Algebra Of Differential Operators Research Articles

Ladder relations for a class of matrix valued orthogonal polynomials

AbstractUsing the theory introduced by Casper and Yakimov, we investigate the structure of algebras of differential and difference operators acting on matrix valued orthogonal polynomials (MVOPs) on, and we derive algebraic and differential relations for these MVOPs. A particular case of importance is that of MVOPs with respect to a matrix weight of the formon the real line, whereis a scalar polynomial of even degree with positive leading coefficient andis a constant matrix.

Reductions of the strict KP hierarchy

Let $$R$$ be a commutative complex algebra and $$ \partial $$ be a $$ \mathbb{C} $$ -linear derivation of $$R$$ such that all powers of $$ \partial $$ are $$R$$ -linearly independent. Let $$R[ \partial ]$$ be the algebra of differential operators in $$ \partial $$ with coefficients in $$R$$ and $$ P{\kern-1.5pt}sd $$ be its extension by the pseudodifferential operators in $$ \partial $$ with coefficients in $$R$$ . In the algebra $$R[ \partial ]$$ , we seek monic differential operators $$ \mathbf{M} _n$$ of order $$n\ge2$$ without a constant term satisfying a system of Lax equations determined by the decomposition of $$ P{\kern-1.5pt}sd $$ into a direct sum of two Lie algebras that lies at the basis of the strict KP hierarchy. Because this set of Lax equations is an analogue for this decomposition of the $$n$$ -KdV hierarchy, we call it the strict $$n$$ -KdV hierarchy. The system has a minimal realization, which allows showing that it has homogeneity properties. Moreover, we show that the system is compatible, i.e., the strict differential parts of the powers of $$M=( \mathbf{M} _n)^{1/n}$$ satisfy zero-curvature conditions, which suffice for obtaining the Lax equations for $$ \mathbf{M} _n$$ and, in particular, for proving that the $$n$$ th root $$M$$ of $$ \mathbf{M} _n$$ is a solution of the strict KP theory if and only if $$ \mathbf{M} _n$$ is a solution of the strict $$n$$ -KdV hierarchy. We characterize the place of solutions of the strict $$n$$ -KdV hierarchy among previously known solutions of the strict KP hierarchy.

Lie algebras of matrix difference differential operators and special matrix functions

We discuss certain models of irreducible representations of Lie algebra sl(2,C) from the special matrix functions point of view. These models are constructed in terms of matrix differential operators and matrix difference differential operators and are connected through a matrix integral transformation. In the process, we find new matrix function identities involving one and two variable special matrix functions.

Logarithmic modules for chiral differential operators of nilmanifolds

We describe explicitly the vertex algebra of (twisted) chiral differential operators on certain nilmanifolds and construct their logarithmic modules. This is achieved by generalizing the construction of vertex operators in terms of exponentiated scalar fields to Jacobi theta functions naturally appearing in these nilmanifolds. This construction furnishes non-trivial examples of logarithmic vertex algebra modules, a notion recently introduced by Bakalov.

Lie–Rinehart and Hochschild cohomology for algebras of differential operators

Let (S,L) be a Lie–Rinehart algebra such that L is S-projective and let U be its universal enveloping algebra. In this paper we present a spectral sequence which converges to the Hochschild cohomology of U with values on a U-bimodule M and whose second page involves the Lie–Rinehart cohomology of the algebra and the Hochschild cohomology of S with values on M. After giving a convenient description of the involved algebraic structures we use the spectral sequence to compute explicitly the Hochschild cohomology of the algebra of differential operators tangent to a central arrangement of three lines.

Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?

The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands made a proposal for developing an analytic theory of automorphic forms on the moduli space of $G$-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of some analogues of Hecke operators. In these notes I show that if $G$ is an abelian group then there are well-defined Hecke operators, and I give a complete description of their eigenfunctions and eigenvalues. For non-abelian $G$, Hecke operators involve integration, which presents some difficulties. However, there is an alternative approach to developing an analytic theory of automorphic forms, based on the existence of a large commutative algebra of global differential operators acting on half-densities on the moduli stack of $G$-bundles. This approach (which implements some ideas of Joerg Teschner) is outlined here, as a preview of a joint work with Pavel Etingof and David Kazhdan.

Intertwining Operators Associated with Dihedral Groups

The Dunkl operators associated to a dihedral group are a pair of differential-difference operators that generate a commutative algebra acting on differentiable functions in $\mathbb{R}^2$. The intertwining operator intertwines between this algebra and the algebra of differential operators. The main result of this paper is an integral representation of the intertwining operator on a class of functions. As an application, closed formulas for the Poisson kernels of $h$-harmonics and sieved Gegenbauer polynomials are deduced when one of the variables is at vertices of a regular polygon, and similar formulas are also derived for several other related families of orthogonal polynomials.

