k2,k3,ℓ2 Si2i3k2k3(θ12) Si1k2j1l2(θ13) Sl2k3j2j3(θ23) =∑k1,k2,l2 Si1i2k1k2(θ23) Sk2i3l2j3(θ13) Sk1l2j1j2(θ12) (4)$

Let us rewrite (4) in a tensor form. We will keep the following notations. Let us consider "multi-matrices" $T=\left({T}_{{i}_{1} {i}_{2} \cdots {i}_{n}}^{{j}_{1} {j}_{2} \cdots {j}_{n}}\right)$ with the multi-indices $I=\left({i}_{1},\cdots ,{i}_{n}\right),$ $J=\left({j}_{1},\cdots ,{j}_{n}\right),$ respectively, of rows and columns $\text{(}1\le {i}_{k},{j}_{k}\le N$ for $1\le k\le n\text{).}$ These $T$ act on "multi-vectors" $x=\left({x}_{{i}_{1}{i}_{2}\cdots {i}_{n}}\right)$ by the natural formula $Tx=\left({\sum }_{J}{T}_{I}^{J}{x}_{J}\right)\text{.}$ If multi-indices are assumed to be (lexicographically) ordered then $x$ and $T$ are usual vectors and matrices for ${ℂ}^{{N}^{n}}$ in place of ${ℂ}^{N}\text{.}$ Given two $N×N\text{-matrices}$ $X=\left({X}_{i}^{j}\right),$ $Y=\left({Y}_{i}^{j}\right)$ (from the matrix algebra ${M}_{N}\text{)}$ one can define the tensor product $T=X\otimes Y:$ $T=\left({T}_{{i}_{1} {i}_{2}}^{{j}_{1} {j}_{2}}\right),$ ${T}_{{i}_{1} {i}_{2}}^{{j}_{1} {j}_{2}}={X}_{{i}_{1}}^{{j}_{1}}{Y}_{{i}_{2}}^{{j}_{2}}\text{.}$ The definition of $X\otimes Y\otimes Z\otimes \cdots$ is quite analogous.

Later on, ${\delta }_{i}^{j}$ will be the Kronecker symbol. Put $kX = (∏m≠kδimjm) Xikjk, klT = (∏m≠k,lδimjm) Tikiljkjl (5)$ for $X=\left({X}_{i}^{j}\right),$ $T=\left({T}_{{i}_{1}{i}_{2}}^{{j}_{1}{j}_{2}}\right),$ $1\le k\ne l\le n,$ $1\le m\le n\text{.}$ These matrices are the natural images of $X,T$ in the ${M}_{N}^{\text{⊗n}}={M}_{{N}^{n}}$ with respect to the indices $k$ and $\left(k,l\right)\text{.}$ Note that ${}^{kl}\left(X\otimes Y\right)={}^{k}{X}^{l}Y$ and ${}^{k}X$ commutes with ${}^{l}Y$ for any $X,Y\in {M}_{N},$ $k\ne l\text{.}$

Let us introduce $S\left(\theta \right)$ as the following matrix (depending on $\theta ={\theta }_{12}\text{)}$ from ${M}_{N}^{\otimes 2}:S=\left({S}_{{i}_{1} {i}_{2}}^{{j}_{1} {j}_{2}}\left(\theta \right)\right)\text{.}$ Now one can represent (4) in the elegant form: $23S(θ12) 12S(θ13) 23S(θ23)= 12S(θ23) 23S(θ13) 12S(θ12). (6)$ If we put $S\left(\theta \right)=PR\left(\theta \right)$ for $P=(Pi1 i2j1 j2), Pi1 i2j1 j2= δi1j2δi2j1 (7)$ we get the Yang-Baxter equation ${}^{12}R(θ12) {}^{13}R(θ13) {}^{23}R(θ23)= {}^{23}R(θ23) {}^{13}R(θ13) {}^{12}R(θ12). (8)$ To check it (to deduce it from (6)) one should carry all the ${}^{12}P,{}^{23}P$ across the other terms in (6) after the substitution $S=PR\text{.}$ Here we have to use the following properties of $P\text{:}$ $23P 12P 23P= 12P 23P 12P, (9a) 12P1X= 2X 12Pfor X∈MN. (9b)$ It is easy either to prove (9) directly or verify them without matrix calculations using the following natural interpretation of ${}^{ij}P\text{.}$