Rings Of Partial Differential Operators Research Articles

Geometric properties of commutative subalgebras of partial differential operators

We investigate further algebro-geometric properties of commutative rings of partial differential operators, continuing our research started in previous articles. In particular, we start to explore the simplest and also certain known examples of quantum algebraically completely integrable systems from the point of view of a recent generalization of Sato's theory, developed by the first author. We give a complete characterization of the spectral data for a class of `trivial' commutative algebras and strengthen geometric properties known earlier for a class of known examples. We also define a kind of restriction map from the moduli space of coherent sheaves with fixed Hilbert polynomial on a surface to an analogous moduli space on a divisor (both the surface and the divisor are part of the spectral data). We give several explicit examples of spectral data and corresponding algebras of commuting (completed) operators, producing as a by-product interesting examples of surfaces that are not isomorphic to spectral surfaces of any (maximal) commutative ring of partial differential operators of rank one. Finally, we prove that any commutative ring of partial differential operators whose normalization is isomorphic to the ring of polynomials is a Darboux transformation of a ring of operators with constant coefficients. Bibliography: 39 titles.

Геометрические свойства коммутативных подалгебр дифференциальных операторов в частных производных

We investigate further alebro-geometric properties of commutative rings of partial differential operators continuing our research started in previous articles. In particular, we start to explore the most evident examples and also certain known examples of algebraically integrable quantum completely integrable systems from the point of view of a recent generalization of Sato's theory which belongs to the second author. We give a complete characterisation of the spectral data for a class of "trivial" rings and strengthen geometric properties known earlier for a class of known examples. We also define a kind of a restriction map from the moduli space of coherent sheaves with fixed Hilbert polynomial on a surface to analogous moduli space on a divisor (both the surface and divisor are part of the spectral data). We give several explicit examples of spectral data and corresponding rings of commuting (completed) operators, producing as a by-product interesting examples of surfaces that are not isomorphic to spectral surfaces of any commutative ring of PDOs of rank one. At last, we prove that any commutative ring of PDOs, whose normalisation is isomorphic to the ring of polynomials $k[u,t]$, is a Darboux transformation of a ring of operators with constant coefficients.

On rings of commuting partial differential operators

We give a natural generalization of the classification of commutative rings of ordinary differential operators, given in works of Krichever, Mumford, Mulase, and determine commutative rings of operators in a completed ring of partial differential operators in two variables (satisfying certain mild conditions) in terms of Parshin's generalized geometric data. It uses a generalization of M.Sato's theory and is constructible in both ways.

Commuting differential operators and higher-dimensional algebraic varieties

Several algebro-geometric properties of commutative rings of partial differential operators (PDOs) as well as several geometric constructions are investigated. In particular, we show how to associate a geometric data by a commutative ring of PDOs, and we investigate the properties of these geometric data. This construction is in some sense similar to the construction of a formal module of Baker–Akhieser functions. On the other hand, there is a recent generalization of Sato’s theory, which belongs to the third author of this paper. We compare both approaches to the commutative rings of PDOs in two variables. As a by-product, we get several necessary conditions on geometric data describing commutative rings of PDOs.

A Constructive Study of the Module Structure of Rings of Partial Differential Operators

The purpose of this paper is to develop constructive versions of Stafford's theorems on the module structure of Weyl algebras A n (k) (i.e., the rings of partial differential operators with polynomial coefficients) over a base field k of characteristic zero. More generally, based on results of Stafford and Coutinho-Holland, we develop constructive versions of Stafford's theorems for very simple domains D. The algorithmization is based on the fact that certain inhomogeneous quadratic equations admit solutions in a very simple domain. We show how to explicitly compute a unimodular element of a finitely generated left D-module of rank at least two. This result is used to constructively decompose any finitely generated left D-module into a direct sum of a free left D-module and a left D-module of rank at most one. If the latter is torsion-free, then we explicitly show that it is isomorphic to a left ideal of D which can be generated by two elements. Then, we give an algorithm which reduces the number of generators of a finitely presented left D-module with module of relations of rank at least two. In particular, any finitely generated torsion left D-module can be generated by two elements and is the homomorphic image of a projective ideal whose construction is explicitly given. Moreover, a non-torsion but non-free left D-module of rank r can be generated by r+1 elements but no fewer. These results are implemented in the Stafford package for D=A n (k) and their system-theoretical interpretations are given within a D-module approach. Finally, we prove that the above results also hold for the ring of ordinary differential operators with either formal power series or locally convergent power series coefficients and, using a result of Caro-Levcovitz, also for the ring of partial differential operators with coefficients in the field of fractions of the ring of formal power series or of the ring of locally convergent power series.

