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 SIGMA, 2012, Volume 8, 004, 10 pages (Mi sigma681)

On a Lie algebraic characterization of vector bundles

Pierre B.A. Lecomte, Thomas Leuther, Elie Zihindula Mushengezi

Institute of Mathematics, Grande Traverse 12, B-4000 Liège, Belgium

Abstract: We prove that a vector bundle $\pi\colon E\to M$ 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 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.

Keywords: vector bundle, algebraic characterization, Lie algebra, differential operators.

DOI: https://doi.org/10.3842/SIGMA.2012.004

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ArXiv: 1109.4772
MSC: 13N10; 16S32; 17B65; 17B63
Received: September 23, 2011; in final form January 23, 2012; Published online January 26, 2012
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Citation: Pierre B.A. Lecomte, Thomas Leuther, Elie Zihindula Mushengezi, “On a Lie algebraic characterization of vector bundles”, SIGMA, 8 (2012), 004, 10 pp.

Citation in format AMSBIB
\Bibitem{LecLeuZih12} \by Pierre B.A. Lecomte, Thomas Leuther, Elie Zihindula Mushengezi \paper On a Lie algebraic characterization of vector bundles \jour SIGMA \yr 2012 \vol 8 \papernumber 004 \totalpages 10 \mathnet{http://mi.mathnet.ru/sigma681} \crossref{https://doi.org/10.3842/SIGMA.2012.004} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2892331} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000299614900001} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84881515091} 

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This publication is cited in the following articles:
1. Grabowski J., Kotov A., Poncin N., “Lie Superalgebras of Differential Operators”, J. Lie Theory, 23:1 (2013), 35–54
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