Deformations and contractions of Lie algebras

We discuss the mutually opposite procedures of deformations and contractions of Lie algebras. Our main purpose is to illustrate the fact that, with appropriate combinations of both procedures, we obtain new Lie algebras. Firstly, we discuss low-dimensional Lie algebras, and these simple examples illustrate that, whereas for every contraction there exists a reverse deformation, the converse is not true in general. We point out that otherwise ordinary members of parameterized families of Lie algebras are singled out by this irreversibility of deformations and contractions. Then, we remind that global deformations of the Witt, Virasoro, and affine Kac-Moody algebras allow one to retrieve Lie algebras of Krichever-Novikov type and show that, in turn, contractions of the latter lead to new infinite dimensional Lie algebras.