Symmetries of Modules of Differential Operators

Gargoubi, Hichem; Mathonet, Pierre; Ovsienko, Valentin

doi:10.2991/jnmp.2005.12.3.4

Cited by 5 publications

(4 citation statements)

References 15 publications

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“…We formulate here the problem of classification of these modules (see [8,10] for the case of S 1 ), as well as existence of the exceptional weights (λ, µ) in the sense of [5]. It would also be interesting to study the corresponding automorphism groups (see [9]). …”

Section: Discussionmentioning

confidence: 99%

Differential Operators on Supercircle: Conformally Equivariant Quantization and Symbol Calculus

Gargoubi¹,

Mellouli

Ovsienko

2006

Lett Math Phys

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We consider the supercircle S 1|1 equipped with the standard contact structure. The conformal Lie superalgebra K(1) acts on S 1|1 as the Lie superalgebra of contact vector fields; it 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 interesting resonant cases where such an isomorphism does not exist.

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Section: Discussionmentioning

confidence: 99%

Differential Operators on Supercircle: Conformally Equivariant Quantization and Symbol Calculus

Gargoubi¹,

Mellouli

Ovsienko

2006

Lett Math Phys

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“…The map T is non-local, that is, for some A ∈ D n,k λ,μ (S 1|n ) vanishing in an open subset U ∈ S 1|n , the operator T (A) does not vanish on U . In the S 1 -case (see [9]), section 6.4), it has been proven that, if T :…”

Section: Local Supersymmetriesmentioning

confidence: 99%

Symmetries of modules of differential operators on the supercircle $$S^{1|n} $$

Boujelben

Safi²,

Saoudi

et al. 2021

Indian J Pure Appl Math

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Let F n λ be the space of tensor densities of degree λ ∈ C on the supercircle S 1|n . We consider the space D n,k λ,μ of k-th order linear differential operators from F n λ to F n μ as a module over the superalgebra K(n) of contact vector fields on S 1|n and we compute the superalgebra of endomrphisms on D n,k λ,μ commuting with the aff(n|1)-action where aff(n|1) is the affine subalgebra of K(n). This result allows us to determine the superalgebra of endomrphisms on D n,k λ,μ commuting with the osp(n|2)-action for n ∈ {1, 2, 3} where osp(n|2) is the orthosymlectic superalgebras of S 1|n .

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“…Differential operators on the space of densities of different weights have been under intensive study, see [8], [1], [2], [10], [3], [9], [4], [5] and the book [11] and citations therein. See also [6] and [7].…”

Section: Introductionmentioning

confidence: 99%

Operator pencil passing through a given operator

Biggs

Khudaverdian

2013

Journal of Mathematical Physics

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Let $\Delta$ be a linear differential operator acting on the space of densities of a given weight $\lo$ on a manifold $M$. One can consider a pencil of operators $\hPi(\Delta)=\{\Delta_\l\}$ passing through the operator $\Delta$ such that any $\Delta_\l$ is a linear differential operator acting on densities of weight $\l$. This pencil can be identified with a linear differential operator $\hD$ acting on the algebra of densities of all weights. The existence of an invariant scalar product in the algebra of densities implies a natural decomposition of operators, i.e. pencils of self-adjoint and anti-self-adjoint operators. We study lifting maps that are on one hand equivariant with respect to divergenceless vector fields, and, on the other hand, with values in self-adjoint or anti-self-adjoint operators. In particular we analyze the relation between these two concepts, and apply it to the study of $\diff(M)$-equivariant liftings. Finally we briefly consider the case of liftings equivariant with respect to the algebra of projective transformations and describe all regular self-adjoint and anti-self-adjoint liftings.Comment: 32 pages, LaTeX fil

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Symmetries of Modules of Differential Operators

Abstract: Let F λ (S 1 ) be the space of tensor densities of degree (or weight) λ on the circle) is a natural module over Diff(S 1 ), the diffeomorphism group of S 1 . We determine the algebra of symmetries of the modules D

Cited by 5 publications

References 15 publications

Differential Operators on Supercircle: Conformally Equivariant Quantization and Symbol Calculus

Differential Operators on Supercircle: Conformally Equivariant Quantization and Symbol Calculus

Symmetries of modules of differential operators on the supercircle $$S^{1|n} $$

Operator pencil passing through a given operator

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