Faculte Saint-Jean,
University of Alberta,
8406-91 Street,
Edmonton AB T6G 4G9,

Deformations and contractions of Lie algebras

I report recent work done in collaboration with A. Fialowski (Eotvos Univ. Budapest), where we have examined the mutually opposite procedures of deformations and contractions of Lie algebras. The main purpose is to show that, with appropriate combinations of both procedures, we obtain new interesting Lie algebras. I discuss low-dimensional Lie algebras, and illustrate that, whereas to every contraction there exists a reverse deformation, the converse is not true in general. I point out that otherwise ordinary members of parameterized families of Lie algebras are singled out by this irreversibility of deformations and contractions. Then, I mention that so-called global deformations of the Witt, Virasoro, and affine Kac-Moody algebras allow one to retrieve Lie algebras of Krichever-Novikov type. In turn, contractions of the latter lead to new infinite dimensional Lie algebras.