Classification of real Lie algebras of dimension six
Abstract
We summarize a coordinate-free classification of six-dimensional real Lie algebras up to isomorphism. Instead of solving the overdetermined Jacobi system of 120 quadratic equations on 90 structure constants modulo GL(6,ℝ), the method uses the Levi--Malcev decomposition, Mostow's nilradical constraints, and the Skjelbred--Sund cohomological central extension technique. In the semisimple sector, the four real forms of A1⊕ A1 are classified by Killing signature invariants, with explicit matrix representatives for sl(2,ℂ)ℝ≅so(3,1). For non-solvable algebras with three-dimensional radical, representation-theoretic rigidity shows that su(2) cannot couple indecomposably with non-abelian radicals, yielding exactly three indecomposable classes governed by so(3) and sl(2,ℝ). In the solvable category, Mostow's theorem restricts the nilradical dimension to {3,4,5,6}. The nilpotent sector in dimension six has twenty-four indecomposable isomorphism classes plus decomposable counterparts. Solvable non-nilpotent algebras are classified by nilradical codimension, with explicit real Jordan matrix representations for codimension one and two that resolve continuous moduli under gauge actions. The work also computes the minimal faithful representation dimension μ(g) under Ado's theorem, relates the classification to invariant complex structures on nilmanifolds, and provides a deterministic, page-breakable identification algorithm.
Related
- Classification of real Lie algebras of dimension one, two and three
- Classification of real Lie algebras of dimension five
- Classification of real Lie algebras of dimension four
- Jacobson radical non-nil algebras of Gel'fand-Kirillov dimension 2
- Nuclear Dimension of Twisted C*-Algebras of Virtually Abelian Groups
BibTeX
@misc{nadjafikhah2026classification,
title = {{Classification of real Lie algebras of dimension six}},
author = {Mehdi Nadjafikhah},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.10009},
eprint = {2611.10009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.10009},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of real Lie algebras of dimension six
AU - Mehdi Nadjafikhah
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.10009
UR - https://arxiv.org/abs/2611.10009
AB - We summarize a coordinate-free classification of six-dimensional real Lie algebras up to isomorphism. Instead of solving the overdetermined Jacobi system of 120 quadratic equations on 90 structure constants modulo GL(6,ℝ), the method uses the Levi--Malcev decomposition, Mostow's nilradical constraints, and the Skjelbred--Sund cohomological central extension technique. In the semisimple sector, the four real forms of A_1⊕ A_1 are classified by Killing signature invariants, with explicit matrix representatives for sl(2,ℂ)_ℝ≅so(3,1). For non-solvable algebras with three-dimensional radical, representation-theoretic rigidity shows that su(2) cannot couple indecomposably with non-abelian radicals, yielding exactly three indecomposable classes governed by so(3) and sl(2,ℝ). In the solvable category, Mostow's theorem restricts the nilradical dimension to {3,4,5,6}. The nilpotent sector in dimension six has twenty-four indecomposable isomorphism classes plus decomposable counterparts. Solvable non-nilpotent algebras are classified by nilradical codimension, with explicit real Jordan matrix representations for codimension one and two that resolve continuous moduli under gauge actions. The work also computes the minimal faithful representation dimension μ(g) under Ado's theorem, relates the classification to invariant complex structures on nilmanifolds, and provides a deterministic, page-breakable identification algorithm.
ER -