Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 28 April 2014

Notes and References

This is an excerpt of the paper Lectures on affine Knizhnik-Zamolodchikov equations, quantum many body problems, Hecke algebras, and Macdonald theory by Ivan Cherednik, in collaboration with Etsuro Date, Kenji Iohara, Michio Jimbo, Masaki Kashiwara, Tetsuji Miwa, Masatoshi Noumi, and Yoshihisa Saito.

The affine Knizhnik-Zamolodchikov equation

We introduce the degenerate affine Hecke algebra and the corresponding affine Knizhnik-Zamolodchikov equation. We show how the former appear as the consistency and invariance conditions for the latter.

The algebra A1 and the hypergeometric equation

In this section, we introduce the affine Knizhnik-Zamolodchikov (AKZ) equation associated with the root system of type A1. It is a first-order differential equation for Φ, where Φ depends on a single variable u and takes values in an infinite-dimensional algebra called the degenerate affine Hecke algebra.

The equation is as follows: Φu= (kseu-1+x) Φ. (2.1) Here k is a parameter, and s and x are operators acting on a vector space where Φ takes its value. We impose the following two relations. s2=1, (2.2) sx+xs=k. (2.3) These relations make (2.1) invariant. Namely, if Φ solves (2.1), then Φ(u)=s Φ(-u) (2.4) also is a solution of (2.1). We claim that (2.1) is integrable in terms of the classical hypergeometric functions. At least, this statement is valid under a certain irreducibility condition.

The AKZ is the equation (2.1) with values in the degenerate affine Hecke algebra A1 of type A1 generated by the elements s and x satisfying the defining relations (2.2) and (2.3): A1= s,x/ {(2.2), (2.3)}. (2.5) Let Φ(u) be a function of u with values in A1. Note that one can multiply Φ(u) by an arbitrary constant element on the right, i.e. if Φ(u) is a solution, then Φ(u)a (aA1) is also a solution. Let us check the invariance of AKZ (see (2.4)).

We plug in -Φ(u)u = s(kse-u-1+x) Φ(-u) = (kse-u-1+sxs) Φ(u) and use 11-e-u= 1eu-1+1. Finally, Φ(u)u= (kseu-1+ks-sxs) Φ(u), where ks-sxs=x.

Now we will integrate (2.1). More generally, let us first consider the equation Φz= (A1-z+Bz) Φ. (2.6) It is (2.1) for z=e-u, A=-ks, and B=-x. The equation (2.6) is much more complicated than the AKZ. However, if A,B are 2×2 matrices acting on the 2-component vector Φ, this equation is nothing but the hypergeometric differential equation. It readily gives the formulas when Φ takes values in irreducible representations of A1, because the latter exist only in dimensions 1 or 2.

Indeed, a generic 2-dimensional representation ρ of A1 is given by ρ(s)= (100-1) ,ρ(x)=k ( s2+ (0ζξ0) ) . (2.7) Because of the gauge transformation ζcζ, ξc-1ξ, it is characterized by ζξ, or by μ=(ζξ+14)1/2. (2.8) Then a solution Φ=(Φ1Φ2) for A=-kρ(s), B=-ρ(x) is given in terms of the Gauss hypergeometric function. The first component is Φ1(u)= z-kμ (1-z)kF ( k(1-2μ),k, 1-2kμ;z ) , (2.9) where z=e-u and F(α,β,γ;z)=n=0(α)n(β)n(γ)nn!zn with (x)n=x(x+1)(x+n-1).

If ζξ=0, then the representation ρ (2.7) is reducible. In this case, solutions are in terms of elementary functions. We note that the parameters α,β,γ in (2.9) are not arbitrary but obey the constraint α+1=β+γ.

The AKZ equation for GLn

In this section, we introduce the AKZ equation of type GLn. It can be obtained as a specialization of the standard Knizhnik-Zamolodchikov (KZ) equation from the conformal field theory. The consistency and invariance conditions give rise to the defining relations of the degenerate affine Hecke algebra GLn introduced by Drinfeld.

Recall that the KZ equation reads Φzi =k(0jnjiΩijzi-zj) Φ(0in). (2.10) In the less sophisticated case Ωij are the permutation matrices [KZa1984]. Let us assume that Ωij are any constant elements (operators) and Ωij=Ωji. We consider Φ(z) (z=(z0,,zn)) taking values in the abstract algebra generated by the elements Ωij. The self-consistency of the system of equations (2.10) means that Ajzi- Aizj= [Ai,Aj], (2.11) where Ai=kji Ωijzi-zj. (2.12) It holds for all values of the complex parameter k if and only if [Ωij,Ωkl]=0, (2.13) [ Ωij, Ωik+ Ωjk ] =0, (2.14) where the indices i,j,k,l are pairwise distinct. The KZ in this form is due to Aomoto [Aom1988] (it was also studied by Kohno [Koh1987]).

