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 affine Hecke algebra and connect them with the degenerate affine Hecke algebra using the monodromy of the AKZ equation. We also establish an isomorphism between the solution space of the AKZ equation and that of a quantum many body problem.
In this section we define induced representations of
For the character of (i.e. a ring homomorphism is an assignment We denote it by
Definition 3.1. We define an as the representation induced from Here is endowed with the structure by the character
We have the Poincaré-Birkhoff-Witt type theorem for Namely any is expressed uniquely in either of the following ways: with The existence results from the relations (2.32)-(2.34) in Hence Thus is as a where the action of is determined by for the identity The action of on other elements of have to be determined using the defining relation (similar to the calculations in the Fock representation).
We also need another construction. Let be induced from the trivial character Then is isomorphic to as a vector space and moreover as a To get finite-dimensional representations from we use the coincidence of the center of with the algebra of polynomials in This theorem is due to Bernstein. The procedure is as follows. Let us fix an element and introduce the ideal in generated by for all polynomials Set Then has a structure of by virtue of the Bernstein theorem.
We will also use the anti-involution on Since the relations of are self-dual it is well-defined. For an we consider its dual The dual has an anti-action (a right action) of Composing it with the anti-automorphism we get a natural (left) action of We denote the resulting module by
We write for
|(a)||is irreducible if and only if for any|
|(b)||There exists a permutation of (i.e. for such that for any Then|
|(c)||For the longest element in|
A key lemma in proving Theorem 3.1 is
Lemma 3.2. is irreducible.
The proof from [Che1995-2] is based on the intertwining operators of degenerate affine Hecke algebras (to be defined below). See also [KLu0862716, Kat1981, Rog1985] and the references therein (the non-degenerate case).
Definition 3.2. For we set For with a reduced decomposition We call the elements intertwiners.
The elements belong to the localization of the degenerate affine Hecke algebra by the polynomials. They give a certain 'baxterization' of and are closely related to the Yang's Let us show that does not depend on the choice of the reduced decomposition of
We have Indeed, which can be rewritten as follows: Using the definition of (2.36), the right hand side of (3.12) is So we come to (2.37). The relations (3.10) fix uniquely up to the multiplication on the right by functions in The leading terms of being they coincide for any reduced decompositions.
To demonstrate the role of intertwiners, let us check the irreducibility of for generic First note that the vectors are common eigenvectors of because For a generic the eigenvalues are simple, hence these vectors are linearly independent. Now, any nonzero of contains at least one eigenvector of By the simplicity of eigenvalues, such an eigenvector must be in the form for some On the other hand are invertible elements. Indeed, Therefore Since is generated by we conclude that
Actually this very reasoning leads to the proof of the Theorem (a),(b). However if is arbitrary one must operate with the intertwiners much more carefully. It is necessary to multiply them by the denominators and remember that the invertibility does not hold for special
Remark 3.1. The of will be interpreted below as certain quotients of the representing the quantum many-body eigenvalue problem. A solution of the AKZ in induces solutions in any of its (if is reducible). It gives a one-to-one correspondence between the (quotients, constituents) of and those of the representing the quantum many-body eigenvalue problem. The description of the latter is an analytical problem. The classification of the former is a difficult question in the representation theory of Hecke algebras. For instance, the multiplicities of the irreducible constituents are described in terms of the Kazhdan-Lusztig polynomials. It is very interesting to combine the two approaches together.
In this section we discuss the monodromy of the AKZ equation, which is a key ingredient in establishing the isomorphism between the AKZ equation in the representation and the quantum many-body problem (QMBP) with the eigenvalue
Let be the open subset of given by The lattice generated by will be denoted by It is isomorphic to and acts on by translations. Namely, where The semi-direct product is the so-called extended affine Weyl group, acting on and leaving invariant. Picking we set The group structure of is described as follows. Given an element let be a path from to in For elements we define the composition of and as the path composed of and the path mapped by (see Fig.5). The class of will be denoted by The map is a homomorphism onto
It is convenient to choose and the generators of as follows. Set Then is a simply connected open subset of Let us take such that for For any element we denote a path from to in by This condition simply means that whenever intersects the imaginary axis it must go through the 'window'
For any element we define an element of to be the image of Since is simply connected, depends only on We set and choose to be a path from to the point with the same coordinates for and The structure of is described in the following theorem from [Lek1981].
