i n f o a r t i c l e r é s u m é Historique de l'article : Reçu le 21 juin 2010 Accepté après révision le 27 octobre 2010 Disponible sur Internet le 19 novembre 2010 Présenté par Michel Duflo En 1999, Kostant introduit un opérateur de Dirac cubique D g/h associé à tout triplet (g, h, B), où g est une algèbre de Lie complexe munie de la forme bilinéaire symétrique ad g-invariante non dégénérée B, et h est une sous-algèbre de Lie de g sur laquelle B est non dégénérée. Kostant montre alors que le carré de D g/h vérifie une formule qui généralise la formule de Parthasarathy (1972). Nous donnons ici une nouvelle démonstration de cette formule. Tout d'abord, au moyen d'une induction par étage, nous montrons qu'il suffit d'établir la formule dans le cas particulier où h = 0. Il apparaît alorsque, dans ce cas, l'annulation du terme d'ordre 1 dans la formule de Kostant pour D 2 g/h est une conséquence de propriétés classiques en cohomologie des algèbres de Lie, tandis que le fait que le carré du terme cubique soit scalaire résulte de telles considérations, ainsi que de l'identité de Jacobi.
a b s t r a c tIn 1999, Kostant introduces a Dirac operator D g/h associated to any triple (g, h, B), where g is a complex Lie algebra provided with an ad g-invariant nondegenerate symmetric bilinear form B, and h is a Lie subalgebra of g such that the bilinear form B is nondegenerate on h. Kostant then shows that the square of this operator satisfies a formula that generalizes the so-called Parthasarathy formula (1972). We give here a new proof of this formula. First we use an induction by stage argument to reduce the proof of the formula to the particular case where h = 0. In this case we show that the vanishing of the first order term in the Kostant formula for D 2 g/h is a consequence of classic properties related to Lie algebra cohomology, and the fact that the square of the cubic term is a scalar follows from such considerations, together with the Jacobi identity.
We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in order to study N -body type Hamiltonians with interactions. More precisely, let Y ⊂ X be a linear subspace of a finite dimensional Euclidean space X, and v Y be a continuous function on X/Y that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the formwhere the subspaces Y ⊂ X belong to some given family S of subspaces. Georgescu and Nistor have considered the case when S consists of all subspaces Y ⊂ X, and Nistor and the authors considered the case when S is a finite semi lattice and Georgescu generalized these results to any families. In this paper, we develop new techniques to prove their results on the spectral theory of the Hamiltonian to the case where S is any family of subspaces also, and extend those results to other operators affiliated to a larger algebra of pseudo-differential operators associated to the action of X introduced by Connes. In addition, we exhibit Fredholm conditions for such elliptic operators. We also note that the algebras we consider answer a question of Melrose and Singer.
We show that the image of the Poisson map, defined by Mehdi and Zierau in [MZ14b], which intertwines principal series representations with a submodule of the kernel of the cubic Dirac operator, commutes with the translation functor. As a byproduct, we obtain a systematic geometric process which produces interacting Weyl fermions with a fixed energy level on homogeneous spacetimes.In honor of Professor Jean LudwigIt is a first order Lorentz-invariant differential operator acting on sections of the spin bundle S over Minkowski spacetime. The spin bundle splits into half-spin bundles S + and S − so that
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