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Emil Horozov

Publications and source records attributed to Emil Horozov.

15 recordsLinked to original sources
arXiv2024

Adelic and Rational Grassmannians for finite dimensional algebras

We develop a theory of Wilson's adelic Grassmannian ${\mathrm{Gr}}^{\mathrm{ad}}(R)$ and Segal-Wilson's rational Grasssmannian ${\mathrm{Gr}}^ {\mathrm{rat}}(R)$ associated to an arbitrary finite dimensional complex algebra $R$. We provide several equivalent descriptions of the former in terms of the indecomposable projective modules of $R$ and its primitive idempotents, and prove that it classifies the bispectral Darboux transformations of the $R$-valued exponential function. The rational Grasssmannian $ {\mathrm{Gr}}^{\mathrm{rat}}(R)$ is defined by using certain free submodules of $R(z)$ and it is proved that it can be alternatively defined via Wilson type conditions imposed in a representation theoretic settings. A canonical embedding ${\mathrm{Gr}}^{\mathrm{ad}}(R) \hookrightarrow {\mathrm{Gr}}^{\mathrm{rat}}(R)$ is constructed based on a perfect pairing between the $R$-bimodule of quasiexponentials with values in $R$ and the $R$-bimodule $R[z]$.

math.CA↗
arXiv2023

A Burchnall-Chaundy-Krichever Theory for Fractional Differential Operators

Fractional differential (and difference) operators play a role in a number of diverse settings: integrable systems, mirror symmetry, Hurwitz numbers, the Bethe ansatz equations. We prove extensions of the three major results on algebras of commuting (ordinary) differentials operators to the setting of fractional differential operators: (1) the Burchnall-Chaundy theorem that a pair of commuting differential operators is algebraically dependent, (2) the classification of maximal commutative algebras of differential operators in terms of Sato's theory and (3) the Krichever correspondence constructing those of rank 1 in an algebro-geometric way. Unlike the available proofs of the Burchnall-Chaundy theorem which use the action of one differential operator on the kernel of the other, our extension to the fractional case uses bounds on orders of fractional differential operators and growth of algebras, which also presents a new and much shorter proof of the original result. The second main theorem is achieved by developing a new tool of the spectral field of a point in Sato's Grassmannian, which carries more information than the widely used notion of spectral curve of a KP solution. Our Krichever type correspondence for fractional differential operators is based on infinite jet bundles.

math.RA↗
arXiv2019

d-orthogonal polynomials, Toda Lattice and Virasoro symmetries

The subject of this paper is a connection between d-orthogonal polynomials and the Toda lattice hierarchy. In more details we consider some polynomial systems similar to Hermite polynomials, but satisfying $d+2$-term recurrence relation, $d >1$. Any such polynomial system defines a solution of the Toda lattice hierarchy. However we impose also the condition that the polynomials are also eigenfunctions of a differential operator, i.e. a bispectral problem. This leads to a solution of the Toda lattice hierarchy, enjoying a number of special properties. In particular the corresponding tau-functions $τ_m$ satisfy the Virasoro constraints. The most spectacular feature of these tau-functions is that all of them are partition functions of matrix models. Some of them are well known matrix models - e.g. Kontsevich model, Kontsevich-Penner models, $r$-spin models, etc. A remarkable phenomenon is that the solution corresponding to $d=2$ contains two famous tau functions describing the intersection numbers on moduli spaces of compact Riemann surfaces and of open Riemann surfaces.

math-ph↗
arXiv2018

In search of higher Bochner theorem

We initiate the study of a natural generalisation of the classical Bochner-Krall problem asking which linear ordinary differential operators possess sequences of eigenpolynomials satisfying linear recurrence relations of finite length; the classical case corresponds to the 3-term recurrence relations with real coefficients subject to some extra restrictions. We formulate a general conjecture and prove it in the first non-trivial case of operators of order 3.

math-ph↗
arXiv2018

$d$-Orthogonal Analogs of Classical Orthogonal Polynomials

Classical orthogonal polynomial systems of Jacobi, Hermite and Laguerre have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a famous theorem by Bochner they are the only systems on the real line with this property. Similar results hold for the discrete orthogonal polynomials. In a recent paper we introduced a natural class of polynomial systems whose members are the eigenfunctions of a differential operator of higher order and which are orthogonal with respect to d measures, rather than one. These polynomial systems, enjoy a number of properties which make them a natural analog of the classical orthogonal polynomials. In the present paper we continue their study. The most important new properties are their hypergeometric representations which allow us to derive their generating functions and in some cases also Mehler-Heine type formulas.

