Notes on Affine Hecke Algebras I. (Degenerated Hecke Algebras and Yangians in Mathematical Physics)

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-Baxter identities

Let us describe a sequence of $n$ particles at the moment $t = t_{0}$ with the $x -coordinates$ $x_{1} < x_{2} < \dots < x_{n}$ (see fig. 2) by the symbol $A_{J} (Θ) = A_{j_{1}} (θ_{1}) \dots A_{j_{n}} (θ_{n}),$ where $Θ = (θ_{1}, \dots, θ_{n})$ and $J = (j_{1}, \dots, j_{n})$ are the corresponding sets of the angles and the colours $(- π / 2 < θ_{k} < π / 2, 1 \leq j_{k} \leq N) .$ Owing to property (c) the previous or subsequent changes of a set of particles depend only on its symbol $A_{J} (Θ)$ (on the colours, angles and on the order of $x -coordinates$ only). Given $A_{I} (Θ')$ one has $\begin{matrix} A_{I} {(Θ')}_{in} = \sum_{J} S_{I}^{J} (Θ, Θ') A_{J} {(Θ)}_{out} & (1) \end{matrix}$ for some $I = (i_{1}, \dots, i_{n}),$ $Θ' = (θ_{1}^{'}, \dots, θ_{n}^{'})$ describing the set of particles $t = t_{0}^{'} < t_{0} .$ We have expressed $A_{I} (Θ')$ considered as an in-state in terms out-states. Every (scalar) coefficient $S_{I}^{J}$ is the $S -matrix$ element (the amplitude) from $A_{I} (Θ')$ to $A_{J} (Θ)$ (by definition). Here $I$ and $J$ can be arbitrary $(1 \leq i_{k}, j_{k} \leq N)$ but not the set $Θ' .$ The $S -matrix$ is nontrivial only if $\begin{matrix} Θ' = w (Θ) = (θ_{w^{- 1} (1)}, θ_{w^{- 1} (2)}, \dots θ_{w^{- 1} (n)}) & (2) \end{matrix}$ for an appropriate permutation $w$ from the symmeric group $S_{n} .$

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

Let us discuss examples. We will omit the indications "in" and "out". One has for $n = 2$ and $θ_{1} < θ_{2},$ $θ_{1}^{'} = θ_{2},$ $θ_{2}^{'} = θ_{1} :$ $\begin{matrix} A_{i_{1}} (θ_{2}) A_{i_{2}} (θ_{1}) = \sum_{i_{1}, i_{2}} S_{i_{1} i_{2}}^{j_{1} j_{2}} (θ_{12}) A_{j_{1}} (θ_{1}) A_{j_{2}} (θ_{2}), & (3) \end{matrix}$ where $θ_{i j} = θ_{i} - θ_{j}$ by definition. Given $I = (i_{1}, i_{2}, i_{3}),$ $J = (j_{1}, j_{2}, j_{3})$ in the case of fig. 3a we obtain the following relations: $\begin{matrix} A_{i_{1}} (θ_{3}) A_{i_{2}} (θ_{2}) A_{i_{3}} (θ_{1}) & = & \sum S_{i_{2} i_{3}}^{k_{2} k_{3}} (θ_{12}) A_{i_{1}} (θ_{3}) A_{k_{2}} (θ_{1}) A_{k_{3}} (θ_{2}) \\ = & \sum S_{i_{2} i_{3}}^{k_{2} k_{3}} (θ_{12}) S_{i_{1} k_{2}}^{j_{1} ℓ_{2}} (θ_{13}) A_{j_{1}} (θ_{1}) A_{ℓ_{2}} (θ_{3}) A_{k_{3}} (θ_{2}) \\ = & \sum S_{i_{2} i_{3}}^{k_{2} k_{3}} (θ_{12}) S_{i_{1} k_{2}}^{j_{1} l_{2}} (θ_{13}) S_{l_{2} k_{3}}^{j_{2} j_{3}} (θ_{23}) A_{J} (Θ), \end{matrix}$ where the sum is over all free indices $(j_{1}, j_{2}, j_{3}, k_{2}, k_{3}, l_{2}) .$ The analogical calculation for fig. 3b should give the same result. We arrive at the identity: $\begin{matrix} \sum_{k_{2}, k_{3}, ℓ_{2}} S_{i_{2} i_{3}}^{k_{2} k_{3}} (θ_{12}) S_{i_{1} k_{2}}^{j_{1} l_{2}} (θ_{13}) S_{l_{2} k_{3}}^{j_{2} j_{3}} (θ_{23}) = \sum_{k_{1}, k_{2}, l_{2}} S_{i_{1} i_{2}}^{k_{1} k_{2}} (θ_{23}) S_{k_{2} i_{3}}^{l_{2} j_{3}} (θ_{13}) S_{k_{1} l_{2}}^{j_{1} j_{2}} (θ_{12}) & (4) \end{matrix}$

