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's S-matrix

Let us discuss the basic example of a factorized S-matrix (a solution of (6)). The following one is the so-called Yang's S-matrix: S(θ)=1+ Pθ,θ=θ12 =θ1-θ2. (17) Here and further we will denote by 1 the unit matrix in MN, 11 and so on. Setting S(θ)η=1η(θ+η)-1+Pθ(θ+η)-1 for any η we get an unitary S-matrix, satisfying (6) and (14). In particular, S(θ)0=P corresponds to a world without any scattering. To verify (6) for S (or Sη) 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 (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 Yi for θi (1in) in Si(Θ)=ii+1S(θi-θi+1), where S is from (17). We assume {Yi} 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), foryi,i+1 (18c) Formulas (18) are equivalent to the relations Yi+1si-si Yi=1=si Yi+1-Yisi (19a) Yjsi=siYj foryi,i+1 (19b) Here and further we will identify Psi=ii+1P with si and 1 with 1. Let us check e.g. the first of these formulas. It results from (18a) that Yii+1= (Yi+1si-siYi) Yii+1,Yij =Yi-Yj. (20) Then one can divide (20) by Yii+1. We see that (19) (18), but the converse holds true only for {Yi} in a "general position".

We have arrived at the following object. Let [Sn]=ww be the group algebra of Snw with the natural multiplication law: w·w=wwSn (e.g. (a+(12))·(b+(23))=ab+a(23)+b(12)+(3,1,2), a,b). It is nothing else but the algebra of formal linear combinations of permutations. Its extension by pairwise commutative symbols Y1,,Yn with relations (19) is called the degenerated affine Hecke algebra (written n). It is due to Murphy and Drinfeld (see [Dri1986]). Starting with Sη instead of S we get n(η), where η should stay for 1 in (19a). The algebra n(η) isomorphic to n for η0 (use the substitution {YiηYi}. But this η is important to understand n as some quantum object.

To differentiate Si(Θ) from ii+1S(Yii+1) (see (13b)) let us denote the latter by: Σi=1+si (Yi-Yi+1), 1i<n The main point is that (16) is equivalent to the following identity in n: ΣiΣi+1 Σi=Σi+1 ΣiΣi+1 (1i<n). (21) To prove the equivalence we need some kind of Wick (or Poincaré-Birckhof-Witt) theorem for n. One can deduce directly from (19) that each element An has the unique representation of the following type: A=wwyw, where wSn, yw are some polynomials in Y1,,Yn. Let us denote this sum for A after the converse substitution Yiθi by A. The only thing we need is to show that ΣiΣi+1Σi and Σi+1ΣiΣi+1 coincide, respectively, with the l.h.s and r.h.s of (16). Let us carry all the Yi,Yi+1,Yi+2 in (21) over Σi,Σi+1 by means of (18) from the left to the right. Then one obtains Yi-Yi+1, Yi-Yi+2 and Yi+1-Yi+2 instead of Yii+1,Yi+1i+2,Yii+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, Ση in the natural notations is involutive in n(η), i.e. (Ση)(Ση)=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 {Σi} from n extends uniquely to the set {Σw,wSn}n with the following properties:

(a) ΣxΣy=Σxy if (xy)=(x)+(y), Σid=1
(b) ΣwYiΣw-1=Yw(i), wSn, 1in.

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 n. 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 {Yi}. Hence, the product ΣxΣy should be equal to Σxy modulo multiplications by some elements from the centralizer (commutant) of {Y1,,Yn} in n. It is not difficult to prove that this centralizer coincides with the algebra [Y1,,Yn] of polynomials of {Yi} (see theorem 3). In particular, (6) for Yang's S has to be true up to a multiplication by a scalar function in θ1,θ2,θ3 (use the Wick theorem for n). 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 n.

page history