Sliced Extensions, Irreducible Extensions, and Associated Graphs: An Analysis of Lie Algebra Extensions. I. General Theory

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Abstract
The Lie algebra extension problem is investigated and incorporated into a formalism which allows for a visualization of the algebraic structures involved. A finer analysis than the usual mathematical one is sometimes required for the physical applications. This brings about the consideration of sliced extensions, i.e., of extensions provided with sections. Some of these, the ω‐sliced extensions, are particularly interesting. They are directly connected with natural Levi decompositions of the Lie algebras obtained from extensions. Graphs are associated with ω‐sliced extensions. This is especially suitable for the study of irreducible extensions, which are the basic ones among the extensions. Their structure may be described in terms of graph theory. A subset is picked out from the set of all irreducible extensions of an arbitrary Lie algebra. Its elements, the primitive extensions, have the simplest extension structure and are characterized by the empty graph or by one‐vertex graphs.

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