Classification of real Lie algebras of dimension five
Abstract
We summarize a coordinate-free classification of five-dimensional real Lie algebras up to isomorphism. Rather than solving the overdetermined quadratic Jacobi system modulo GL(5,ℝ), the method uses Levi--Malcev theory and nilradical stratification. In the non-solvable case, the Levi factor is three-dimensional, so(3) or sl(2,ℝ), acting on a two-dimensional radical; this yields exactly five classes, including the indecomposable sl(2,ℝ)ltimesℝ2. In the solvable case, Schur's bound on commutative subalgebras of gl(2,ℝ) implies dimn∈{3,4,5}, eliminating one- and two-dimensional nilradicals. The nilpotent case dimn=5 has nine classes: three decomposable and six indecomposable, including h5 and two filiform algebras. For dimn=4, classification reduces to normal forms of outer derivations for ℝ4, h3⊕ℝ, and n4 under automorphism and scaling. For dimn=3, it uses two-dimensional abelian subalgebras of OutDer(n) containing no nonzero nilpotent operators. The work also computes minimal faithful representation dimensions μ(g), gives a dictionary with classical tables, and designs a deterministic identification algorithm.
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BibTeX
@misc{nadjafikhah2026classification,
title = {{Classification of real Lie algebras of dimension five}},
author = {Mehdi Nadjafikhah},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06971},
eprint = {2611.06971},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06971},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of real Lie algebras of dimension five
AU - Mehdi Nadjafikhah
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06971
UR - https://arxiv.org/abs/2611.06971
AB - We summarize a coordinate-free classification of five-dimensional real Lie algebras up to isomorphism. Rather than solving the overdetermined quadratic Jacobi system modulo GL(5,ℝ), the method uses Levi--Malcev theory and nilradical stratification. In the non-solvable case, the Levi factor is three-dimensional, so(3) or sl(2,ℝ), acting on a two-dimensional radical; this yields exactly five classes, including the indecomposable sl(2,ℝ)ltimesℝ^2. In the solvable case, Schur's bound on commutative subalgebras of gl(2,ℝ) implies dimn∈{3,4,5}, eliminating one- and two-dimensional nilradicals. The nilpotent case dimn=5 has nine classes: three decomposable and six indecomposable, including h_5 and two filiform algebras. For dimn=4, classification reduces to normal forms of outer derivations for ℝ^4, h_3⊕ℝ, and n_4 under automorphism and scaling. For dimn=3, it uses two-dimensional abelian subalgebras of OutDer(n) containing no nonzero nilpotent operators. The work also computes minimal faithful representation dimensions μ(g), gives a dictionary with classical tables, and designs a deterministic identification algorithm.
ER -