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See detailSECOND ORDER SYMMETRIES OF THE CONFORMAL LAPLACIAN
Michel, Jean-Philippe ULg; Radoux, Fabian ULg; Silhan, Josef

in Symmetry, Integrability and Geometry: Methods and Applications [=SIGMA] (2014), 10(016), 26

Let (M,g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by ... [more ▼]

Let (M,g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3. We determine the form of all the conformal symmetries of the conformal (or Yamabe) Laplacian on (M,g), which are given by differential operators of second order. They are constructed from conformal Killing 2-tensors satisfying a natural and conformally invariant condition. As a consequence, we get also the classification of the second order symmetries of the conformal Laplacian. Our results generalize the ones of Eastwood and Carter, which hold on conformally flat and Einstein manifolds respectively. We illustrate our results on two families of examples in dimension three. [less ▲]

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See detailspo(2|2)-equivariant quantizations on the supercircle S^{1|2}
Mellouli, Najla; Nibirantiza, Aboubacar; Radoux, Fabian ULg

in Symmetry, Integrability and Geometry: Methods and Applications [=SIGMA] (2013), 9(055), 17

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See detailOn a Lie Algebraic Characterization of Vector Bundles
Lecomte, Pierre ULg; Leuther, Thomas ULg; Zihindula Mushengezi, Elie ULg

in Symmetry, Integrability and Geometry: Methods and Applications [=SIGMA] (2012)

We prove that a vector bundle E -> M is characterized by the Lie algebra generated by all differential operators on E which are eigenvectors of the Lie derivative in the direction of the Euler vector ... [more ▼]

We prove that a vector bundle E -> M is characterized by the Lie algebra generated by all differential operators on E which are eigenvectors of the Lie derivative in the direction of the Euler vector field. Our result is of Pursell-Shanks type but it is remarkable in the sense that it is the whole f ibration that is characterized here. The proof relies on a theorem of [Lecomte P., J. Math. Pures Appl. (9) 60 (1981), 229{239] and inherits the same hypotheses. In particular, our characterization holds only for vector bundles of rank greater than 1. [less ▲]

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See detailConformally Equivariant Quantization - a Complete Classification
Michel, Jean-Philippe ULg

in Symmetry, Integrability and Geometry: Methods and Applications [=SIGMA] (2012), 8(022), 20

Conformally equivariant quantization is a peculiar map between symbols of real weight d and differential operators acting on tensor densities, whose real weights are designed by l and l+d. The existence ... [more ▼]

Conformally equivariant quantization is a peculiar map between symbols of real weight d and differential operators acting on tensor densities, whose real weights are designed by l and l+d. The existence and uniqueness of such a map has been proved by Duval, Lecomte and Ovsienko for a generic weight d. Later, Silhan has determined the critical values of d for which unique existence is lost, and conjectured that for those values of d existence is lost for a generic weight l. We fully determine the cases of existence and uniqueness of the conformally equivariant quantization in terms of the values of d and l. Namely, (i) unique existence is lost if and only if there is a nontrivial conformally invariant differential operator on the space of symbols of weight d, and (ii) in that case the conformally equivariant quantization exists only for a finite number of l, corresponding to nontrivial conformally invariant differential operators on l-densities. The assertion (i) is proved in the more general context of IFFT (or AHS) equivariant quantization. [less ▲]

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See detailNatural and Projectively Invariant Quantizations on Supermanifolds
Leuther, Thomas ULg; Radoux, Fabian ULg

in Symmetry, Integrability and Geometry: Methods and Applications [=SIGMA] (2011)

The existence of a natural and projectively invariant quantization in the sense of P. Lecomte [Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132] was proved by M. Bordemann [math.DG/0208171], using ... [more ▼]

The existence of a natural and projectively invariant quantization in the sense of P. Lecomte [Progr. Theoret. Phys. Suppl. (2001), no. 144, 125-132] was proved by M. Bordemann [math.DG/0208171], using the framework of Thomas-Whitehead connections. We extend the problem to the context of supermanifolds and adapt M. Bordemann's method in order to solve it. The obtained quantization appears as the natural globalization of the pgl(n+1|m)-equivariant quantization on Rn|m constructed by P. Mathonet and F. Radoux in [arXiv:1003.3320]. Our quantization is also a prolongation to arbitrary degree symbols of the projectively invariant quantization constructed by J. George in [arXiv:0909.5419] for symbols of degree two. [less ▲]

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