Theorem 3.3. Here for we put
Fig.6 proves the relation (3.17). It shows the only, which is sufficient for this relation.
Let us introduce the affine Hecke algebra associated with a root system as a quotient of the group algebra of by the quadratic relations.
Definition 3.3. The affine Hecke algebra associated with a root system is an associative generated by with the following relations: The monomials are defined as in (3.18), Here and above we mean the homogeneous Coxeter relations: factors on each side, where whenever the corresponding vertices in the Dynkin diagram are connected by laces.
Let be an invertible solution of the AKZ equation associated with defined in a neighborhood of Then, for is defined near (see (2.30)). Let be a path in from to Denote by the analytic continuation of back to along the path where denotes the class of in the fundamental group We will also use the projection homomorphism sending to for Using this homomorphism we can extend the action of from (2.30) to multiplying on the left by
Let us define the monodromy to be the ratio Here dot means the product in Since and both satisfy the same AKZ equation, does not depend on So it is an invariant of the homotopy class of and is always invertible. If we choose and the paths in as above, then for are well-defined. The monodromy is a homomorphism from (but not from which readily results from the definition.
As a preparation for an explicit computation of in the next section, we shall introduce a special class of solutions
Proposition 3.4. For generic there exists a unique solution of the AKZ equation such that where and are independent of
We call the solution in the proposition the asymptotically free solution. To be more exact, we need either to complete or restrict ourselves with finite-dimensional representations of this algebra. Then establishing the the (local) convergence is easy. In these notes we will follow the second way. We give general formulas, which are quite rigorous in finite-dimensional representations (say, in the induced representations).
Let us examine the condition necessary for the existence of the asymptotically free solutions in the case of A general consideration follows the same lines. In this case, The equation (2.39) leads to Comparing the coefficients of Given a representation of we find assuming that is invertible for any in this representation. Therefore, setting the conditions ensure the existence of the asymptotically free solutions. The convergence estimates are straightforward. These conditions are fulfilled in generic induced representations.
In this section we establish an isomorphism between and using the monodromy of the AKZ equation.
Let us fix an invertible solution of the AKZ system in a neighborhood of The functions will be extended to through Since is infinite-dimensional, we have to consider all formulas in finite dimensional representations. Once we get the final expressions it is not difficult to find a proper completion of the degenerate Hecke algebra for them.
Theorem 3.5 ([Che1991-4]). There exists a homomorphism from to given by where If is sufficiently general (say, not a root of unity), then it is an isomorphism at the level of finite dimensional representations or after a proper completion.
Under the notation (3.23), and Hence the relations (3.19)-(3.21) result from Theorem 3.3, and only the quadratic relations (3.22) need to be proved. We skip a simple direct proof since these relations follow from the exact formulas below.
Let us find the formulas for and for the asymptotically free solution Given we set and define analogously.
([Che1991-3]). Let us choose the asymptotically free solution
We will give a sketch of the proof of Theorem 3.6. The statement (a) is immediate, since To prove the statement (b), we reduce the problem to the case. Let us fix the index Set so that Let us define as follows: where The AKZ system for reads: Reduction procedure. Since the monodromy does not depend on the point and the path connecting and may be replaced by any deformations in or their limits. Provided the existence, the resulting monodromy coincides with For instance, equals
Indeed, the latter is the limiting monodromy for a path with approaching the infinity. We note that if does.
In the reduced equations (3.29) and (3.30), we may diminish the values, considering the subalgebra of generated by and In this algebra, the following elements are central: Hence, if we define by it enjoys the following properties:
To explain the structure of the formula for let us involve the intertwiners of They are defined similar to those in the degenerate case:
It readily results from the definition of (cf. 3.10).
