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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.{
©2017 American Institute of Physics}

MSC:
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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References:
[1] Cabras, A.; Vinogradov, M. A., Extension of the Poisson bracket to differential forms and multi-vector fields, J. Geom. Phys., 9, 1, 75-100, (1992) · Zbl 0748.58008
[2] Filippov, T. V., N-ary Lie algebras, Sib. Math. J., 9, 126-140, (1985) · Zbl 0585.17002
[3] Hanlon, P.; Wachs, L. M., On Lie k-algebras, Adv. Math., 113, 206-236, (1995) · Zbl 0844.17001
[4] Humphreys, E. J., Introduction to Lie Algebras and Representation Theory, (1978), Springer-Verlag: Springer-Verlag, New York · Zbl 0447.17001
[5] Jacobson, N., Lie Algebras, (1962), Wiley International: Wiley International, New York, London · JFM 61.1044.02
[6] Kosmann-Schwarzbach, Y., Poisson manifolds, Lie algebroids, modular classes: A survey, Symmetry, Integrability Geom.: Methods Appl., 4, 30, (2008) · Zbl 1147.53067
[7] Koszul, J.-L., Crochet de Schouten-Nijenhuis et cohomologie, in the mathematical heritage of Élie Cartan (Lyon, 1984), Astérisque, 257-291, (1985)
[8] Mackenzie, K., General Theory of Lie Groupoids and Lie Algebroids, (2005), Cambridge University Press · Zbl 1078.58011
[9] Magri, F., A simple model of the integrable Hamiltonian equation, J. Math. Phys., 19, 1156-1162, (1978) · Zbl 0383.35065
[10] Marmo, G.; Vilasi, G.; Vinogradov, M. A., The local structure of n-Poisson and n-Jacobi manifolds, J. Geom. Phys., 25, 141-182, (1998) · Zbl 0978.53126
[11] Moreno, G., The Bianchi variety, Differ. Geom. Appl., 28, 6, 705-721, (2010) · Zbl 1201.53084
[12] Nestruev, J., Smooth Manifolds and Observables, (2002), Springer · Zbl 1021.58001
[13] Vinogradov, M. A., The union of the Schouten and Nijenhuis brackets, cohomology, and superdifferential operators (Russian), Mat. Zametki, 47, 6, 138-140, (1990) · Zbl 0712.58059
[14] Vinogradov, M. A., “Particle-like structure of Lie algebras II: Coaxial algebras” (unpublished). · Zbl 1422.17009
[15] Vinogradov, M. A.; Krasil’shchik, I. S., What is the Hamiltonian formalism?, Uspehi Mat. Nauk, 30, 1, 173-198, (1975); Vinogradov, M. A.; Krasil’shchik, I. S., What is the Hamiltonian formalism?, Uspehi Mat. Nauk, 30, 1, 173-198, (1975); Vinogradov, M. A.; Krasil’shchik, I. S., What is the Hamiltonian formalism?, Uspehi Mat. Nauk, 30, 1, 173-198, (1975); · Zbl 0461.35078
[16] Vinogradov, M. A.; Vinogradov, M. M., On multiple generalization of Lie algebras and Poisson manifolds, Contemp. Math., 219, 273-287, (1998) · Zbl 1074.17501
[17] Vinogradov, M. A.; Vinogradov, M. M., Graded multiple analogs of Lie algebras, Acta Appl. Math., 72, 183-197, (2002) · Zbl 1020.17003
[18] Weinstein, A., The local structure of Poisson manifolds, J. Differ. Geom., 18, 3, 523-557, (1983) · Zbl 0524.58011
[19] P_{f} is, in fact, the opposite of the usual Hamiltonian vector field but more convenient in the context of this paper.
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