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Second-Order Conformally Equivariant Quantization in Dimension $1|2$
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper is the next step of an ambitious program to develop conformally equivariant quantization on supermanifolds. This problem was considered so far in (super)dimensions 1 and $1|1$. We will show that the case of several odd variables is much more difficult. We consider the supercircle $S^{1|2}$ equipped with the standard contact structure. The conformal Lie superalgebra $\mathcal K(2)$ of contact vector fields on $S^{1|2}$ contains the Lie superalgebra $\mathrm{osp}(2|2)$. We study the spaces of linear differential operators on the spaces of weighted densities as modules over $\mathrm{osp}(2|2)$. We prove that, in the non-resonant case, the spaces of second order differential operators are isomorphic to the corresponding spaces of symbols as $\mathrm{osp}(2|2)$-modules. We also prove that the conformal equivariant quantization map is unique and calculate its explicit formula.
Mots-clés : equivariant quantization; conformal superalgebra. 
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     author = {Najla Mellouli},
     title = {Second-Order {Conformally} {Equivariant} {Quantization} in {Dimension} $1|2$},
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     url = {http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a110/}
}
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Najla Mellouli. Second-Order Conformally Equivariant Quantization in Dimension $1|2$. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a110/

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