Particle-like structure of Lie algebras. (English) Zbl 1422.17008

Summary: If a Lie algebra structure \(\mathbf{\mathfrak{g}}\) on a vector space is the sum of a family of mutually compatible Lie algebra structures \(\mathbf{\mathfrak{g}}_i\)’s, we say that \(\mathbf{\mathfrak{g}}\) is simply assembled from the \(\mathbf{\mathfrak{g}}_i\)’s. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the \(\mathbf{\mathfrak{g}}_i\)’s, one obtains a Lie algebra assembled in two steps from \(\mathbf{\mathfrak{g}}_i\)’s, and so on. We describe the process of modular disassembling of a Lie algebra into a unimodular and a non-unimodular part. We then study two inverse questions: which Lie algebras can be assembled from a given family of Lie algebras, and from which Lie algebras can a given Lie algebra be assembled. We develop some basic assembling and disassembling techniques that constitute the elements of a new approach to the general theory of Lie algebras. The main result of our theory is that any finite-dimensional Lie algebra over an algebraically closed field of characteristic zero or over \(\mathbb{R}\) can be assembled in a finite number of steps from two elementary constituents, which we call dyons and triadons. Up to an abelian summand, a dyon is a Lie algebra structure isomorphic to the non-abelian 2-dimensional Lie algebra, while a triadon is isomorphic to the 3-dimensional Heisenberg Lie algebra. As an example, we describe constructions of classical Lie algebras from triadons.{
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17B05 Structure theory for Lie algebras and superalgebras
17B81 Applications of Lie (super)algebras to physics, etc.
81R12 Groups and algebras in quantum theory and relations with integrable systems
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