• DOI:10.1063/1.4991657
• Corpus ID: 119316544

Particle-like structure of Lie algebras

@article{Vinogradov2017ParticlelikeSO,
  title={Particle-like structure of Lie algebras},
  author={Alexandre M. Vinogradov},
  journal={Journal of Mathematical Physics},
  year={2017},
  volume={58},
  pages={071703}
}

• A. Vinogradov
• Published 13 July 2017
• Mathematics
• Journal of Mathematical Physics

If a Lie algebra structure 𝔤 on a vector space is the sum of a family of mutually compatible Lie algebra structures 𝔤i’s, we say that 𝔤 is simply assembled from the 𝔤i’s. Repeating this procedure with a number of Lie algebras, themselves simply assembled from the 𝔤i’s, one obtains a Lie algebra assembled in two steps from 𝔤i’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 ...