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}


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


Zbl 1422.17008
Full Text: DOI


[1] Cabras, A.; Vinogradov, A. M., Extension of the Poisson bracket to differential forms and multi-vector fields, J. Geom. Phys., 9, 1, 75-100, (1992) · Zbl 0748.58008
[2] Fialowski, A.; Fuchs, D., Construction of miniversal deformations of Lie algebras, J. Funct. Anal., 161, 76-110, (1999) · Zbl 0944.17015
[3] Filippov, V. T., N-ary Lie algebras, Sibirskii Math. J., 24, 126-140, (1985) · Zbl 0585.17002
[4] Hanlon, P.; Wachs, M. L., On Lie k-algebras, Adv. Math., 113, 206-236, (1995) · Zbl 0844.17001
[5] Kac, V. G., Infinite-Dimensional Lie Algebras, (1994), Cambridge University Press
[6] Lévy-Nahas, M., Deformation and contraction of Lie algebras, J. Math. Phys., 8, 1211-1222, (1967) · Zbl 0175.24803
[7] Marmo, G.; Vilasi, G.; Vinogradov, A. M., The local structure of n-Poisson and n-Jacobi manifolds, J. Geom. Phys., 25, 141-182, (1998) · Zbl 0978.53126
[8] Moreno, G., The Bianchi variety, Differ. Geom. Appl., 28, 6, 705-772, (2010) · Zbl 1201.53084
[9] Nijenhuis, A.; Richardson, R. W., Deformations of Lie algebra structures, J. Math. Mech., 17, 89-105, (1967) · Zbl 0166.30202
[10] Onishchik, A. L.; Vinberg, E. B., Lie Groups and Lie Algebras, (1991), Springer
[11] Vinogradov, A. M., Particle-like structure of Lie algebras, J. Math. Phys., 58, 071703, (2017) · Zbl 1422.17008
[12] Vinogradov, A. M.; Vinogradov, M. M., On multiple generalisations of Lie algebras and Poisson manifolds, Contemp. Math., 219, 273-287, (1998) · Zbl 1074.17501
[13] Weimar-Woods, E., Contractions, generalized Inönü-Wigner contractions and deformations of finite-dimensional Lie algebras, Rev. Math. Phys., 12, 1505-1529, (2000) · Zbl 1030.17002
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