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.

Towards the CFT

There are several possibilities to generalize and extend the above constructions. I shall try to outline only some of them connected with the two-dimensional conformal field theory.

First of all, one can substitute everywhere the q-analog of Yang's S. It is written as Siq(θ)=Ti+ (q-q-1)/ (q2θ-1), 1in (37) where q, {Ti} are the generators of the Hecke algebra Hnq. They satisfy the following defining relations (Ti-q) (Ti+q-1)=0, TiTi+1Ti= Ti+1TiTi+1, [Ti,Tj]=0 (38) for |i-j|2, 1i,jn. For q=1 we arrive at [Sn].

The function Sq was found independently in (one-dimensional) mathematical physics as some solution of (16) and in the theory of representations of p-adic affine Hecke algebras as an interwiner (see [Che1987] for some details). The latter are defined as n but with the term (q2-1)Yi+1 in place of 1 in formula (19a), where one should substitute Ti for si. In p-adic papers q=pm for a prime p,m. This way of definition is due to Bernstein, Zelevinsky. In many works {Ti} are considered in some natural representation of Hnq in (N)n (Wenzl, Baxter).

Since T21 we have two elements T,T-1 being on equal grounds. We omit the arguments, but it results in two possible pictures for two-particle S-matrices instead of the only one above. We can consider intersecting as a passage of a particle over or under the other.

The Sq from (36) is unitary after a proper normalization. But in other non-unitary theories this note can be important.

We have assumed the two-particle intersections to be the only elementary processes. But one can disagree with this assumption. Look at fig. 3a. There is a certain process between the intersections (θ1,θ2) and (θ1,θ2) of the corresponding particles. The particle with the angle θ1 should move away from the particle with θ2 after the first intersection and approach particle θ3. This transference may be quantum as well. (In fact, any movement can be quantum in some general theory).

Let us consider the arranged symbols AˆJ(Θ), which are AJ(Θ) from sec. 1 with some complete set of brackets between some Ajk(θjk). For example, (Aj1(θ1)Aj2(θ2))Aj3(θ3), (Aj1(θ1)Aj2(θ2))(Aj3(θ3)Aj4(θ4)) are complete but (A1A2)(A3A4)A5 is not. The correct arranged symbol should be either ((A1A2)(A3A4))A5 or (A1A2)((A3A4)A5). Here we have omitted j,θ. Physically, the last symbol can be interpreted as follows: the particles A3,A4 are very close one to another, A5 is close to A3 or A4 (it is all the same, since A3 and A4 are very close, more close than A5 to each of them), A1 is close to A2, the pair A1,A2 is not close to the triple A3,A4,A5. In fact, we have the ordered sequence of relations "not close, close, very close, very very close and so on" on the set of AJ(Θ).

Formally speaking, a system of brackets is not complete if Aˆ contains a segment of type (Aˆ1)(Aˆ2)(Aˆ3) for some arranged symbols Aˆ1,Aˆ2,Aˆ3. In this case Aˆ1 and Aˆ3 are at the same level (close, very close, ...) with respect to Aˆ2. We forbid it, that resembles very much the Pauli principle in quantum mechanics.

Given some arranged AˆJ(Θ) for any Θ=(θ1,,θn), we can define to following two elementary operations. One can interchange two adjacent terms Ajk(θk) and Ajk+1(θk+1), but only if they are in brackets. The corresponding quantum Sk will be introduced like in sec. 1-3. Another operation (written Φk) is the passage AˆJ= (Aˆ1(Ajk(θk)Aˆ2)) ((Aˆ1Ajk(θk))Aˆ2) (39) for some arranged Aˆ1,Aˆ2 or the analogous transformation from (Aˆ1Ak)Aˆ2 to Aˆ1(AkAˆ2). Given k the corresponding Aˆ1,Aˆ2 (if any) can be found uniquely. For example, Ai1(θ1) (Ai2(θ2)Ai3(θ3))= ΣJΦIJ (θ1,θ2,θ3) (Aj1(θ1)Aj2(θ2)) Aj3(θ3), where Φi1i2i3j1j2j3(Θ) are the amplitudes from the in-state, where A2 is more close to A3 than to A1, to the out-state, where A2 is more close to A1; Φ=(ΦIJ).

In a contrast with sec. 1 these two operations (processes) exist only for some k. E.g. let us consider Aˆ= ((A1A2)(A3A4)) (A5(A6(A7A8))). One can apply only the operations S1,S3,S7,Φ2,Φ3,Φ5,Φ6,Φ7 to this Aˆ. It is quite natural to postulate the identities [Φi,Φj]= [Si,Sj]= [Si,Φk]=0, |i-j|>1, ki,i+1. The reason is that ΦiΦj and ΦjΦi induce for |i-j|>1 the same changes of brackets. It holds true for the permutations and changes of brackets in the case [S,S] or [S,Φ] as well. The other relations are of the following type (see fig. 3). We begin with AˆI=A1(A2A3) and use here the abbreviations θ12=(θ1,θ2), θ123=(θ1,θ2,θ3) and so on. One has S2(θ12) Φ2(θ312) S1(θ13) Φ2(θ132) S2(θ23) Φ2(θ123) = Φ2(θ321) S2(θ23) Φ2(θ231) S2(θ13) Φ2(θ213) S2(θ12). This equality is quite analogous to identity (2.6) from [MSe1989] (see also [Dri1989-2]). It is small wonder since our symbolic language and appropriate pictures are very close to these of [MSe1989,Dri1989-2].

