On Deformations and Contractions of Lie Algebras

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2006
• Main Authors: Fialowski, A., Marc de Montigny
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2006
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/146177
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: On Deformations and Contractions of Lie Algebras / A. Fialowski, Marc de Montigny // Symmetry, Integrability and Geometry: Methods and Applications. — 2006. — Т. 2. — Бібліогр.: 14 назв. — англ.

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.