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$\mathfrak{spo}(2|2)$-Equivariant Quantizations on the Supercircle $S^{1|2}$
Najla Melloulia,
Aboubacar Nibirantizab,
Fabian Radouxb a University of Sfax, Higher Institute of Biotechnology,
Route de la Soukra km 4, B.P. ¹ 1175, 3038 Sfax, Tunisia
b University of Liège, Institute of Mathematics,
Grande Traverse, 12-B37, B-4000 Liège, Belgium
Abstract:
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.
MSC: 53D10;
17B66;
17B10 Received: February 18, 2013; in final form
August 15, 2013; Published online
August 23, 2013
Language: English
DOI:
10.3842/SIGMA.2013.055
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