Space Of Linear Differential Operators Research Articles

English

Let $${{\cal D}_{\lambda ,\mu }}$$ be the space of linear differential operators on weighted densities from $${{\cal F}_\lambda }$$ to $${{\cal F}_\mu }$$ as module over the orthosymplectic Lie superalgebra $$\mathfrak{osp}(3\left| 2 \right.)$$ , where $${{\cal F}_\lambda }$$ , $$\lambda \in \mathbb{C}$$ is the space of tensor densities of degree λ on the supercircle S1∣3. We prove the existence and uniqueness of projectively equivariant quantization map from the space of symbols to the space of differential operators. An explicite expression of this map is also given.

Cohomology of $$\mathfrak {aff}(n|1)$$ acting on the spaces of linear differential operators on the superspace $$\mathbb {R}^{1|n}$$

We compute the first differential cohomology of the affine Lie superalgebra $$\mathfrak {aff}(n|1)$$ with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1, n)-dimensional real superspace. We also compute the same, but $$\mathfrak {aff}(n-1|1)$$-relative cohomology. We explicitly give 1-cocycles spanning these cohomology groups.

Differential operators on the supercircle S1|2 and symbol map

We consider the supercircle [Formula: see text] equipped with the standard contact structure. The conformal Lie superalgebra [Formula: see text] acts on [Formula: see text] as the Lie superalgebra of contact vector fields; it contains the M[Formula: see text]bius superalgebra [Formula: see text]. We study the space of linear differential operators on weighted densities as a module over [Formula: see text]. We introduce the canonical isomorphism between this space and the corresponding space of symbols. This result allows us to give, in contrast to the classical setting, a classification of the [Formula: see text]-modules [Formula: see text] of linear differential operators of order [Formula: see text] acting on the superspaces of weighted densities. This work is the simplest superization of a result by Gargoubi and Ovsienko [Modules of differential operators on the real line, Funct. Anal. Appl. 35(1) (2001) 13–18.]

Cohomology of 𝔞𝔣𝔣(2|1) acting on the space of linear differential operators on the superspace ℝ1|2

We compute the first cohomology of the affine Lie superalgebra [Formula: see text] on the (1,2)-dimensional real superspace with coefficients in the superspace [Formula: see text] of linear differential operators acting on weighted densities. We also compute the same, but [Formula: see text]-relative, cohomology. We explicitly give [Formula: see text]-cocycles spanning these cohomology.

Cohomology of osp(2|2) acting on spaces of linear differential operators on the superspace ℝ1|2

We compute the first differential cohomology of the orthosymplectic Lie superalgebra osp(2|2) with coefficients in the superspace of linear differential operators acting on the space of weighted densities on the (1, 2)-dimensional real superspace. We also compute the same, but osp(1|2)-relative, cohomology. We explicitly give 1-cocycles spanning these cohomologies. This work is the simplest generalization of a result from [1].

Mixed hyperbolic/trigonometric spaces for design

We investigate the class of five-dimensional null spaces of linear differential operators with constant coefficients and odd characteristic polynomials. One of the advantages of this class is that it permits to mix trigonometric and hyperbolic functions within the same space, and we will more specially focus on this interesting blending. Whenever necessary we determine the critical lengths for design. This yields the largest possible intervals on which existence of Bernstein bases is guaranteed, such bases being then automatically the optimal normalised totally positive bases. This also enables us to show the interest of this class of spaces for geometric design.

Second-Order Conformally Equivariant Quantization in Dimension 1|2

This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (su- per)dimensions 1 and 1|1. We will show that the case of several odd variables is much more difficult. We consider the supercircle S 1|2 equipped with the standard contact structure. The conformal Lie superalgebra K(2) of contact vector fields on S 1|2 contains the Lie su- peralgebra osp(2|2). We study the spaces of linear differential operators on the spaces of weighted densities as modules over osp(2|2). We prove that, in the non-resonant case, the spaces of second order differential operators are isomorphic to the corresponding spaces of symbols as osp(2|2)-modules. We also prove that the conformal equivariant quantization map is unique and calculate its explicit formula.

Differential Operators on Supercircle: Conformally Equivariant Quantization and Symbol Calculus

We consider the supercircle S 1|1 equipped with the standard contact structure. The Lie superalgebra \({\fancyscript{K}(1)}\) of contact vector fields contains the Möbius superalgebra osp(1|2). We study the space of linear differential operators on weighted densities as a module over osp(1|2). We introduce the canonical isomorphism between this space and the corresponding space of symbols and find all cases where such an isomorphism does not exist.

Cohomology of Groups of Diffeomorphisms Related to the Modules of Differential Operators on a Smooth Manifold

Let M be a manifold and T * M be the cotangent bundle. We introduce a 1-cocycle on the group of diffeomorphisms of M with values in the space of linear differential operators acting on C ∞(T * M). When M is the n-dimensional sphere, S n, we use this 1-cocycle to compute the first-cohomology group of the group of diffeomorphisms of S n, with coefficients in the space of linear differential operators acting on contravariant tensor fields.

A New Higher-Dimensional Generalization of the Schwarzian Derivative

A new formula for the multi-dimensional Schwarzian derivative on a locally projective manifold is introduced. This derivative is a 1-cocycle on the group of diffeomorphisms with values in the space of linear differential operators acting from (symmetric contravariant) twice-tensor fields to functions.

Projectively equivariant symbol calculus

The spaces of linear differential operators \(D_\lambda (\mathbb{R}^n )\) acting on λ-densities on \(\mathbb{R}^n\) and the space \({\text{Pol(}}T^* \mathbb{R}^n {\text{)}}\) of functions on \(T^* \mathbb{R}^n\) which are polynomial on the fibers are not isomorphic as modules over the Lie algebra Vect (ℝn) of vector fields of ℝn. However, these modules are isomorphic as sl(n + 1,ℝ)-modules where \({\text{sl(}}n + 1,\mathbb{R}{\text{)}} \subset {\text{Vect(}}\mathbb{R}^n {\text{)}}\) is the Lie algebra of infinitesimal projective transformations. In addition, such an \({\text{sl}}_{n + 1}\)-equivariant bijection is unique (up to normalization). This leads to a notion of projectively equivariant quantization and symbol calculus for a manifold endowed with a (flat) projective structure. We apply the \({\text{sl}}_{n + 1}\)-equivariant symbol map to study the \({\text{Vect(}}M{\text{) - modules}}D_\lambda ^k (M)\) of kth-order linear differential operators acting on λ-densities, for an arbitrary manifold M and classify the quotient-modules \(D_\lambda ^k (M)/D_\lambda ^l (M)\).

Space of Second-Order Linear Differential Operators as a Module over the Lie Algebra of Vector Fields

The space of linear differential operators on a smooth manifoldMhas a natural one-parameter family of Diff(M)- (and Vect(M)-) module structures, defined by their action on the space of tensor densities. It is shown that, in the case of second-order differential operators, the Vect(M)-module structures are equivalent, for any degree of tensor densities except for three critical values; {0,12,1}. A second-order analogue of the Lie derivative appears as an intertwining operator between the spaces of second-order differential operators on tensor densities.

Space of linear differential operators on the real line as a module over the Lie algebra of vector fields

One of the basic structures on the space of linear differential operators is a natural family of module structures over the group of diffeomorphisms Diff(M) (and of the Lie algebra of vector fields Vect(M)). These Diff(M)(and Vect(M))-module structures are defined if one considers the arguments of differential operators as tensor-densities of degree λ on M. In this paper we consider the space of differential operators on R.1 Denote by D the space of kth-order linear differential operators