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SIGMA 9 (2013), 055, 17 pages arXiv:1302.3727
https://doi.org/10.3842/SIGMA.2013.055
$\mathfrak{spo}(2|2)$-Equivariant Quantizations on the Supercircle $S^{1|2}$
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 $\mathcal{D}_{\lambda\mu}$ 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
$\mathbb{R}^{p|q}$.
Key words:
equivariant quantization; supergeometry; contact geometry; orthosymplectic Lie superalgebra.
pdf (474 kb)
tex (21 kb)
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