
Journal of Lie Theory 34 (2024), No. 2, 437451 Copyright Heldermann Verlag 2024 On Higher Dimensional Milnor Frames Hayden Hunter College of Liberal Arts and Sciences, University of Florida, Gainesville, Florida, U.S.A. haydenhunter@ufl.edu [Abstractpdf] A classic result of Milnor shows that any 3dimensional unimodular metric Lie algebra admits an orthonormal frame with at most three nontrivial structure constants. These frames are referred to as Milnor frames. We define extensions of Milnor frames into higher dimensions and refer to these higher dimensional analogues as Lie algebras with Milnor frames. We determine that $n$dimensional, $n \geq 4$, Lie algebras with Milnor frames are isomorphic to the direct sum of 3dimensional Heisenberg Lie algebras $\mathfrak{h}^3$, 4dimensional 3step nilpotent Lie algebras $\mathfrak{h}^4$, and an abelian Lie algebra $\mathfrak{a}$. Moreover, for any Lie algebra $\mathfrak{g}\not\cong \mathfrak{h}^3 \oplus \mathfrak{a}$ with a Milnor frame, there exists an inner product structure $g$ on $\mathfrak{g}$ such that $(\mathfrak{g}, g)$ does not admit an orthonormal Milnor frame. Keywords: Lie algebras, nilmanifolds, unimodular, Ricci flow. MSC: 22Exx. [ Fulltextpdf (144 KB)] for subscribers only. 