Symbolic and Iterative Computation of Quasi-Filiform Nilpotent Lie Algebras of Dimension Nine

Mercedes Pérez,Francisco Pérez,Emilio Jiménez

doi:10.3390/sym7041788

Mercedes Pérez, Francisco Pérez + Show 1 more

Open Access

https://doi.org/10.3390/sym7041788

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Journal: Symmetry	Publication Date: Oct 1, 2015
Citations: 20	License type: CC BY 4.0

Affiliation: University of La Rioja, Universidad de Sevilla

Abstract

This paper addresses the problem of computing the family of two-filiform Lie algebra laws of dimension nine using three Lie algebra properties converted into matrix form properties: Jacobi identity, nilpotence and quasi-filiform property. The interest in this family is broad, both within the academic community and the industrial engineering community, since nilpotent Lie algebras are applied in traditional mechanical dynamic problems and current scientific disciplines. The conditions of being quasi-filiform and nilpotent are applied carefully and in several stages, and appropriate changes of the basis are achieved in an iterative and interactive process of simplification. This has been implemented by means of the development of more than thirty Maple modules. The process has led from the first family formulation, with 64 parameters and 215 constraints, to a family of 16 parameters and 17 constraints. This structure theorem permits the exhaustive classification of the quasi-filiform nilpotent Lie algebras of dimension nine with current computational methodologies.

Highlights

State of the ArtLie algebras have been used in physics in the context of symmetry groups of dynamical systems, as a powerful tool to study the underlying conservation laws [1,2]
Hamiltonian mechanics describes the state of a dynamic system with 2n variables (n coordinates and n momenta), and the other interesting observable physics quantities are functions of them
We present the development of the structure theorem of the family of laws of complex quasi-filiform Lie algebras (QFLA) of dimension nine

Summary

State of the Art

Lie algebras have been used in physics in the context of symmetry groups of dynamical systems, as a powerful tool to study the underlying conservation laws [1,2]. A description in quantum mechanics is obtained by an algebra of Hermitian operators in a Hilbert space with the bracket product as the commutator In such a case, the Heisenberg algebra arises if n is one, and the generalized Heisenberg algebra results for other values of n, since the traditional canonical variables preserve the Poisson bracket. Numerous researchers have tackled the problem of nilpotent Lie algebra classification Their studies were restricted to the filiform case, due to the difficulties arising from a nilpotence index higher than the dimension, providing a great number of parameters without restrictions among them. The present paper tackles the proof of the structure theorem of quasi-filiform Lie algebras of dimension nine.

Terminology

Structure Theorem

Concluding Remarks