Topological Classifying Spaces of Lie Algebras and the Natural Completion of Contractions

Abstract

The space K^n of all n-dimensional { Lie} algebras has a natural non-Hausdorff topology k^n, which has characteristic limits, called transitions, A -> B, between distinct Lie algebras A and B. The entity of these transitions are the natural transitive completion of the well known Inonu-Wigner contractions and their partial generalizations by Saletan. Algebras containing a common ideal of codimension 1 can be characterized by homothetically normalized Jordan normal forms of one generator of their adjoint representation. For such algebras, transitions A -> B can be described by limit transitions between corresponding normal forms. The topology k^n is presented in detail for n < 5. Regarding the orientation of the algebras as vector spaces has a non-trivial effect for the corresponding topological space K^n_or: There exist both, selfdual points and pairs of dual points w.r.t. orientation reflection.