The Bianchi variety

Moreno, Giovanni

doi:10.1016/j.difgeo.2010.07.004

Cited by 2 publications

(2 citation statements)

References 9 publications

(27 reference statements)

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“…See also [11] for an explicit description of the algebraic variety Lie(3) of all Lie algebra structures on a 3-dimensional vector space.…”

Section: Statement Of the Problem Simple Disassemblings And Lieonsmentioning

confidence: 99%

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Particle-like structure of Lie algebras

Vinogradov

2017

Journal of Mathematical Physics

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If a Lie algebra structure g on a vector space is the sum of a family of mutually compatible Lie algebra structures g i 's, we say that g is simply assembled from the g i 's. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the g i 's, one obtains a Lie algebra assembled in two steps from the 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 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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“…See also [11] for an explicit description of the algebraic variety Lie(3) of all Lie algebra structures on a 3-dimensional vector space.…”

Section: Statement Of the Problem Simple Disassemblings And Lieonsmentioning

confidence: 99%

“…This suggests that we use this "web structure" in order to find a description of Lie(V ). An instructive example of this kind is given in [11]. Also, a deeper understanding of the structure of Lie(V ) for algebraically closed ground fields could shed some light on the general disassembling problem.…”

Section: 2mentioning

confidence: 99%

Particle-like structure of Lie algebras

Vinogradov

2017

Journal of Mathematical Physics

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Particle-like structure of coaxial Lie algebras

Vinogradov

2018

Journal of Mathematical Physics

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This paper is a natural continuation of Vinogradov [J. Math. Phys. 58, 071703 (2017)] where we proved that any Lie algebra over an algebraically closed field or over 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.

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The Bianchi variety

Cited by 2 publications

References 9 publications

Particle-like structure of Lie algebras

Particle-like structure of Lie algebras

Particle-like structure of coaxial Lie algebras

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