Symbolic computation of optimal systems of subalgebras of three- and four-dimensional real Lie algebras

TL;DR

This paper presents a computational method using Wolfram Mathematica to classify all optimal systems of subalgebras in three- and four-dimensional real Lie algebras, confirming previous theoretical results efficiently.

Contribution

It introduces a symbolic computation approach with a specialized program to analyze subalgebra systems, streamlining classification tasks in Lie algebra theory.

Findings

The program efficiently reproduces known classifications.

Results validate the computational approach against classical classifications.

Analysis confirms the completeness of the optimal systems obtained.

Abstract

The complete optimal systems of subalgebras of all nonisomorphic three- and four-dimensional real Lie algebras are analyzed by the program \symbolie running in the computer algebra system \emph{Wolfram Mathematica}\texttrademark. The approach uses the definition of $p$-families of Lie subalgebras whose set can be partitioned by introducing a binary relation (reflexive and transitive, though not necessarily symmetric) induced by inner automorphisms of the Lie algebra. The results, produced in a few minutes by \symbolie, represent a good test for the program; in fact, except for minor differences that are discussed, the results confirm those given in 1977 in a paper by Patera and Winternitz.