Here we assume that S, Φ depend on the corresponding parameters in the natural order. In general, Φ can depend on many indices and parameters. E.g. Φ6 for (39) may have 4 matrix indices and be a function of θ5678=(θ5,θ6,θ7,θ8). In some sense the order of A7 and A8 is not important for Φ6 since they both are at the same level with respect to A6. In particular, the dependence of Φ6 on the indices of A7,A8 should be symmetric. The development of this point can give some version of the axiom system from [MSe1989], where any Φ are defined by means of the comultiplication in terms of the least possible Φ (with 3 matrix indices). The penthagone relation arises in this way. We note that our angles are, in fact, parallel to the conformal dimensions (see e.g. [Dri1989-2]).

I'd like to give another example of connections between the two-dimensional conformal theory and the affine Hecke algebras. The so-called Knizhnik-Zamolodchikov equation for the n-point function of the Wess-Zumino-Witten model can be written in terms of [Sn] only. It has the following natural "affine" generalization κdG/dzi= ( j(ij) (zi-zj)-1 +xizi-1 ) G, (40) where 1ijn, G(z1,,zn) takes its values in the algebra 𝒜 generated by [Sn] and some operators {xi} with the relations wκiw-1=xw(i), wSn. Here (ij) are the usual transposition, κ. It is easy to show (see [Che1991]), that the cross-derivative integrability conditions for (40) are equivalent to the relations [xi,xj+(ij)] =0=[(ij),xi+xj]. (41) The latter (together with the conditions wxiw-1=xw(i)) coincide with the defining relations for Yi (sec. 5) if Yi=-xi- nj>i (ij),1in. Hence, this 𝒜 should be 6ome quotient of the degenerated affine Hecke algebra n.

To get the usual Knizhnik-Zamolodchikov equation one should put xi=0 for any i. It gives us the so-called Murphy surjection Yi-nj>i(ij) of n onto [Sn] (see [Dri1986]). This homorphism of algebras is important in the theory of Sn. For example, the centre of [Sn] is generated by symmetric polynomials in the images of {Yi} (cf. theorem 3 and [Che1987]).

To finish this part of my notes I will describe without going into detail some quantum counterpart of (41).

Let us consider the space with the glass at the point x=0. It is transparent for particles from sec. 1, but passing through this glass is assumed to be quantum. One can connect with this process two one-index matrices X(+θ), Xˆ(-θ) respectively for θ<0 and θ>0 (see fig. 9). E.g. for θ<0 Ai(θ)in =glassj Xij(θ)Aj (θ)out The factorization relations are close to (6). We will write them down in term s of R=PS: 1X(θ1) 12R(θ12) 2Xˆ(-θ2) = 2Xˆ(-θ2) 12R(θ12) 1X(θ1), 12R(θ12) 1X(θ1) 2X(θ2) = 2X(θ2) 1X(θ1) 12R(θ12), 12R(θ12) 2Xˆ(-θ2) 1Xˆ(-θ1) = 1Xˆ(-θ1) 2Xˆ(-θ2) 12R(θ12). (42) Of course, R should be a solution of (8) as well.

These equalities hold true (follow from (8)) if one formally substitute X=10R(θ10), Xˆ=01R(θ01), θ0=0, where 0 is some other tensor index. Really, the transmission through the glass can be interpreted as intersecting with the particle of angle θ0=0 and colour =0, where the latter does not change its colour in any quantum interactions.

The natural problem is to combine S,Φ, mirrors (not more than 2), polarizations (any number) and glasses (any number) in one picture. Then to consider more complicated spaces (circumferences, elliptic curves) and find interesting examples. Only some fragments of this heavy construction are clear (see [Che1984,Che1991,MSe1989]).

Let R=Rη=θ(θ+η)-1+η(θ+η)-1P (see sec. 5), X=Xˆ. We can consider (42) over [Sn] in a natural manner. One identifies permutations with the corresponding matrices and supposes iX to be some undeterminate functions with the following action of Sn:wiXw-1=w(i)X. We have Ri=ii+1R=1+ηsi/θi+θ(η) as η0. Let us impose the analogical restrictions Xi=1+ηxi/θ+o(η) on Xi=iX (in particular, xw(i)=wxiw-1). Then (42) results in (41).

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