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

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

1995-08-15

arxiv.org/abs/alg-geom/9508006v1

Mathematics

Algebraic Geometry

48 pages, latex, 2 figures

Scientific paper

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.

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