Let us rewrite (4) in a tensor form. We will keep the following notations. Let us consider "multi-matrices" $T = (T_{i_{1} i_{2} \dots i_{n}}^{j_{1} j_{2} \dots j_{n}})$ with the multi-indices $I = (i_{1}, \dots, i_{n}),$ $J = (j_{1}, \dots, j_{n}),$ respectively, of rows and columns $(1 \leq i_{k}, j_{k} \leq N$ for $1 \leq k \leq n).$ These $T$ act on "multi-vectors" $x = (x_{i_{1} i_{2} \dots i_{n}})$ by the natural formula $T x = (\sum_{J} T_{I}^{J} x_{J}) .$ 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 $ℂ^{N} .$ Given two $N \times N -matrices$ $X = (X_{i}^{j}),$ $Y = (Y_{i}^{j})$ (from the matrix algebra $M_{N})$ one can define the tensor product $T = X \otimes Y :$ $T = (T_{i_{1} i_{2}}^{j_{1} j_{2}}),$ $T_{i_{1} i_{2}}^{j_{1} j_{2}} = X_{i_{1}}^{j_{1}} Y_{i_{2}}^{j_{2}} .$ The definition of $X \otimes Y \otimes Z \otimes \dots$ is quite analogous.

Later on, $δ_{i}^{j}$ will be the Kronecker symbol. Put $\begin{matrix} ^{k} X & = & (\prod_{m \neq k} δ_{i_{m}}^{j_{m}}) X_{i_{k}}^{j_{k}}, \\ ^{k l} T & = & (\prod_{m \neq k, l} δ_{i_{m}}^{j_{m}}) T_{i_{k} i_{l}}^{j_{k} j_{l}} & (5) \end{matrix}$ for $X = (X_{i}^{j}),$ $T = (T_{i_{1} i_{2}}^{j_{1} j_{2}}),$ $1 \leq k \neq l \leq n,$ $1 \leq m \leq n .$ These matrices are the natural images of $X, T$ in the $M_{N}^{\otimesn} = M_{N^{n}}$ with respect to the indices $k$ and $(k, l) .$ Note that $^{k l} (X \otimes Y) =^{k} X^{l} Y$ and $^{k} X$ commutes with $^{l} Y$ for any $X, Y \in M_{N},$ $k \neq l .$

Let us introduce $S (θ)$ as the following matrix (depending on $θ = θ_{12})$ from $M_{N}^{\otimes 2} : S = (S_{i_{1} i_{2}}^{j_{1} j_{2}} (θ)) .$ Now one can represent (4) in the elegant form: $\begin{matrix} ^{23} S (θ_{12})^{12} S (θ_{13})^{23} S (θ_{23}) =^{12} S (θ_{23})^{23} S (θ_{13})^{12} S (θ_{12}) . & (6) \end{matrix}$ If we put $S (θ) = P R (θ)$ for $\begin{matrix} P = (P_{i_{1} i_{2}}^{j_{1} j_{2}}), P_{i_{1} i_{2}}^{j_{1} j_{2}} = δ_{i_{1}}^{j_{2}} δ_{i_{2}}^{j_{1}} & (7) \end{matrix}$ we get the Yang-Baxter equation $\begin{matrix} ^{12} R (θ_{12})^{13} R (θ_{13})^{23} R (θ_{23}) =^{23} R (θ_{23})^{13} R (θ_{13})^{12} R (θ_{12}) . & (8) \end{matrix}$ 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 $S = P R .$ Here we have to use the following properties of $P :$ $\begin{matrix} ^{23} P^{12} P^{23} P =^{12} P^{23} P^{12} P, & (9a) \\ ^{12} P^{1} X =^{2} X^{12} P for X \in M_{N} . & (9b) \end{matrix}$ It is easy either to prove (9) directly or verify them without matrix calculations using the following natural interpretation of $^{i j} P .$

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