
SIGMA 8 (2012), 004, 10 pages arXiv:1109.4772
http://dx.doi.org/10.3842/SIGMA.2012.004
On a Lie Algebraic Characterization of Vector Bundles
Pierre B.A. Lecomte, Thomas Leuther and Elie Zihindula Mushengezi
Institute of Mathematics, Grande Traverse 12, B4000 Liège, Belgium
Received September 23, 2011, in final form January 23, 2012; Published online January 26, 2012
Abstract
We prove that a vector bundle π: 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 PursellShanks 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), 229239] 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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References
 Gel'fand I.M., Kolmogorov A.N., On rings of continuous functions on topological
spaces, Dokl. Akad. Nauk SSSR 22 (1939), no. 1, 710.
 Grabowski J., Kotov A., Poncin N.,
Lie superalgebras of differential operators,
arXiv:1011.1804.
 Grabowski J., Kotov A., Poncin N.,
The Lie superalgebra of a supermanifold,
J. Lie Theory 20 (2010), 739749.
 Grabowski J., Poncin N.,
Automorphisms of quantum and classical Poisson algebras,
Compos. Math. 140 (2004), 511527,
math.RA/0211175.
 Grabowski J., Poncin N., Lie algebraic characterization of manifolds,
Cent. Eur. J. Math. 2 (2004), 811825,
math.DG/0310202.
 Grabowski J., Poncin N.,
On quantum and classical Poisson algebras, in Geometry and Topology of Manifolds,
Banach Center Publ., Vol. 76, Polish Acad. Sci., Warsaw, 2007, 313324,
math.DG/0510031.
 Lecomte P.,
On the infinitesimal automorphisms of a vector bundle,
J. Math. Pures Appl. (9) 60 (1981), 229239.
 Shanks M.E., Pursell L.E.,
The Lie algebra of a smooth manifold,
Proc. Amer. Math. Soc. 5 (1954), 468472.

