Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 004, 10 pages      arXiv:1109.4772

On a Lie Algebraic Characterization of Vector Bundles

Pierre B.A. Lecomte, Thomas Leuther and Elie Zihindula Mushengezi
Institute of Mathematics, Grande Traverse 12, B-4000 Liège, Belgium

Received September 23, 2011, in final form January 23, 2012; Published online January 26, 2012

We prove that a vector bundle π: EM 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 fibration 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.

Key words: vector bundle; algebraic characterization; Lie algebra; differential operators.

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