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 $\mathrm{GL}(5,\mathbb{R})$, the method uses Levi--Malcev theory and nilradical stratification. In the non-solvable case, the Levi factor is three-dimensional, $\mathfrak{so}(3)$ or $\mathfrak{sl}(2,\mathbb{R})$, acting on a two-dimensional radical; this yields exactly five classes, including the indecomposable $\mathfrak{sl}(2,\mathbb{R})\ltimes\mathbb{R}^2$. In the solvable case, Schur's bound on commutative subalgebras of $\mathfrak{gl}(2,\mathbb{R})$ implies $\dim\mathfrak{n}\in\{3,4,5\}$, eliminating one- and two-dimensional nilradicals. The nilpotent case $\dim\mathfrak{n}=5$ has nine classes: three decomposable and six indecomposable, including $\mathfrak{h}_5$ and two filiform algebras. For $\dim\mathfrak{n}=4$, classification reduces to normal forms of outer derivations for $\mathbb{R}^4$, $\mathfrak{h}_3\oplus\mathbb{R}$, and $\mathfrak{n}_4$ under automorphism and scaling. For $\dim\mathfrak{n}=3$, it uses two-dimensional abelian subalgebras of $\mathrm{OutDer}(\mathfrak{n})$ containing no nonzero nilpotent operators. The work also computes minimal faithful representation dimensions $μ(\mathfrak{g})$, gives a dictionary with classical tables, and designs a deterministic identification algorithm.