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Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 9 (2013), 055, 17 pages      arXiv:1302.3727      https://doi.org/10.3842/SIGMA.2013.055

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

Najla Mellouli a, Aboubacar Nibirantiza b and Fabian Radoux b
a) University of Sfax, Higher Institute of Biotechnology, Route de la Soukra km 4, B.P. no 1175, 3038 Sfax, Tunisia
b) University of Liège, Institute of Mathematics, Grande Traverse, 12 - B37, B-4000 Liège, Belgium

Received February 18, 2013, in final form August 15, 2013; Published online August 23, 2013

Abstract
We consider the space of differential operators Dλμ acting between λ- and μ-densities defined on S1|2 endowed with its standard contact structure. This contact structure allows one to define a filtration on Dλμ 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 Dλμ and the associated graded space of symbols Sδ (δ=μλ) can be considered as spo(2|2)-modules, where spo(2|2) is the Lie superalgebra of contact projective vector fields on S1|2. We show in this paper that there is a unique isomorphism of spo(2|2)-modules between Sδ and Dλμ 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 Rp|q.

Key words: equivariant quantization; supergeometry; contact geometry; orthosymplectic Lie superalgebra.

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