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Particle-like structure of coaxial Lie algebras. (English) Zbl 1422.17009
Summary: This paper is a natural continuation of the author [J. Math. Phys. 58, No. 7, 071703, 49 p. (2017; Zbl 1422.17008)] where we proved that any Lie algebra over an algebraically closed field or over \(\mathbb{R}\) can be assembled in a number of steps from two elementary constituents, called dyons and triadons. Here we consider the problems of the construction and classification of those Lie algebras which can be assembled in one step from base dyons and triadons, called coaxial Lie algebras. The base dyons and triadons are Lie algebra structures that have only one non-trivial structure constant in a given basis, while coaxial Lie algebras are linear combinations of pairwise compatible base dyons and triadons. We describe the maximal families of pairwise compatible base dyons and triadons called clusters, and, as a consequence, we give a complete description of the coaxial Lie algebras. The remarkable fact is that dyons and triadons in clusters are self-organised in structural groups which are surrounded by casings and linked by connectives. We discuss generalisations and applications to the theory of deformations of Lie algebras.{
©2018 American Institute of Physics}

MSC:
17B05 Structure theory for Lie algebras and superalgebras
17B30 Solvable, nilpotent (super)algebras
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