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

SIGMA 2 (2006), 048, 10 pages      math.RT/0605091

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

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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