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$\mathfrak{spo}(2|2)$-Equivariant Quantizations on the Supercircle $S^{1|2}$
Symmetry, integrability and geometry: methods and applications, Tome 9 (2013) Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider the space of differential operators $\mathcal{D}_{\lambda\mu}$ acting between $\lambda$- and $\mu$-densities defined on $S^{1|2}$ endowed with its standard contact structure. This contact structure allows one to define a filtration on $\mathcal{D}_{\lambda\mu}$ 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 $\mathcal{D}_{\lambda\mu}$ and the associated graded space of symbols $\mathcal{S}_{\delta}$ ($\delta=\mu-\lambda$) can be considered as $\mathfrak{spo}(2|2)$-modules, where $\mathfrak{spo}(2|2)$ 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 $\mathfrak{spo}(2|2)$-modules between $\mathcal{S}_{\delta}$ and $\mathcal{D}_{\lambda\mu}$ that preserves the principal symbol (i.e.an {$\mathfrak{spo}(2|2)$-equivariant} quantization) for some values of $\delta$ 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 $\mathfrak{spo}(2|2)$-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 $\mathfrak{pgl}(p+1|q)$-equivariant quantization on $\mathbb{R}^{p|q}$.
Keywords: equivariant quantization; supergeometry; contact geometry; orthosymplectic Lie superalgebra. 
@article{SIGMA_2013_9_a54,
     author = {Najla Mellouli and Aboubacar Nibirantiza and Fabian Radoux},
     title = {$\mathfrak{spo}(2|2)${-Equivariant} {Quantizations} on the {Supercircle} $S^{1|2}$},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2013},
     volume = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a54/}
}
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Najla Mellouli; Aboubacar Nibirantiza; Fabian Radoux. $\mathfrak{spo}(2|2)$-Equivariant Quantizations on the Supercircle $S^{1|2}$. Symmetry, integrability and geometry: methods and applications, Tome 9 (2013). http://geodesic.mathdoc.fr/item/SIGMA_2013_9_a54/

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