Lie Subalgebras of Differential Operators in One Variable

Let $$\mathrm{Witt}$$ be the Lie algebra generated by the set $$\{L_i\,\vert \, i \in {{\mathbb {Z}}}\}$$ and $$\mathrm{Vir}$$ its universal central extension. Let $$\mathrm{Diff}(V)$$ be the Lie algebra of differential operators on $$V={{\mathbb {C}}}(\!(z)\!)$$, or $$V={{\mathbb {C}}}(z)$$. We explicitly describe all Lie algebra homomorphisms from $$\mathfrak {sl}(2)$$, $$\mathrm{Witt}$$, and $$\mathrm{Vir}$$ to $$\mathrm{Diff}(V)$$, such that $$L_0$$ acts on V as a first-order differential operator.

Regularity of twisted spectral triples and pseudodifferential calculi

Motivated by examples coming from the theory of quantum groups, we investigate the regularity condition for twisted spectral triples. This condition is equivalent to the existence of an appropriate pseudodifferential calculus associated to the spectral triple. A natural approach to obtain such a calculus is to start with a twisted algebra of abstract differential operators. Under an appropriate algebraic condition on the twisting, we obtain a pseudodifferential calculus which admits an asymptotic expansion, similarly to the untwisted case. We discuss some examples coming from the theory of quantum groups. Finally we discuss zeta functions and the residue (twisted) traces on differential operators.

Algebras of invariant differential operators

We prove that an invariant subalgebra AnW of the Weyl algebra An is a Galois order over an adequate commutative subalgebra Γ when W is a two-parameters irreducible unitary reflection group G(m,1,n), m≥1, n≥1, including the Weyl group of type Bn, or alternating group An, or the product of n copies of a cyclic group of fixed finite order. Earlier this was established for the symmetric group in [15]. In each of the cases above, except for the alternating groups, we show that AnW is free as a right (left) Γ-module. Similar results are established for the algebra of W-invariant differential operators on the n-dimensional torus where W is a symmetric group Sn or orthogonal group of type Bn or Dn. As an application of our technique we prove the quantum Gelfand-Kirillov conjecture for Uq(sl2), the first Witten deformation and the Woronowicz deformation.

Darboux transformations from the Appell-Lauricella operator

We define two isomorphic algebras of differential operators: the first algebra consists of ordinary differential operators and contains the hypergeometric differential operator, while the second one consists of partial differential operators in d variables and contains the Appell-Lauricella partial differential operator. Using this isomorphism, we construct partial differential operators which are Darboux transformations from polynomials of the Appell-Lauricella operator. We show that these operators can be embedded into commutative algebras of partial differential operators, containing d mutually commuting and algebraically independent partial differential operators, which can be considered as quantum completely integrable systems. Moreover, these algebras can be simultaneously diagonalized on the space of polynomials leading to extensions of the Jacobi polynomials orthogonal with respect to the Dirichlet distribution on the simplex.

KP-KdV Hierarchy and Pseudo-Differential Operators

The study of KP-KdV equations are the archetype of integrable systems and are one of the most fundamental equations of soliton phenomena and a topic of active mathematical research. Our purpose here is to give a motivated and a sketchy overview of this interesting subject. One of the objectives of this paper is to study the KdV equation and the inverse scattering method (based on Schrödinger and Gelfand-Levitan equations) used to solve it exactly. We study some generalities on the algebra of infinite order differential operators. The algebras of Virasoro and Heisenberg, nonlinear evolution equations such as the KdV, Boussinesq and KP play a crucial role in this study. We make a careful study of some connection between pseudo-differential operators, symplectic structures, KP hierarchy and tau functions based on the Sato-Date-Jimbo-Miwa-Kashiwara theory. A few other connections and ideas concerning the KdV and Boussinesq equations, the Gelfand-Dickey flows, the Heisenberg and Virasoro algebras are given.

Green's functions of recurrence relations with reflection

In this work we develop an algebraic theory of linear recurrence equations and systems with constant coefficients and reflection. We obtain explicit solutions and the Green's functions associated to different problems under general linear boundary conditions. Furthermore, we establish different relations between the algebras of recurrence and differential operators, showing the similarities and differences between them.

Quantum-Classical Correspondence on Associated Vector Bundles Over Locally Symmetric Spaces

Abstract For a compact Riemannian locally symmetric space $\mathcal M$ of rank 1 and an associated vector bundle $\mathbf V_{\tau }$ over the unit cosphere bundle $S^{\ast }\mathcal M$, we give a precise description of those classical (Pollicott–Ruelle) resonant states on $\mathbf V_{\tau }$ that vanish under covariant derivatives in the Anosov-unstable directions of the chaotic geodesic flow on $S^{\ast }\mathcal M$. In particular, we show that they are isomorphically mapped by natural pushforwards into generalized common eigenspaces of the algebra of invariant differential operators $D(G,\sigma )$ on compatible associated vector bundles $\mathbf W_{\sigma }$ over $\mathcal M$. As a consequence of this description, we obtain an exact band structure of the Pollicott–Ruelle spectrum. Further, under some mild assumptions on the representations $\tau$ and $\sigma$ defining the bundles $\mathbf V_{\tau }$ and $\mathbf W_{\sigma }$, we obtain a very explicit description of the generalized common eigenspaces. This allows us to relate classical Pollicott–Ruelle resonances to quantum eigenvalues of a Laplacian in a suitable Hilbert space of sections of $\mathbf W_{\sigma }$. Our methods of proof are based on representation theory and Lie theory.

