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

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=t0 with the x-coordinates x1<x2<<xn (see fig. 2) by the symbol AJ(Θ)=Aj1(θ1)Ajn(θn), where Θ=(θ1,,θn) and J=(j1,,jn) are the corresponding sets of the angles and the colours (-π/2<θk<π/2,1jkN). Owing to property (c) the previous or subsequent changes of a set of particles depend only on its symbol AJ(Θ) (on the colours, angles and on the order of x-coordinates only). Given AI(Θ) one has AI(Θ)in= JSIJ(Θ,Θ) AJ(Θ)out (1) for some I=(i1,,in), Θ=(θ1,,θn) describing the set of particles t=t0<t0. We have expressed AI(Θ) considered as an in-state in terms out-states. Every (scalar) coefficient SIJ is the S-matrix element (the amplitude) from AI(Θ) to AJ(Θ) (by definition). Here I and J can be arbitrary (1ik,jkN) but not the set Θ. The S-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 Sn.

In this formula some misunderstanding is possible. I will comment on it. Any permutation w:(1,2,,n)(1,2,,n) acts on an ordered set s=(x,y,z,,) 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,,n)21 interchanges the content of the first and second places. This definition results in the natural formula v(w(s))=(v·w)(s), w,vSn. 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 1234545132 in the two-line notation, i.e. takes (12345) to (35412); w-1=(1234535412)=(3,5,4,1,2).

Let us discuss examples. We will omit the indications "in" and "out". One has for n=2 and θ1<θ2, θ1=θ2, θ2=θ1: Ai1(θ2) Ai2(θ1)= i1,i2 Si1i2j1j2 (θ12)Aj1 (θ1)Aj2 (θ2), (3) where θij=θi-θj by definition. Given I=(i1,i2,i3), J=(j1,j2,j3) 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) Si1k2j12 (θ13)Aj1 (θ1)A2 (θ3)Ak3 (θ2) = Si2i3k2k3 (θ12) Si1k2j1l2 (θ13) Sl2k3j2j3 (θ23) AJ(Θ), where the sum is over all free indices (j1,j2,j3,k2,k3,l2). The analogical calculation for fig. 3b should give the same result. We arrive at the identity: k2,k3,2 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=(Ti1i2inj1j2jn) with the multi-indices I=(i1,,in), J=(j1,,jn), respectively, of rows and columns (1ik,jkN for 1kn). These T act on "multi-vectors" x=(xi1i2in) by the natural formula Tx=(JTIJxJ). If multi-indices are assumed to be (lexicographically) ordered then x and T are usual vectors and matrices for Nn in place of N. Given two N×N-matrices X=(Xij), Y=(Yij) (from the matrix algebra MN) one can define the tensor product T=XY: T=(Ti1i2j1j2), Ti1i2j1j2=Xi1j1Yi2j2. The definition of XYZ is quite analogous.

Later on, δij will be the Kronecker symbol. Put kX = (mkδimjm) Xikjk, klT = (mk,lδimjm) Tikiljkjl (5) for X=(Xij), T=(Ti1i2j1j2), 1kln, 1mn. These matrices are the natural images of X,T in the MNn=MNn with respect to the indices k and (k,l). Note that kl(XY)=kXlY and kX commutes with lY for any X,YMN, kl.

Let us introduce S(θ) as the following matrix (depending on θ=θ12) from MN2:S=(Si1i2j1j2(θ)). 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(θ)=PR(θ) for P=(Pi1i2j1j2), Pi1i2j1j2= δi1j2δi2j1 (7) we get the Yang-Baxter equation 12R(θ12) 13R(θ13) 23R(θ23)= 23R(θ23) 13R(θ13) 12R(θ12). (8) To check it (to deduce it from (6)) one should carry all the 12P,23P across the other terms in (6) after the substitution S=PR. Here we have to use the following properties of P: 23P 12P 23P= 12P 23P 12P, (9a) 12P1X= 2X 12Pfor XMN. (9b) It is easy either to prove (9) directly or verify them without matrix calculations using the following natural interpretation of ijP.

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