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SIGMA 2 (2006), 048, 10 pages math.RT/0605091
https://doi.org/10.3842/SIGMA.2006.048
On Deformations and Contractions of Lie Algebras
Alice Fialowski a and Marc de Montigny b
a) Institute of Mathematics, Eötvös Loránd University,
Pázmány Péter sétány 1/C, H-1117, Budapest, Hungary
b) Campus Saint-Jean and Theoretical Physics Institute,
University of Alberta, 8406 - 91 Street, Edmonton, Alberta, T6C 4G9, Canada
Received February 24, 2006, in final form April 25, 2006; Published online May 03, 2006
Abstract
In this contributed presentation, we
discuss and compare the mutually
opposite procedures of deformations and contractions
of Lie algebras. We suggest
that with appropriate combinations of both procedures one may
construct new Lie algebras. We first discuss low-dimensional Lie
algebras and illustrate thereby that whereas for every contraction
there exists a reverse deformation, the converse is not true
in general. Also we note that some Lie algebras belonging to
parameterized families are singled out by the irreversibility of
deformations and contractions. After reminding that global
deformations of the Witt, Virasoro, and affine Kac-Moody algebras
allow one to retrieve Lie algebras of Krichever-Novikov type, we
contract the latter to find new infinite dimensional Lie algebras.
Key words:
Lie algebras; deformations; contractions; Kac-Moody algebras.
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