Remarques a propos de l'operateur de Dirac cubique

TL;DR

This paper discusses properties of Kostant's Dirac operator associated with complex Lie algebras, providing a new proof of a formula relating its square to algebraic invariants, using cohomological methods.

Contribution

It offers a novel proof of Kostant's Dirac operator formula, simplifying the argument through induction and Lie algebra cohomology insights.

Findings

The square of the Dirac operator satisfies a generalized Parthasarathy formula.

The vanishing of certain terms is explained via Lie algebra cohomology.

The scalar nature of the cubic term's square is established using algebraic identities.

Abstract

Remarks on the Kostant Dirac operator In 1999, Kostant [Kos99] indroduces 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 non degenerate nsymetric bilinear form B, and h is a Lie subalgebra of g such that the bilinear form B is non degenerate on h. Kostant then shows that the square of this operator safisties a formula that generalizes the so-called Parthasarathy formula [Par72]. 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 ordrer term in the Kostant formula for D2_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.