Last update: 30 April 2014
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.
Let us now turn to the We introduce the quantum affine Knizhnik-Zamolodchikov (QAKZ) equation, and show that there is an isomorphism between solutions of the QAKZ equation and solutions of the generalized Macdonald eigenvalue problem.
In this section we recall the definition of the affine Hecke algebra in the case of
Let be a parameter. Then is the algebra defined over by the following set of generators and relations: The relations (4.1) are called the quadratic relations, (4.2)-(4.3) the Coxeter relations, (4.4) the commutativity, and (4.5),(4.6) the cross relations.
Set It follows from the defining relations (4.1)-(4.6) that the right hand side is independent of and therefore equals to
The algebra can be presented as
where the quotient is by the quadratic relations (4.1), the Coxeter relations (4.2)-(4.3) and the following:
Notice that in terms of we have The relations (a) and (b) readily follow from (4.7) and the defining relations (4.1)-(4.6). For instance,
To establish (4.8), we start with and introduce the elements by We must check the commutativity etc. using (a), (b). The first reads We plug in the above formula for and move to the left. The commutativity with 'distant' is obvious. The other relations formally follows from these ones. We leave the verifications to the reader as an exercise.
In this section, we introduce the QAKZ equation.
Definition 4.1. For we define the intertwiners by
They satisfy The second relation can be deduced from Lemma 3.7 as we did for the degenerate Hecke algebra.
The quantum affine Knizhnik-Zamolodchikov (QAKZ) equation is the following system of difference equations for a function that takes values in (or any Here is a new parameter.
Theorem 4.2. The QAKZ system (4.12) is self-consistent. It is invariant in the following sense: if is a solution, then so is
This follows from (4.10), (4.11). Later we will make it quite obvious.
Let us discuss the quasi-classical limit of the QAKZ system. Setting let The generators are supposed to have the form where by we mean terms of order The relations of the degenerate affine Hecke algebra for can be readily verified. Using the formula we find that Hence the AKZ equation (3.42) is a semi-classical limit of the QAKZ equation.
To make the QAKZ equations more transparent, let us discuss the action of the affine Weyl group. The affine Weyl group of type is the semi-direct product where is a free abelian group of rank Define the action of on a vector by We also introduce Its action on and the coordinates reads as
Lemma 4.3. can be presented as where the relations are and
It is convenient to represent the elements graphically.
Fig.7 shows a reduced decomposition of
For a function with values in let
Theorem 4.4 ([Che1992-2]). The formulas (4.13), (4.14) can be extended to an action of
We denote this action by For instance, Hence the QAKZ equation simply means the invariance of with respect to the pairwise commuting elements
Let us connect QAKZ with the q-KZ introduced by Smirnov and Frenkel-Reshetikhin [Smi1986, FRe1991]. We fix an complex vector space and introduce by due to Baxter and Jimbo. The algebra acts on by where and One can check that this action is well-defined by a direct calculation.
For let be the subspace. Here denote the standard basis of It is easy to see that this subspace is closed under the action of We state the next proposition without proof.
Proposition 4.5. If and is generic, then the space is isomorphic to
Writing down AQKZ in we get the q-KZ (for and in the fundamental representation). Combining this observation with the isomorphism with the Macdonald eigenvalue problem (our next aim) we can explain why the Macdonald polynomials appear in many calculations involving the vertex operators.
Let be a solution of the QAKZ equation. Thanks to (4.15), is also a solution of the QAKZ equation for any We define by and call it the monodromy cocycle. It follows From (4.13) and (4.14) that and Here stands for
The QAKZ equation implies that Hence depends only on the image of in
Let be the set of function on Next we define two anti-actions of where and Lemma 4.6 means exactly that is an anti-action For instance,
We note that in the difference theory the monodromy can be always made trivial. Indeed, the 1-cocycle is always a co-boundary because of the Hilbert 90 theorem. Hence conjugating solutions of AQKZ we can always get rid of the monodromy. So the above actions are not too much different in contrast to the differential theory.
This argument can be applied to the AQKZ itself, although the group is infinite. We can formally solve the QAKZ equation as follows. Let Then the infinite sum where satisfies the AQKZ, provided the convergence. For example, if is rapidly decreasing, then one can check that is convergent.
We see, that constructing solution to QAKZ for finite-dimensional poses no problem. What is more difficult is to ensure a proper asymptotical behavior.
In this subsection, we introduce the Macdonald eigenvalue problem and prove its equivalence to the QAKZ equation. This is a of the relation between AKZ and QMBP discussed in §3.4.
Let be a solution of the QAKZ equation with values in for a We assume that it is invertible for sufficiently general Setting we get from (4.18) and (4.9): Let us introduce the operator by Then and (see (4.19)). The operators and commute with the left multiplication by and any elements from Using all these: We come to the following definition:
Since and commute with each other, By the construction, the operators act in functions. However if we understand them formally the commutativity can be deduced from the relations The latter means that
Let be a polynomial in variables. Then and we can represent where are pure difference operators, which do not contain
For symmetric we introduce a difference operator of Macdonald type by
Let be a function on The system will be called the Macdonald eigenvalue problem. The operators can be calculated for instead of As in the differential case, the result will be the same.