math.CA↗
arXiv2017

Vector orthogonal polynomials with Bochner's property

Classical orthogonal polynomial systems of Jacobi, Hermite, Laguerre and Bessel have the property that the polynomials of each system are eigenfunctions of a second order ordinary differential operator. According to a classical theorem by Bochner they are the only systems with this property. Similarly, the polynomials of Charlier, Meixner, Kravchuk and Hahn are both orthogonal and are eigenfunctions of a suitable difference operator of second order. We recall that according to the famous theorem of Favard-Shohat, the condition of orthogonality is equivalent to the 3-term recurrence relation. Vector orthogonal polynomials (VOP) satisfy finite-term recurrence relation with more terms, according to a theorem by J. Van Iseghem and this characterizes them. Motivated by Bochner's theorem we are looking for VOP that are also eigenfunctions of a differential (difference) operator. We call these simultaneous conditions Bochner's property. The goal of this paper is to introduce methods for construction of VOP which have Bochner's property. The methods are purely algebraic and are based on automorphisms of non-commutative algebras. They also use ideas from the so called bispectral problem. Applications of the abstract methods include broad generalizations of the classical orthogonal polynomials, both continuous and discrete. Other results connect different families of VOP, including the classical ones, by linear transforms of purely algebraic origin, despite of the fact that, when interpreted analytically, they are integral transformations.

math.CA↗
arXiv2016

Noncommutative bispectral Darboux transformations

We prove a general theorem establishing the bispectrality of noncommutative Darboux transformations. It has a wide range of applications that establish bispectrality of such transformations for differential, difference and q-difference operators with values in all noncommutative algebras. All known bispectral Darboux transformations are special cases of the theorem. Using the methods of quasideterminants and the spectral theory of matrix polynomials, we explicitly classify the set of bispectral Darboux transformations from rank one differential operators and Airy operators with values in matrix algebras. These sets generalize the classical Calogero-Moser spaces and Wilson's adelic Grassmannian.

math.CA↗
arXiv2016

Automorphisms of Algebras and Bochner's Property for Vector Orthogonal Polynomials

We construct new families of vector orthogonal polynomials that have the property to be eigenfunctions of some differential operator. They are extensions of the Hermite and Laguerre polynomial systems. A third family, whose first member has been found by Y. Ben Cheikh and K. Douak is also constructed. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary vector orthogonal polynomial systems which are eigenfunctions of a differential operator into other systems of this type.

math.CA↗
arXiv2016

Automorphisms of algebras and Bochner`s property for discrete vector orthogonal polynomials

We construct new families of discrete vector orthogonal polynomials that have the property to be eigenfunctions of some difference operator. They are extensions of Charlier, Meixner and Kravchuk polynomial systems. The ideas behind our approach lie in the studies of bispectral operators. We exploit automorphisms of associative algebras which transform elementary (vector) orthogonal polynomial systems which are eigenfunctions of a difference operator into other systems of this type. While the extension of Charlier polynomilas is well known it is obtained by different methods. The extension of Meixner polynomial system is new.

math.CA↗
arXiv2008

Trigonometric Darboux transformations and Calogero-Moser matrices

We characterize in terms of Darboux transformations the spaces in the Segal-Wilson rational Grassmannian, which lead to commutative rings of differential operators having coefficients which are rational functions of e^x. The resulting subgrassmannian is parametrized in terms of trigonometric Calogero-Moser matrices.

math.QA↗
arXiv2007

The real loci of Calogero-Moser spaces, representations of rational Cherednik algebras and the Shapiro conjecture

We prove a criterion for the reality of irreducible representations of the rational Cherednik algebras H_{0,1}(S_n). This is shown to imply a criterion for the real loci of the Calogero-Moser spaces C_n in terms of the Etingof-Ginzburg finite maps Υ\colon C_n \to C^n/S_n \times C^n/S_n, recovering a result of Mikhin, Tarasov, and Varchenko [MTV2]. As a consequence we obtain a criterion for the real locus of the Wilson's adelic Grassmannian of rank one bispectral solutions of the KP hierarchy. Using Wilson's first parametrisation of the adelic Grassmannian, we give a new proof of a result of [MTV2] on real bases of spaces of quasi polynomials. The Shapiro Conjecture for Grassmannians is equivalent to a special case of our result for Calogero-Moser spaces, namely for the fibres of Υover C^n/S_n \times 0.

math.RT↗
arXiv2002

Strictly nilpotent elements and bispectral operators in the Weyl algebra

In this paper we give another characterization of the strictly nilpotent elements in the Weyl algebra, which (apart from the polynomials) turn out to be all bispectral operators with polynomial coefficients. This also allows to reformulate in terms of bispectral operators the famous conjecture, that all the endomorphisms of the Weyl algebra are automorphisms (Dixmier, Kirillov, etc).

math-ph↗
arXiv2002

Bispectral operators of prime order

The aim of this paper is to solve the bispectral problem for bispectral operators whose order is a prime number. More precisely we give a complete list of such bispectral operators. We use systematically the operator approach and in particular - Dixmier ideas on the first Weyl algebra. When the order is 2 the main theorem is exactly the result of Duistermaat-Grünbaum . On the other hand our proofs seem to be simpler.

math-ph↗
arXiv1998

Darboux Transformations of Bispectral Quantum Integrable Systems

We present an approach to higher dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the algebro-geometric definition of quantum integrability, we utilize the bispectral duality of quantum Hamiltonian systems to construct non-trivial Darboux transformations between completely integrable quantum systems. As an application, we are able to construct new quantum integrable systems as the Darboux transforms of trivial examples (such as symmetric products of one dimensional systems) or by Darboux transformation of well-known bispectral systems such as quantum Calogero-Moser.

math-ph↗