The trigonometric KZ (and the elliptic ones) where introduced for the first time in [Che1989], To be more exact, in this paper the equations which I called the r-matrix KZ were defined, the connections with Kac-Moody algebras were established, and the reduction procedure was applied to the monodromy (see below). The equations corresponding to the simplest trigonometric r-matrces (there are many of them due to Belavin and Drinfeld) are closely connected with the AKZ. There were doubts about the importance of my trigonometric KZ in physics when they appeared. Now they are quite common for both mathematicians and physicists.

Consider the group algebra [𝕊n] of the permutation group 𝕊n of the set {1,,n}. We denote by sij the transposition of i and j. If we set Ωij=sij (0i,jn), the relations (2.13)-(2.14) are satisfied.

Setting z0=0, (2.15) Ωij=sij (i,j0), (2.16) Ω0i=k-1 Ωi, (2.17) the equation (2.10) turns into Φzi= [ k(1jnjisijzi-zj) +Ωizi ] Φ(1in), (2.18) and the relations (2.13),(2.14) read as follows: [sij,Ωi+Ωj] =0, (2.19) [ksij+Ωi,Ωj] =0, (2.20) [sij,Ωl] =0, (2.21) where the indices i,j,l are pairwise distinct.

Substituting zi=evi, (2.22) we come to Φvi= ( kji sij1-evj-vi +Ωi ) Φ. (2.23) Using the elements yi=Ωi+ kj>i sij, (2.24) Φvi= ( kj>i sijevi-vj-1 -kj<i sijevj-vi-1 +yi ) Φ. (2.25) The elements {y} are convenient since in the limit v1v2vn, we get the system Φvi =yiΦ. (2.26) The consistency of these equations is equivalent to the commutativity [yi,yj]=0. (2.27) We claim that (2.27), a 'limiting self-consistency', together with the relations [si,yj]=0 ifji,i+1, (2.28) siyi- yi+1si =k, (2.29) where si=sii+1 (1in-1), ensure (2.19)-(2.21).

It can be put in the following way. Let us introduce the degenerate affine Heche algebra of type GLn as an algebraic span of [𝕊n] and yi (1in) with the relations (2.27), (2.28) and (2.29), denoting it by GLn, or simply by n. We call the system (2.25) with the values in n the AKZ of type GLn. It is well-defined, i.e. self-consistent.

The 𝕊n-invariance

In this section, we discuss the 𝕊n-symmetry of the AKZ equation introduced in the previous section and clarify in full the definition of n.

The group 𝕊n acts on n naturally by v=(v1,,vn) nw(v)= (vi1,,vin) n for w-1=(i1,i2,,in)𝕊n. Given a function Φ(v) of vn with values in n, we define the action of w[𝕊n] on Φ(v) by (w(Φ))(v) =w·Φ(w-1(v)). (2.30) Here the dot means the product in n. It follows from (2.28) and (2.29) that if Φ solves (2.25), then so does w(Φ). Just conjugate the equations by {si}. Moreover, the invariance is exactly equivalent to the relations (2.28) and (2.29). Thus the invariance and the limiting self-consistency (2.27) give the self-consistency of our system for all k.

Degenerate affine Hecke algebras

In this section, we fix notations for root systems and define the degenerate affine Hecke algebra for an arbitrary root system.

Let Σ be a root system in n with the inner product (,). Choose a system of simple roots α1,,αn of Σ and denote by Σ+ the set of positive roots. For a root αΣ, define the coroot α by α=2α(α,α) and the reflection sα by sα(u)=u- (α,u)α (un). We will denote sαi simply by si. The fundamental coweights bi are as follows: (bi,αj)= δij. We also use the notation ai=αi. For un, the coordinates will be ui=(u,αi). We also set uα=(u,α) for αΣ. Check that uαui= ναi=(bi,α) =the multiplicity ofαiinα.

Let W be the Weyl group of Σ: W=sα,αΣ=s1,,sn. Define the action of W on functions on n by wf(u)=j (w-1(u)) (un). (2.31)

Then we have wuα= (w-1(u),α) =uw(α).

Now we can define the degenerate affine Hecke algebra Σ associated with Σ. This definition is due to Lusztig [Lus1989] (he calls it the graded affine Hecke algebra, considering k as a formal parameter). Drinfeld introduced this algebra in the GLn-case in [Dri1986-2] prior to Lusztig. These algebras are natural degenerations of the corresponding p-adic ones.