The image of in with respect to the homomorphism constructed in Theorem 3.5 can be represented as for a function of Indeed, which gives the proportionality. Recall that Here must be of the form for a function in one variable, and can be calculated using the hypergeometric equation (3.31). We omit the details (see [Che1991-3]).
We note, that the quadratic relations for can be made quite obvious using the same reduction (the exact formulas above are not necessary). Set to simplify the indices. We switch from (3.31) to (2.18) with two variables and a parameter When the substitutions are as follows The monodromy corresponding to the transposition of and for coincides with It does not depend on up to a conjugation (the same reduction argument applied to the KZ-equation with three variables). Sending to infinity we eliminate the The monodromy of the resulting equation can be calculated immediately. Since it is conjugated to we get the desired quadratic relations.
Heckman in [Hec1987] used a similar reduction approach when calculating the monodromy of the quantum many-body problem (also called the Heckman-Opdam system). Our next aim is establishing an isomorphism of AKZ and the latter. Combining Heckman's formulas and mine for the AKZ, which coincide since the representation of is the same, we readily conclude that these equations are isomorphic for generic This will be made much more constructive below. We will also consider any
Remark 3.2. Let us apply Theorem 3.6 to the standard rational KZ equation in the case. We calculated the monodromy of Taking special and substituting we come to It corresponds to the simplest in (2.18). By the way, these induce a homomorphism from to due to Drinfeld. Diagonalizing the commuting elements we recover the monodromy computed by Tsuchiya-Kanie [TKa1988]. It also gives an explicit example of the general results on the monodromy of the rational KZ over Lie algebras due to Drinfeld and Kohno (see [Koh1987]).
Remark 3.3. In Theorems 3.5 and 3.6, we established the isomorphism where and represented it as a relation between the intertwiners of the degenerate and non-degenerate affine Hecke algebras: This construction can be naturally generalized. In fact we need only a very mild restriction on to get such a homomorphism. Normalizing the intertwiners to make them 'unitary' we come to the simplest possible map:
Actually here we have four formulas in one since we can put the denominators on the right and on the left. One of them was found by Lusztig in [Lus1989].
Here we present the isomorphism between the AKZ equation and the quantum many-body problem (QMBP). The latter will appear as a 'trace' of the first.
We will need a variant of the general notion of monodromy by A. Grothendieck. Let us fix the notations:
Given a finite union of affine real closed half-hyperplanes, we set assuming that
|(i)||does not contain 'bad hyperplanes'|
|(ii)||is simply connected,|
Let us fix a system of cutoffs and Then for each there is a path (unique up to homotopy) joining and So the choice of implies a choice of representatives in the fundamental group Here is the complement of the union of 'bad hyperplanes' (3.13).
We pick a solution of the AKZ equation in and define the monodromy function Here is invertible at least at one point and is extended analytically to the whole The values are in the endomorphisms of any finite-dimensional representation of (we will apply the construction to the induced representations).
The monodromy satisfies the following:
|(b)||and hence is locally constant.|
Next let us introduce the operators acting on functions on The relations for the operators are the same as for the permutations Note that the property a) follows from the condition for Indeed,
Let be the space of solutions of the AKZ equation with values in When we consider the AKZ equation on a finite-dimensional we will denote the space of its solutions by Starting with AKZ let us go to QMBP. In what follows, or In the latter case all operators act on functions.
|(1)||Using we rewrite the AKZ equation: Let us denote: The local invertibility of and the relations result in the commutativity Here one can use that the commutators do not contain the derivatives, which readily results from the relations for Moreover, the commutativity follows from these relations algebraically. It was proved in [Che1992] (see [Che1994-3] for a more conceptual proof based on the induced representations). It also follows from the corresponding difference theory, where this and similar statements are much simpler (and completely conceptual).|
|(2)||Since the multiplication by commutes with we get for any polynomial|
|(3)||For let us take an with the following properties:|
Let be a polynomial. Using the commutation relations (3.35), we can write where are differential operators (they do not contain They are scalar and commute with Thus Now, we assume that is Applying tr (see (3.36) and (3.37)), we come to where is a function. The differential operators are which follows from the same construction (we will reprove this algebraically below).
