Lie algebra of homogeneous operators of a vector bundle

P. B. A. Lecomte, Elie Zihindula Mushengezi

Published 2020-07-29Version 1

We prove that for a vector bundle E, the Lie algebra De(E) generated by all differential operators on E which are eigenvectors of the Lie derivative in the direction of the Euler vector field of E and the Lie algebra Dg(E) obtained by Grothendieck construction over the R-algebra A(E):= Pol(E), coincide. This allows us to compute all the derivations of A(E) and to obtain a Lie algebraic charaterization of the vector bundle E with the Lie algebra of zero-weight derivations of the R-algebra A(E).