spo(2|2)-Equivariant Quantizations on the Supercircle S¹|²

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2013
• Hauptverfasser: Mellouli, N., Nibirantiza, A., Radoux, F.
• Format: Artikel
• Sprache: English
• Veröffentlicht: Інститут математики НАН України 2013
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/149350
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: spo(2|2)-Equivariant Quantizations on the Supercircle S¹|² / N. Mellouli, A. Nibirantiza, F. Radoux // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 27 назв. — англ.

We consider the space of differential operators Dλμ acting between λ- and μ-densities defined on S¹|² endowed with its standard contact structure. This contact structure allows one to define a filtration on Dλμ 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 Dλμ and the associated graded space of symbols Sδ (δ=μ−λ) can be considered as spo(2|2)-modules, where spo(2|2) is the Lie superalgebra of contact projective vector fields on S¹|². We show in this paper that there is a unique isomorphism of spo(2|2)-modules between Sδ and Dλμ that preserves the principal symbol (i.e. an spo(2|2)-equivariant quantization) for some values of δ 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 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 pgl(p+1|q)-equivariant quantization on Rp|q.