Macdonald Operators Research Articles

Ruijsenaars’ commuting difference operators and invariant subspace spanned by theta functions

We study a family of mutually commutative difference operators introduced by Ruijsenaars. The conjugations of these operators with an appropriate function give the Hamiltonians of some relativistic quantum systems. These operators can be regarded as elliptic analogs of the Macdonald operators and their coefficients consist of the Jacobi theta functions. We show that these operators act on the space of meromorphic functions on the Cartan subalgebra of affine Lie algebras and that the space spanned by characters of a fixed positive level is invariant under the action of these operators.

Yangian Gelfand-Zetlin Bases, $\glN$ -Jack Polynomials and Computation of Dynamical Correlation Functions in the Spin Calogero-Sutherland Model

We consider the gl(N)-invariant Calogero-Sutherland Models with N=1,2,3,... in a unified framework, which is the framework of Symmetric Polynomials. By the framework we mean an isomorphism between the space of states of the gl(N)-invariant Calogero-Sutherland Model and the space of Symmetric Laurent Polynomials. In this framework it becomes apparent that all gl(N)-invariant Calogero-Sutherland Models are manifestations of the same entity, which is the commuting family of Macdonald Operators. Macdonald Operators depend on two parameters $q$ and $t$. The Hamiltonian of gl(N)-invariant Calogero-Sutherland Model belongs to a degeneration of this family in the limit when both $q$ and $t$ approach the N-th elementary root of unity. This is a generalization of the well-known situation in the case of Scalar Calogero-Sutherland Model (N=1). In the limit the commuting family of Macdonald Operators is identified with the maximal commutative sub-algebra in the Yangian action on the space of states of the gl(N)-invariant Calogero-Sutherland Model. The limits of Macdonald Polynomials which we call gl(N)-Jack Polynomials are eigenvectors of this sub-algebra and form Yangian Gelfand-Zetlin bases in irreducible components of the Yangian action. The gl(N)-Jack Polynomials describe the orthogonal eigenbasis of gl(N)-invariant Calogero-Sutherland Model in exactly the same way as Jack Polynomials describe the orthogonal eigenbasis of the Scalar Model (N=1). For each known property of Macdonald Polynomials there is a corresponding property of gl(N)-Jack Polynomials. As a simplest application of these properties we compute two-point Dynamical Spin-Density and Density Correlation Functions in the gl(2)-invariant Calogero-Sutherland Model at integer values of the coupling constant.

Algebraic Integrability of Macdonald Operators and Representations of Quantum Groups

In this paper we construct examples of commutative rings of difference operators with matrix coefficients from representation theory of quantum groups, generalizing the results of our previous paper [ES] to the q-deformed case. A generalized Baker–Akhiezer function Ψ is realized as a matrix character of a Verma module and is a common eigenfunction for a commutative ring of difference operators. In particular, we obtain the following result in Macdonald theory: at integer values of the Macdonald parameter k, there exist difference operators commuting with Macdonald operators which are not polynomials of Macdonald operators. This result generalizes an analogous result of Chalyh and Veselov for the case q=1, to arbitrary q. As a by-product, we prove a generalized Weyl character formula for Macdonald polynomials (= Conjecture 8.2 from [FV]), the duality for the Ψ-function, and the existence of shift operators.

Notes on the Elliptic Ruijsenaars Operators

We construct the spaces that the elliptic Ruijsenaars operators act on. It is shown that they are extensible to nonnegative selfadjoint operators on a space of square integrable functions, or preserve a finite dimensional vector space of entire functions. These facts are shown in terms of the R-operators which satisfy the Yang–lBaxter equation. The elliptic Ruijsenaars operators are considered as the elliptic analogues of the Macdonald operators or the difference analogues of the Lame operators.

Affine R-Matrix and the Generalized Elliptic Ruijsenaars Models

Commutative elliptic difference operators associated with the affine root systems are constructed in terms of affine R-matrices. These operators describe the Ruijsenaars models with elliptic potentials and reduce to the Macdonald operators in the trigonometric limit.

Quantum integrability of the generalized elliptic Ruijsenaars models

The quantum integrability of the generalized elliptic Ruijsenaars models is shown. These models are mathematically related to the Macdonald operator and the Macdonald - Koornwinder operator, which appeared in the q-orthogonal polynomial theories. We construct these integrable families by using the Yang - Baxter equation and the reflection equation.

Affine Hecke algebra, Macdonald polynomials, and quantum many-body systems

The Macdonald operators associated with the classical root systems are constructed based on the infinite-dimensional representation for solutions of the Yang - Baxter equation and the reflection equation.

Collective field theory, Calogero-Sutherland model and generalized matrix models

On the basis of the collective field method, we analyze the Calogero-Sutherland model (CSM) and the Selberg-Aomoto integral, which defines, in particular case, the partition function of the matrix models. Vertex opertor realizations for some of the eigenstates (the Jack polynomials) of the CSM Hamiltonian are obtained. We derive Virasoro constraint for the generalized matrix models and indicate relations with the CSM operators. Similar results are presented for the q-deformed case (the Macdonald operator and polynomials), which gives the generating functional of infinitely many conserved charges in the CSM.

Macdonald's operator from the center of the quantized universal enveloping algebra Uq gl( N )

Journal Article Macdonald's operator from the center of the quantized universal enveloping algebra U q gl( N ) Get access Katsuhisa Mimachi Katsuhisa Mimachi Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1994, Issue 10, 1994, Pages 415–424, https://doi.org/10.1155/S1073792894000450 Published: 01 May 1994 Article history Published: 01 May 1994 Received: 26 September 1994 Revision received: 07 October 1994

Quantum Knizhnik-Zamolodchikov equations and affine root systems

Quantum (difference) Knizhnik-Zamolodchikov equations [S1, FR] are generalized for theR-matrices from [Ch1] with the arguments in arbitrary root systems (and their formal counterparts). In particular, QKZ equations with certain boundary conditions are introducted. The self-consistency of the equations from [FR] and the cross-derivative integrability conditions for ther-matrix KZ equations from [Ch2] are obtained as corollaries. A difference counterpart of the quantum many-body problem connected with Macdonald's operators is defined as an application.

Double affine hecke algebras, knizhnik-zamolodchikov equations, and macdonald’s operators

Double affine Hecke algebras, Knizhnik-Zamolodchikov equations, and Macdonald's operators Get access Ivan Cherednik Ivan Cherednik Search for other works by this author on: Oxford Academic Google Scholar International Mathematics Research Notices, Volume 1992, Issue 9, 1992, Pages 171–180, https://doi.org/10.1155/S1073792892000199 Published: 01 May 1992 Article history Published: 01 May 1992 Received: 28 July 1992