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ЖУРНАЛЫ // Symmetry, Integrability and Geometry: Methods and Applications // Архив

SIGMA, 2013, том 9, 055, 17 стр. (Mi sigma838)

Эта публикация цитируется в 5 статьях

$\mathfrak{spo}(2|2)$-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. № 1175, 3038 Sfax, Tunisia
b University of Liège, Institute of Mathematics, Grande Traverse, 12-B37, B-4000 Liège, Belgium

Аннотация: 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 $\mathbb{R}^{p|q}$.

Ключевые слова: equivariant quantization; supergeometry; contact geometry; orthosymplectic Lie superalgebra.

MSC: 53D10; 17B66; 17B10

Поступила: 18 февраля 2013 г.; в окончательном варианте 15 августа 2013 г.; опубликована 23 августа 2013 г.

Язык публикации: английский

DOI: 10.3842/SIGMA.2013.055



Реферативные базы данных:
ArXiv: 1302.3727


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