Fix We take a left with the following properties:
|(1)||for any symmetric polynomial in variables and all|
|(2)||there exists a map such that for all and|
As always, we fix a (local) invertible solution of the QAKZ equation with values in Note that all solutions of the QAKZ equation can be written in the form for function
Theorem 4.7. Let be an with the above properties, be the space of solutions of the QAKZ equation with values in and the space of solutions of the Macdonald eigenvalue problem (4.29). Then
Let Then For a reduced decomposition of Since commutes with the left action of we have Using the commutativity of with we represent as a sum for some functions Finally Applying this relation to and taking tr, we conclude:
Let us now consider The definition is quite similar to the differential case. We start with Here is the one-dimensional representation sending to and is the ideal generated by As in §3.1, stands for the dual module defined via the anti-involution of
The main result of this subsection is the following theorem from [Che1992, Che1994-2].
Theorem 4.8. If then the map from to is injective.
The theorem results from the following two lemmas.
Lemma 4.9. Let be a of Then implies
The proof repeats that in the differential case (see ((3.38))).
Lemma 4.10. Let be a solution of the QAKZ equation. Assume Then
First for all Then and Therefore, representing we have where are function. Hence and
Now we shall prove that for all by induction. Assume that for Since it is enough to see that Since is a solution of the QAKZ equation we have On the other hand, where and are some scalar functions. Therefore where is a sequence of integers such that and is some scalar function. If then there are the following possibilities:
case (1): As we have
case (2): Since for By the induction hypothesis, Hence
case (3): In this case must be of the form By induction, So for all
Because of the relations between and it remains to check that for any One can show this by induction on
We set Here are from (4.20),
Switching from to Let be the elementary symmetric polynomial in variables. We represent for difference operators and define All these operators are which results from the following lemmas.
Lemma 4.11. Consider the algebra generated by Then extends to an algebra isomorphism Moreover, if is a symmetric polynomial in variables, then is a central element in
Actually this observation is the key point (it can be checked directly or with some representation theory). We note that the formulas for generalize the so-called Demazure operations and the Bernstein-Gelfand-Gelfand operations. They were also studied by Lusztig and in a paper by Kostant-Kumar.
From now on we identify with
Lemma 4.12. Let be a function on Then is symmetric if and only if for all
Lemma 4.13. Let be a symmetric polynomial in variables. Then acts on the space of the symmetric polynomials in
This follows immediately from Lemma 4.11 and 4.12.
Let us calculate Since is symmetric, it is enough to find the coefficient of Using the it is easy to see that does not appear in The of is equal to
After the symmetrization we get the formula:
Similarly, where is a sequence of integers such that
To recapitulate, let us consider the classical limit of the Macdonald operators. Setting and we have
Remark 4.1. Take a solution of the QAKZ equation in an assuming that has the trivial monodromy. Then, for any polynomial we have where are the difference Dunkl operators defined before. Note that can be replaced by because the monodromy of is trivial. We also need a linear functional for a vector such that for any Given any element let us define a scalar-valued function setting Then the formula (4.34) implies Thus, the scalar-valued function solves the Dunkl eigenvalue problem.
Remark 4.2. Arbitrary root systems. Let be any reduced root system of rank (of type or and the corresponding affine Hecke algebra. The baxterization (a parametric deformation satisfying the Yang-Baxter relations) of will be given by for each We also have to use the element corresponding to the simple affine root for being the highest root. Its baxterization is quite similar: where The functions satisfy the Yang-Baxter equations associated with the extended Dynkin diagram. For example, in the case of we have The arguments of can also be determined graphically by means of the equivalent pictures of the reflection of two particles (see [Che1992-2]).
Using the affine Hecke algebra has an alternative representation where is a certain finite abelian group. The group is isomorphic to It is the set of all elements of the extended affine Weyl group preserving the set of the simple affine roots. It gives the embedding of into the automorphism group of the extended Dynkin diagram. The action of on is by the affine reflections and the corresponding shifts in the for
Lemma 4.14. with
The group can be embedded into the affine Hecke algebra. The images of the elements permute in the same way as do in with The baxterization of the elements in is trivial: for each
Keeping the notations of the previous sections, we have the following theorem.
Theorem 4.15. Given any function the formulas for all and for all induce a representation of
The QAKZ equation for is the invariance condition for all It can be shown that this equation is equivalent to the difference QMBP associated with the root system defined via similar Dunkl operators. A conceptual proof of this isomorphism theorem is given by means of the intertwiners of double affine Hecke algebras (see [Che1992, Che1994-2]).