Last update: 28 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.
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.
In this section, we introduce the affine Knizhnik-Zamolodchikov (AKZ) equation associated with the root system of type It is a first-order differential equation for where depends on a single variable and takes values in an infinite-dimensional algebra called the degenerate affine Hecke algebra.
The equation is as follows: Here is a parameter, and and are operators acting on a vector space where takes its value. We impose the following two relations. These relations make (2.1) invariant. Namely, if solves (2.1), then 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 of type generated by the elements and satisfying the defining relations (2.2) and (2.3): Let be a function of with values in Note that one can multiply by an arbitrary constant element on the right, i.e. if is a solution, then is also a solution. Let us check the invariance of AKZ (see (2.4)).
We plug in and use Finally, where
Now we will integrate (2.1). More generally, let us first consider the equation It is (2.1) for and The equation (2.6) is much more complicated than the AKZ. However, if are matrices acting on the vector this equation is nothing but the hypergeometric differential equation. It readily gives the formulas when takes values in irreducible representations of because the latter exist only in dimensions or
Indeed, a generic representation of is given by Because of the gauge transformation it is characterized by or by Then a solution for is given in terms of the Gauss hypergeometric function. The first component is where and with
If then the representation (2.7) is reducible. In this case, solutions are in terms of elementary functions. We note that the parameters in (2.9) are not arbitrary but obey the constraint
In this section, we introduce the AKZ equation of type 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 introduced by Drinfeld.
Recall that the KZ equation reads In the less sophisticated case are the permutation matrices [KZa1984]. Let us assume that are any constant elements (operators) and We consider taking values in the abstract algebra generated by the elements The self-consistency of the system of equations (2.10) means that where It holds for all values of the complex parameter if and only if where the indices 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 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 (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 of the permutation group of the set We denote by the transposition of and If we set the relations (2.13)-(2.14) are satisfied.
Setting the equation (2.10) turns into and the relations (2.13),(2.14) read as follows: where the indices are pairwise distinct.
Substituting we come to Using the elements The elements are convenient since in the limit we get the system The consistency of these equations is equivalent to the commutativity We claim that (2.27), a 'limiting self-consistency', together with the relations where ensure (2.19)-(2.21).
It can be put in the following way. Let us introduce the degenerate affine Heche algebra of type as an algebraic span of and with the relations (2.27), (2.28) and (2.29), denoting it by or simply by We call the system (2.25) with the values in the AKZ of type It is well-defined, i.e. self-consistent.
In this section, we discuss the of the AKZ equation introduced in the previous section and clarify in full the definition of
The group acts on naturally by for Given a function of with values in we define the action of on by Here the dot means the product in It follows from (2.28) and (2.29) that if solves (2.25), then so does Just conjugate the equations by 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
In this section, we fix notations for root systems and define the degenerate affine Hecke algebra for an arbitrary root system.
Let be a root system in with the inner product Choose a system of simple roots of and denote by the set of positive roots. For a root define the coroot by and the reflection by We will denote simply by The fundamental coweights are as follows: We also use the notation For the coordinates will be We also set for Check that
Let be the Weyl group of Define the action of on functions on by
Then we have
Now we can define the degenerate affine Hecke algebra associated with This definition is due to Lusztig [Lus1989] (he calls it the graded affine Hecke algebra, considering as a formal parameter). Drinfeld introduced this algebra in the in [Dri1986-2] prior to Lusztig. These algebras are natural degenerations of the corresponding ones.
Definition 2.1. Let be the associative algebra generated by and with the following relations Here is a complex number and
Introducing we can express the right hand side of (2.35) as More generally,
Later we will use the following partial derivatives For instance,
In this section we introduce the AKZ equation associated with the root system and give several examples.
Let us consider the following system of partial differential equations Here is a complex number. We denote the right hand side of (2.39) as First we assume that takes values in an associative algebra generated by and We say that the system (2.39) is self-consistent provided that It is called invariant if, for any solution of (2.39) and any element of (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 satisfy the relations (2.32), (2.33), (2.34) defining
We introduce the AKZ equation associated with to be the system (2.39) for functions with values in
Using the notation and from (2.36) and (2.38), the system (2.39) can be expressed as
Remark 2.1. The parameter may depend on the lengths of roots. Generally speaking the AKZ equation is as follows:
Let us write down the explicit forms of the AKZ equation in the simplest cases.
Example 2.1. When the AKZ equation is exactly (2.1).
Example 2.2. For the AKZ equation is where denotes the transposition of and In this case
Example 2.3. The root system is realized in the following way. Let and form an orthonormal basis of Then the set of positive roots consists of the following vectors: Let and Then and satisfy (the Coxeter relation for and In this case The AKZ equation reads as follows: Note the appearance of the coefficient 2 in the latter. In the case of the coefficients are from to (otherwise they are less than
In this section, we will show that the AKZ equation of type discussed in §2.2 reduces to the AKZ equation for the root system of type
First note that is central in the algebra By setting we have an embedding of where is the root system of type into We put The space is identified with the quotient space of where is the orthonormal basis and From (2.25) we have Therefore, the function is well-defined on the quotient space Now it is straightforward to see that (2.25) reduces to the AKZ equation of type for