Definition 2.1. Let Σ be the associative algebra generated by [W] and x1,,xn with the following relations [xi,xj]=0, i,j, (2.32) [si,xj]=0, ifij, (2.33) sixi-xˆi si=k. (2.34) Here k is a complex number and xˆi=xi- j=1n (αi,αj) xj. (2.35)

Introducing xb=i=1n (b,αi)xi= i=1nki xiforb= i=1nki bi, (2.36) we can express the right hand side of (2.35) as xsi(bi)=xbi-xai. More generally, sixb-xsi(b) si=xbsi-si xsi(b)=k (b,αi). (2.37)

Later we will use the following partial derivatives b(uα)= (α,b). (2.38) For instance, ui= i=bi

The AKZ equation associated with Σ

In this section we introduce the AKZ equation associated with the root system Σ, and give several examples.

Let us consider the following system of partial differential equations Φui= ( kαΣ+ ναi sαeuα-1 +xi ) Φ(1in). (2.39) Here k is a complex number. We denote the right hand side of (2.39) as AiΦ. First we assume that Φ takes values in an associative algebra generated by [W] and x1,,xn. We say that the system (2.39) is self-consistent provided that [ ui-Ai, uj-Aj ] =0. (2.40) It is called invariant if, for any solution Φ of (2.39) and any element w of W, w(Φ) (see 2.30) is again a solution of (2.39).

Theorem 2.1. The system (2.39) is self-consistent and invariant if and only if s1,,sn, x1,,xn satisfy the relations (2.32), (2.33), (2.34) defining Σ.

We introduce the AKZ equation associated with Σ to be the system (2.39) for functions Φ with values in Σ.

Using the notation xb and b from (2.36) and (2.38), the system (2.39) can be expressed as bΦ= ( kαΣ+ (b,α) sαeuα-1 +xb ) Φ. (2.41)

Remark 2.1. The parameter k may depend on the lengths of roots. Generally speaking the AKZ equation is as follows: Φui= ( αΣ+ k|α| ναisαeuα-1 +xi ) Φ. (2.42)

Let us write down the explicit forms of the AKZ equation in the simplest cases.

Example 2.1. When Σ=A1, the AKZ equation is exactly (2.1).

Example 2.2. For A2, the AKZ equation is Φu1 = { k ( s12eu1-1+ s13eu1+u2-1 ) +x1 } Φ, Φu2 = { k ( s23eu2-1+ s13eu1+u2-1 ) +x2 } Φ, where sij denotes the transposition of i and j. In this case xˆ1=x2-x1, xˆ2=x1-x2.

Example 2.3. The root system B2 is realized in the following way. Let ϵ1 and ϵ2 form an orthonormal basis of 2. Then the set of positive roots consists of the following vectors: α1=ϵ1-ϵ2, α2=ϵ2, α1+α2=ϵ1, α1+2α2=ϵ1 +ϵ2. Let s=s1 and t=s2. Then s and t satisfy tsts=stst (the Coxeter relation for WB2=WC2) and s2=1, t2=1. In this case xˆ1=x2-x1, xˆ2=2x1-x2. The AKZ equation reads as follows: Φu1 = { k ( seu1-1+ stseu1+u2-1+ tsteu1+2u2-1 ) +x1 } Φ Φu2 = { k ( seu2-1+ stseu1+u2-1+ tsteu1+2u2-1 ) +x2 } Φ. Note the appearance of the coefficient 2 in the latter. In the case of E8 the coefficients are from 1 to 6 (otherwise they are less than 6).

The An-1 case

In this section, we will show that the AKZ equation of type GLn discussed in §2.2 reduces to the AKZ equation for the root system Σn-1 of type An-1.

First note that x=y1++yn (2.43) is central in the algebra n. By setting xi=y1++yi -inx, (2.44) we have an embedding of Σ, where Σ is the root system of type An-1, into n. We put ui=vi-vi+1 (1in-1). (2.45) The space n-1 is identified with the quotient space of n={i=1nviϵi|vi} n-1 i=1nϵi /ϵ where {ϵi}1in is the orthonormal basis and ϵ=ϵ1++ϵn. From (2.25) we have i=1n Φvi =xΦ. Therefore, the function Φ(v)= e-x·1n(v1++vn) Φ(v) is well-defined on the quotient space n-1. Now it is straightforward to see that (2.25) reduces to the AKZ equation of type An-1 for Φ(v) Φui= ( kji<l sjleuj++ui-1-1 +xi ) Φ (1in-1).

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