Bispectrality of Charlier type polynomials

ABSTRACTGiven a finite set of positive integers G and polynomials , , with degree of equal to g, we associate to them a sequence of Charlier type polynomials defined from the Charlier polynomials by using certain Casoratian determinants (whose entries are the polynomials ). Charlier type polynomials are eigenfunctions of higher order difference operators. When , the polynomials are also orthogonal, and then satisfy a three term recurrente relation. In this paper, we prove that whatever the polynomials 's are, the Charlier type polynomials always satisfy higher order recurrence relations. We also introduce and characterize the algebra of difference operators associated to the recurrence relations satisfied by the sequence . Our characterization is constructive and surprisingly simple. As a consequence, we prove that is, essentially, the unique choice such that the polynomials are orthogonal with respect to a measure.

On the p-supports of a holonomic mathcal {D}-module

For a smooth variety Y over a perfect field of positive characteristic, the sheaf D_Y of crystalline differential operators on Y (also called the sheaf of PD-differential operators) is known to be an Azumaya algebra over T^*_{Y'}, the cotangent space of the Frobenius twist Y' of Y. Thus to a sheaf of modules M over D_Y one can assign a closed subvariety of T^*_{Y'}, called the p-support, namely the support of M seen as a sheaf on T^*_{Y'}. We study here the family of p-supports assigned to the reductions modulo primes p of a holonomic mathcal {D}-module. We prove that the Azumaya algebra of differential operators splits on the regular locus of the p-support and that the p-support is a Lagrangian subvariety of the cotangent space, for p large enough. The latter was conjectured by Kontsevich. Our approach also provides a new proof of the involutivity of the singular support of a holonomic mathcal {D}-module, by reduction modulo p.

Spherical analysis on homogeneous vector bundles

Given a Lie group G, a compact subgroup K and a representation τ∈Kˆ, we assume that the algebra of End(Vτ)-valued, bi-τ-equivariant, integrable functions on G is commutative. We present the basic facts of the related spherical analysis, putting particular emphasis on the rôle of the algebra of G-invariant differential operators on the homogeneous bundle Eτ over G/K. In particular, we observe that, under the above assumptions, (G,K) is a Gelfand pair and show that the Gelfand spectrum for the triple (G,K,τ) admits homeomorphic embeddings in some Cn.In the second part, we develop in greater detail the spherical analysis for G=K⋉H with H nilpotent. In particular, for H=Rn and K⊂SO(n) and for H equal to the Heisenberg group Hn and K⊂U(n), we characterize the representations τ∈Kˆ giving a commutative algebra.

Integrating quantum groups over surfaces

We apply the mechanism of factorization homology to construct and compute category-valued two-dimensional topological field theories associated to braided tensor categories, generalizing the (0,1,2)-dimensional part of Crane–Yetter–Kauffman four-dimensional TFTs associated to modular categories. Starting from modules for the Drinfeld–Jimbo quantum group U q ( g ) we obtain in this way an aspect of topologically twisted four-dimensional N = 4 super Yang–Mills theory, the setting introduced by Kapustin–Witten for the geometric Langlands program. For punctured surfaces, in particular, we produce explicit categories which quantize character varieties (moduli of G-local systems) on the surface; these give uniform constructions of a variety of well-known algebras in quantum group theory. From the annulus, we recover the reflection equation algebra associated to U q ( g ) , and from the punctured torus we recover the algebra of quantum differential operators associated to U q ( g ) . From an arbitrary surface we recover Alekseev's moduli algebras. Our construction gives an intrinsically topological explanation for well-known mapping class group symmetries and braid group actions associated to these algebras, in particular the elliptic modular symmetry (difference Fourier transform) of quantum D-modules.

Surprising examples of nonrational smooth spectral surfaces

In this paper we study necessary and sufficient algebro-geometric conditions for the existence of a nontrivial commutative subalgebra of rank in , a completion of the algebra of partial differential operators in two variables, which can be thought of as a simple algebraic analogue of the algebra of analytic pseudodifferential operators on a manifold. These are conditions on a projective (spectral) surface; they are encoded in a new notion of pre-spectral data. For smooth surfaces the sufficient conditions look especially simple. On a smooth projective surface there should exist 1) an ample integral curve with and ; 2) a divisor with , , , and . Amazingly, there are examples of such surfaces for which the corresponding commutative subalgebras do not admit isospectral deformations. Bibliography: 45 titles.

On divisors of small canonical degree on Godeaux surfaces

Pre-spectral data coding the rank-1 commutative subalgebras of a certain completion of the algebra of differential operators , where is an algebraically closed field of characteristic 0, are shown to exist. Here is a Godeaux surface, is an effective ample divisor represented by a smooth curve, and is a divisor on satisfying the conditions , for and . Bibliography: 26 titles.