Classification of real Lie algebras of dimension four
Abstract
We present an exhaustive, coordinate-free, and self-contained classification of four-dimensional real Lie algebras up to isomorphism. In dimension three, classification schemes frequently exploit Bryant's classical exterior duality isomorphism Λ2g ≅ g* ⊗ Λ3g, which reduces the Jacobi identity to an elegant linear algebraic condition on a single 3× 3 matrix. We explicitly demonstrate that this accidental isomorphism breaks down in dimension four due to the dimensional mismatch dim Λ2g = 6 ≠ 4 = dim(g* ⊗ Λ4g), transforming the Jacobi identity into a non-linear system of 16 quadratic constraints on 24 structure constants. To overcome this geometric obstruction without relying on brute-force polynomial elimination, we employ an intrinsic Lie-theoretic reduction: the Levi--Malcev decomposition isolates the non-solvable sector into direct products with three-dimensional simple ideals, while the solvable sector is classified by anchoring the structure to its nilradical n, which is an intrinsic characteristic ideal. For codimension-one nilradicals (dimn = 3), the problem is reduced to classifying outer derivations of ℝ3 and the Heisenberg algebra h3 under the action of the automorphism group modulo global rescaling and central translations. For two-dimensional nilradicals, the structure is resolved via maximal abelian sub-algebras of gl(2,ℝ) lacking nilpotent elements. Complete proofs are supplied for all reductions, precise moduli parameter ranges are established to eliminate redundancies, and an explicit isomorphism dictionary is provided bridging our structural models with classical tabulations in mathematical physics.
Related
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BibTeX
@misc{nadjafikhah2026classification,
title = {{Classification of real Lie algebras of dimension four}},
author = {Mehdi Nadjafikhah},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23008},
eprint = {2610.23008},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23008},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of real Lie algebras of dimension four
AU - Mehdi Nadjafikhah
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23008
UR - https://arxiv.org/abs/2610.23008
AB - We present an exhaustive, coordinate-free, and self-contained classification of four-dimensional real Lie algebras up to isomorphism. In dimension three, classification schemes frequently exploit Bryant's classical exterior duality isomorphism Λ^2g ≅ g^* ⊗ Λ^3g, which reduces the Jacobi identity to an elegant linear algebraic condition on a single 3× 3 matrix. We explicitly demonstrate that this accidental isomorphism breaks down in dimension four due to the dimensional mismatch dim Λ^2g = 6 ≠ 4 = dim(g^* ⊗ Λ^4g), transforming the Jacobi identity into a non-linear system of 16 quadratic constraints on 24 structure constants. To overcome this geometric obstruction without relying on brute-force polynomial elimination, we employ an intrinsic Lie-theoretic reduction: the Levi--Malcev decomposition isolates the non-solvable sector into direct products with three-dimensional simple ideals, while the solvable sector is classified by anchoring the structure to its nilradical n, which is an intrinsic characteristic ideal. For codimension-one nilradicals (dimn = 3), the problem is reduced to classifying outer derivations of ℝ^3 and the Heisenberg algebra h_3 under the action of the automorphism group modulo global rescaling and central translations. For two-dimensional nilradicals, the structure is resolved via maximal abelian sub-algebras of gl(2,ℝ) lacking nilpotent elements. Complete proofs are supplied for all reductions, precise moduli parameter ranges are established to eliminate redundancies, and an explicit isomorphism dictionary is provided bridging our structural models with classical tabulations in mathematical physics.
ER -