The computation of purity filtrations over commutative noetherian rings of operators and their application to behaviors

Due to the theoretical work and computer implementations of, for instance, Barakat, Quadrat and Robertz and their coauthors the theory of finitely generated (f.g.) modules over non-commutative regular noetherian rings of partial differential operators with variable coefficients like the Weyl algebras and over other similar rings has become constructive in recent years. In particular these authors compute the purity or grade filtration of a f.g. module by homological means and discuss its significance for the associated behavior. Pommaret and Quadrat noted this significance already in 1999. In this note it is shown that over an arbitrary commutative noetherian ring of operators the purity filtration of a finitely generated module can be easily computed by means of the the primary decomposition of its zero submodule and indeed, with smaller complexity, by inductively computing equidimensional parts. In most books on Constructive Commutative Algebra the connection between this primary decomposition, the equidimensional parts and the purity filtration is implicitly stated for cyclic modules. For many commutative rings of operators the standard signal modules are injective cogenerators. In this case the purity filtration of the module gives rise to a corresponding filtration of the dual behavior, and the primary decomposition induces additional sum representations of the pure dimensional factors of this filtration. For non-commutative rings of operators the standard signal modules are in general neither injective nor cogenerators, and for such signal modules the usefulness of the purity filtration of the module for the determination of the behavior and its structural properties is not obvious. It is also shown by a counter-example that dimensional purity of the module or behavior does not imply dimensional purity of the initial conditions according to Riquier of the associated homogeneous Cauchy problem.

Stafford's Reduction of Linear Partial Differential Systems

It is well-known that linear systems theory can been studied by means of module theory. In particular, to a linear ordinary/partial differential system corresponds a finitely presented left module over a ring of ordinary/partial differential operators. The structure of modules over rings of partial differential operators was investigated in Stafford's seminal work [18]. The purpose of this paper is to make some results obtained in [18] constructive. Our results are implemented in the Maple package Stafford. Finally, we give system-theoretic interpretations of Stafford's results within the behavioural approach (e.g., minimal representations, autonomous behaviours, direct decomposition of behaviours, differential flatness).

Baker–Akhiezer Modules on the Intersections of Shifted Theta Divisors

The restriction, on the spectral variables, of the Baker-Akhiezer (BA) module of a g-dimensional principally polarized abelian variety with the non-singular theta divisor to an intersection of shifted theta divisors is studied. It is shown that the restriction to a k-dimensional variety becomes a free module over the ring of differential operators in k variables. The remaining g − k derivations define evolution equations for generators of the BA-module. As a corollary new examples of commutative ring of partial differential operators with matrix coefficients and their non-trivial evolution equations are obtained. e-mail: cho@math.kyushu-u.ac.jp e-mail: mironov@math.nsc.ru e-mail: 6vertex@math.kyushu-u.ac.jp

Factorization and Resultants of Partial Differential Operators

Comparatively little is known about commutative rings of partial differential operators, while in the ordinary case, concrete examples and an algebraic(-geometric) structure can be algorithmically determined for large classes. In this note, by the calculation of the partial μ-shifted differential resultant which we defined in a previous paper, we produce algebraic equations of spectral surfaces for commutative rings in two variables, and Darboux transformations of Airy-type operators that correspond to morphisms of surfaces. There are, however, many elementary differential-algebraic statements that we only observe experimentally, thus we offer open questions which seem to us quite significant in differential algebra, and access to Mathematica code to enable further experimentation.

Ideal intersections in rings of partial differential operators

An important step in solving linear differential equations in closed form is its factorization and generating the Loewy decomposition from it. For ordinary equations this is fairly straightforward because all operators involved generate principal ideals in the corresponding ring of operators. This is different for linear partial differential equations and the operators associated with them; its so-called non-unique factorizations have created some confusion in the past. Fundamental to Loewyʼs theory is the concept of a completely reducible operator; it is defined to be the left intersection of its irreducible constituents. Consequently a systematic investigation of the intersection ideals in the corresponding rings of differential operators is an indispensable requirement for generalizing Loewyʼs theory. The article at hand gives a thorough description of the possible intersection ideals of two first-order operators in two or three variables. Furthermore, it is shown how software provided on the website www.alltypes.de may be applied for solving concrete problems.

The Algebraic Integrability of the Quantum Toda Lattice and the Radon Transform

We study the maximal commutative ring of partial differential operators which includes the quantum completely integrable system defined by the quantum Toda lattice. Kostant shows that the image of the generalized Harish-Chandra homomorphism of the center of the enveloping algebra is commutative (Kostant in Invent. Math. 48:101–184, 1978). We demonstrate the commutativity of the ring of partial differential operators whose principal symbols are \({\bar{N}}\) -invariant. Our commutative ring includes the commutative system of Kostant (Invent. Math. 48:101–184, 1978). The main tools in this paper are Fourier integral operators and Radon transforms.