Let us introduce the trigonometric Dunkl operators replacing by Repeating the above construction, define for a polynomial by Since in the construction of and we use only the commutation relations (3.35) for and these operators just coincide. The trigonometric Dunkl operators are from [Che1991-2]. Dunkl introduced their rational counterparts (see also [Che1994-3] and references therein). When defining my operators I also used [Hec1991]. Heckman's 'global Dunkl operators' are sufficient to introduce QMBP, but do not commute.
We are now in a position to introduce the QMBP with the eigenvalue It is the following system of differential equations for a function It is known [HOp1987] (and easy to see by looking at the leading terms of that the dimension of the space of solutions is
Summarizing, we come to the theorem.
Theorem 3.8. Applying we get a homomorphism Here denotes the space of solutions to QMBP with the eigenvalue
We can say more for concrete represenatations, especially for the induced representations (see (3.4)). We define the 'trace' as the map dual to the embedding sending to Here denotes the ideal generated by One easily checks that tr satisfies the conditions (3.37).
Theorem 3.9 ([Che1991-4]) For any gives an isomorphism
The key observation: Indeed, if then there exists a polynomial such that However
To prove Theorem 3.9, it is enough to show the injectivity of tr, since the surjectivity will then follow by comparing the dimensions of the solution spaces (both of them are So let us suppose that for identically We will show that Differentiating (3.39), By the of Hence Differentiating this equation by we have Using the commutation relations of and we deduce from (3.39), (3.41) that Proceeding in the same way, we establish that for any Combining this with the of tr, we get (3.40).
For each consider the submodule Then and from the key observation above, we deduce that This completes the proof of Theorem 3.9.
The map from Theorem 3.9 was found by Matsuo [Mat1992] for induced representations He proved his theorem algebraically (without the passage through the trigonometric Dunkl operators discussed above) using an explicit presentation for AKZ in The isomorphism for (or for with properly ordered - (3.6)) was established independently and simultaneously by Matsuo and the author in [Che1994-3]. He proved that a certain determinant is non-zero for properly ordered I used the modules Matsuo was the first to conjecture that the QMBP (the Heckman-Opdam system) and a certain specialization of the trigonometric KZ from [Che1989] are isomorphic. The affine KZ were defined in full generality a bit later (in [Che1991-3]).
Let us give the formula for the simplest
Example 3.1. Let Then we have It was studied in [OPe1983].
Remark 3.4. More generally, let be a and As before, is the ideal generated by Then the following holds where now the right hand side means a matrix version of QMBP (sometimes it is called spin-QMBP). It was introduced in [Che1994-3] for the first time. It is a ceratin unification of the Haldane-Shastry model and that by Calogero-Sutherland.
For example, the corresponding to above reads where by we mean the image of in
Let us describe AKZ and QMBP in the case.
In §2.4, we introduced the degenerate affine Hecke algebra of type It is the algebra subject to the following relations: As in §2.4, we will use the coordinates
To prepare the passage to the difference case, we conjugate the AKZ for by the function for A The equation becomes as follows: Only in this form it can be quantized (see §4.2). The system is consistent and
The corresponding Dunkl operators are given by the formula Here stands for the transpositions of the coordinates: Similarly, means the permutation of the coordinates corresponding to
The main point of the theory is that they satisfy the relations from the degenerate Hecke algebra: It holds for any root systems. This statement is from [Che1994-3]. In these notes we will deduce these relations from the difference theory (where they are almost obvious). These relations readily give that and the corresponding are for the polynomials. Use the description of the center of to see this.
In the case of given symmetric where are scalar differential operators, Let us take the elementary symmetric polynomials: as setting Clearly The next operator is:
When we replace by the corresponding is conjugated (by to the original Sutherland operator up to a constant term [Sut1971]. For special values of the parameter these operators are the radial parts of the Laplace operators on the symmetric spaces. A particular case was considered by Koornwinder. The rational counterpart is due to Calogero. It is equivalent to a rational variant of the AKZ (an extension of the rational KZ from [Che1989] by the Here the cannot be represented as the theorem holds in terms of only (see [Che1994-3]).