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Leuther, Thomas; Mathonet, Pierre; Radoux, Fabian

doi:10.1016/j.geomphys.2011.09.003

Cited by 6 publications

(3 citation statements)

References 36 publications

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“…If we denote by ϕ the isomorphism defined by the formulae (17) and because of the form (14), we obtain (18) ϕ :…”

Section: Filtration Of Heisenberg Of D λµ (R 2l+1|n )mentioning

confidence: 99%

Heisenberg order of differential operators on the superspaces $\mathbb{R}^{2l+1|n}$

Nibirantiza

2016

Preprint

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We study in this paper, the existence of tree types of filtrations of the space D λµ (R 2l+1|n ) of differential operators on the superspaces R 2l+1|n endowed with the standard contact structure α. On this space D λµ (R 2l+1|n ), we have the first filtration called canonical and because of the existence of the contact structure on superspaces R 2l+1|n we obtain the second filtration on the space D λµ (R 2l+1|n ) called filtration of Heisenberg and thus the space D λµ (R 2l+1|n ) is therefore denoted by H λµ (R 2l+1|n ). We have also a new filtration induced on D λµ (R 2l+1|n ) by the two filtrations and it calls bifiltration. Explicitly, the space D λµ (R 2l+1|n ) of differential operators is filtered canonically by the order of its differential operators and the order is k ∈ N. When it is filtered by order of Heisenberg, the order of any differential operator is equal to d ∈ 1 2 N. This study is the generalization, in super case, of the model studied by C.H.Conley and V.Ovsienko in [3]. Finally, we show that the spo(2l + 2|n)-module structure on the space D λµ (R 2l+1|n ) of differential operators is induced on the space H λµ (R 2l+1|n ) and therefore on the associated space S δ (R 2l+1|n ) of normal symbols, on the space P δ (R 2l+1|n ) of symbols of Heisenberg and on the space of fine symbol Σ δ (R 2l+1|n ).

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“…If we denote by ϕ the isomorphism defined by the formulae (17) and because of the form (14), we obtain (18) ϕ :…”

Section: Filtration Of Heisenberg Of D λµ (R 2l+1|n )mentioning

confidence: 99%

Heisenberg order of differential operators on the superspaces $\mathbb{R}^{2l+1|n}$

Nibirantiza

2016

Preprint

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“…The results of [15] were generalized in many directions and recently several papers dealt with the problem of equivariant quantizations in the context of supergeometry. For example, the thesis [9] dealt with conformally equivariant quantizations over supercotangent bundles, the papers [12] and [18] exposed and solved respectively the problems of the pgl(p + 1|q)-equivariant quantization over R p|q and of the osp(p + 1, q + 1|2r)equivariant quantization over R p+q|2r , whereas in [19], the authors define the problem of the natural and projectively invariant quantization on arbitrary supermanifolds and show the existence of such a map. Another type of equivariant quantization was studied over the super circles S 1|1 and S 1|2 endowed with the canonical contact structures in [7,11,10].…”

Section: Introductionmentioning

confidence: 99%

The fine $\spo(2|n)$-equivariant quantizations on the super circles $S^{1|n}$

Nibirantiza¹

2016

Preprint

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In this paper, we generalize the known results on the super circles S 1|1 and S 1|2 . We construct the fine equivariant quantization on the super circle S 1|n for n 3. The equivariant Lie superalgebra is spo(2|n) which is constituted of the contact projective vector fields on S 1|n . In order to construct the fine equivariant quantization on S 1|n , we use the model developed, in the purely even case, by Charles H. Conley and Valentin Ovsienko in [Linear Differential Operators on Contact manifolds, www.arxiv:math-Ph/ 1205.6562v1,24p, 2012]. We also use the technical of Casimir operators to prove the uniqueness of the fine quantization on S 1|n . The technical of Casimir operators used here is the same as the one used by P. Mathonet and F. Radoux in [Lett. Math. Phys. 98 (2011),311-331 ] to prove the existence of a pgl(p + 1|q)-equivariant quantization on R p|q .

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“…Recently, several papers dealt with the problem of equivariant quantizations in the context of supergeometry: the thesis [24] dealt with conformally equivariant quantizations over super-cotangent bundles, the papers [22] and [16] exposed and solved respectively the problems of the pgl(p + 1|q)-equivariant quantization over R p|q and of the osp(p + 1, q + 1|2r)-equivariant quantization over R p+q|2r , whereas in [17], the authors define the problem of the natural and projectively invariant quantization on arbitrary supermanifolds and show the existence of such a map.…”

Section: Introductionmentioning

confidence: 99%

spo(2|2) -Equivariant Quantizations on the Supercircle S^1|2

Mellouli

Nibirantiza

Radoux

2013

SIGMA

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Abstract. We consider the space of differential operators D λµ acting between λ-and µ-densities defined on S 1|2 endowed with its standard contact structure. This contact structure allows one to define a filtration on D λµ which is finer than the classical one, obtained by writting a differential operator in terms of the partial derivatives with respect to the different coordinates. The space D λµ and the associated graded space of symbols S δ (δ = µ−λ) can be considered as spo(2|2)-modules, where spo(2|2) is the Lie superalgebra of contact projective vector fields on S 1|2 . We show in this paper that there is a unique isomorphism of spo(2|2)-modules between S δ and D λµ that preserves the principal symbol (i.e. an spo(2|2)-equivariant quantization) for some values of δ called non-critical values. Moreover, we give an explicit formula for this isomorphism, extending in this way the results of [Mellouli N., SIGMA 5 (2009), 111, 11 pages] which were established for second-order differential operators. The method used here to build the spo(2|2)-equivariant quantization is the same as the one used in [Mathonet P., Radoux F., Lett. Math. Phys. 98 (2011), 311-331] to prove the existence of a pgl(p + 1|q)-equivariant quantization on R p|q .

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On osp(p+1,q+1|2r)-equivariant quantizations

Cited by 6 publications

References 36 publications

Heisenberg order of differential operators on the superspaces $\mathbb{R}^{2l+1|n}$

Heisenberg order of differential operators on the superspaces $\mathbb{R}^{2l+1|n}$

The fine $\spo(2|n)$-equivariant quantizations on the super circles $S^{1|n}$

spo(2|2) -Equivariant Quantizations on the Supercircle S^1|2

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On osp(p+1,q+1|2r)-equivariant quantizations

Cited by 6 publications

References 36 publications

Heisenberg order of differential operators on the superspaces $\mathbb{R}^{2l+1|n}$

Heisenberg order of differential operators on the superspaces $\mathbb{R}^{2l+1|n}$

The fine $\spo(2|n)$-equivariant quantizations on the super circles $S^{1|n}$

spo(2|2) -Equivariant Quantizations on the Supercircle S1|2

Contact Info

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About

spo(2|2) -Equivariant Quantizations on the Supercircle S^1|2