Deformations and contractions of Lie algebras
Abstract:
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.