Annihilating ideals for an algebraic local cohomology class

Hypersurface isolated singularities are considered in the context of algebraic analysis. A method for computing relative Čech cohomology representations of algebraic local cohomology classes supported at the isolated singular point is described. An effective method to test membership and a method to compute the normal form for the Jacobi ideal are presented. The main purpose of this paper is to provide an effective algorithm for computing annihilating ideals, in the ring of partial differential operators, of the algebraic local cohomology class that generates the dual vector space to the local Milnor algebra.

A new characterization for the m-quasiinvariants of [formula omitted] and explicit basis for two row hook shapes

In 2002, Feigin and Veselov [M. Feigin, A.P. Veselov, Quasiinvariants of Coxeter groups and m-harmonic polynomials, Int. Math. Res. Not. 10 (2002) 521–545] defined the space of m-quasiinvariants for any Coxeter group, building on earlier work of [O.A. Chalykh, A.P. Veselov, Commutative rings of partial differential operators and Lie algebras, Comm. Math. Phys. 126 (1990) 597–611]. While many properties of those spaces were proven in [P. Etingof, V. Ginzburg, On m-quasi-invariants of a Coxeter group, Mosc. Math. J. 3 (2002) 555–566; M. Feigin, A.P. Veselov, Quasiinvariants of Coxeter groups and m-harmonic polynomials, Int. Math. Res. Not. 10 (2002) 521–545; G. Felder, A.P. Veselov, Action of Coxeter groups on m-harmonic polynomials and Knizhnik–Zamolodchikov equations, Mosc. Math. J. 4 (2003) 1269–1291; A. Garsia, N. Wallach, The non-degeneracy of the bilinear form of m-quasi-invariants, Adv. in Appl. Math. 3 (2006) 309–359. [7]] from this definition, an explicit computation of a basis was only done in certain cases. In particular, in [M. Feigin, A.P. Veselov, Quasiinvariants of Coxeter groups and m-harmonic polynomials, Int. Math. Res. Not. 10 (2002) 521–545], bases for m-quasiinvariants were computed for dihedral groups, including S 3 , and Felder and Veselov [G. Felder, A.P. Veselov, Action of Coxeter groups on m-harmonic polynomials and Knizhnik–Zamolodchikov equations, Mosc. Math. J. 4 (2003) 1269–1291] also computed the non-symmetric m-quasiinvariants of lowest degree for general S n . In this paper, we provide a new characterization of the m-quasiinvariants of S n , and use this to provide a basis for the isotypic component indexed by the partition [ n − 1 , 1 ] . This builds on a previous paper, [J. Bandlow, G. Musiker, Quasiinvariants of S 3 , J. Combin. Theory Ser. A 109 (2005) 281–298], in which we computed a basis for S 3 via combinatorial methods.

The problem of lacunas and analysis on root systems

A lacuna of a linear hyperbolic differential operator is a domain inside its propagation cone where a proper fundamental solution vanishes identically. Huygens’ principle for the classical wave equation is the simplest important example of such a phenomenon. The study of lacunas for hyperbolic equations of arbitrary order was initiated by I. G. Petrovsky (1945). Extending and clarifying his results, Atiyah, Bott and Gårding (1970–73) developed a profound and complete theory for hyperbolic operators with constant coefficients. In contrast, much less is known about lacunas for operators with variable coefficients. In the present paper we study this problem for one remarkable class of partial differential operators with singular coefficients. These operators stem from the theory of special functions in several variables related to finite root systems (Coxeter groups). The underlying algebraic structure makes it possible to extend many results of the Atiyah-Bott-Gårding theory. We give a generalization of the classical Herglotz-Petrovsky-Leray formulas expressing the fundamental solution in terms of Abelian integrals over properly constructed cycles in complex projective space. Such a representation allows us to employ the Petrovsky topological condition for testing regular (strong) lacunas for the operators under consideration. Some illustrative examples are constructed. A relation between the theory of lacunas and the problem of classification of commutative rings of partial differential operators is discussed.

D-Modules and Darboux Transformations

A method of G. Wilson for generating commutative algebras of ordinary differential operators is extended to higher dimensions. Our construction, based on the theory of D-modules, leads to a new class of examples of commutative rings of partial differential operators with rational spectral varieties. As an application, we briefly discuss their link to the bispectral problem and to the theory of lacunas.

Lacunae of Hyperbolic Riesz Kernels and Commutative Rings of Partial Differential Operators

The purpose of this Letter is to demonstrate a close connection between the problem of describing supercomplete commutative rings of partial differential operators (in the sense of Krichever–Veselov) and the theory of lacunae for hyperbolic Riesz kernels. As an application, we give a simple and explicit construction of additional quantum integrals for the generalized Calogero–Moser problem associated with a finite reflection group W and W-invariant integer root multiplicities mα. These quantum integrals are not W-invariant and cannot be constructed directly using the standard technique of Dunkl operators.

Commutative rings of partial differential operators and Lie algebras

Commutative rings of partial differential operators and Lie algebras