Projectively Equivariant Quantizations over the Superspace $${\mathbb{R}^{p|q}}$$

Mathonet, Pierre; Radoux, Fabian

doi:10.1007/s11005-011-0474-0

Cited by 9 publications

(40 citation statements)

References 32 publications

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“…The Lie superalgebra spo(2|2) is the intersection of the Lie superalgebra K(2) and the Lie superalgebra of projective vector fields pgl(2|2) exposed in [22]. The Lie superalgebra spo(2|2) is thus a 4|4-dimensional Lie superalgebra spanned by the contact vector fields associated with the following contact Hamiltonians:…”

Section: The Lie Superalgebra Spo(2|2)mentioning

confidence: 99%

“…In order to tackle the problem of spo(2|2)-equivariant quantization, we will need to adapt the tools used in [22] for the case where g = pgl(p + 1|q). The main ingredients are the affine quantization map and the difference between the representations (S δ , L) and (S δ , L) of spo(2|2) measured by the map γ (see Section 3.2).…”

Section: Tools Used To Build the Quantizationmentioning

confidence: 99%

“…The method that we are going to use here to build the spo(2|2)-equivariant quantization is linked to the Casimir operators, as in [22]. Actually, the method is based on the comparison of the Casimir operators C and C of spo(2|2) associated with the representations L and L on S δ .…”

Section: Casimir Operatorsmentioning

confidence: 99%

“…As in [22], we define an operator which measures the difference between the Casimir operators C and C. This map will be called the operator N , as in the previous reference. From the expression of the map γ, it is easy to deduce an explicit formula for N .…”

Section: Lemma 2 the K-dual Basis Corresponding To The Basismentioning

confidence: 99%

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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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Section: The Lie Superalgebra Spo(2|2)mentioning

confidence: 99%

Section: Tools Used To Build the Quantizationmentioning

confidence: 99%

Section: Casimir Operatorsmentioning

confidence: 99%

Section: Lemma 2 the K-dual Basis Corresponding To The Basismentioning

confidence: 99%

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

Mellouli

Nibirantiza

Radoux

2013

SIGMA

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“…As in the projective setting [28], the situation depends on the superdimension of the space under consideration.…”

Section: Introductionmentioning

confidence: 99%

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

Leuther

Mathonet

Radoux

2012

Journal of Geometry and Physics

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We investigate the concept of equivariant quantization over the superspace R p+q|2r , with respect to the orthosymplectic algebra osp(p + 1, q + 1|2r). Our methods and results vary upon the superdimension p + q − 2r. When the superdimension is nonzero, we manage to obtain a result which is similar to the classical theorem of Duval, Lecomte and Ovsienko: we prove the existence and uniqueness of the equivariant quantization except in some resonant situations. To do so, we have to adapt their methods to take into account the fact that the Casimir operator of the orthosymplectic algebra on supersymmetric tensors is not always diagonalizable, when the superdimension is negative and even. When the superdimension is zero, the situation is always resonant, but we can show the existence of a one-parameter family of equivariant quantizations for symbols of degree at most two. MSC(2010) : 17B66, 58A50.

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