8-dimensional 2-step nilpotent Lie algebras over algebraically closed fields of char $ e 2, 3$

We provide a self contained, elementary, and geometrically-flavored classification of $8$-dimensional $2$-step nilpotent Lie algebras over algebraically closed fields of characteristic $\ne 2,3$, using the algebro-geometric arguments from \cite{B} and elementary linear algebra.

💡 Research Summary

The paper “8‑dimensional 2‑step nilpotent Lie algebras over algebraically closed fields of characteristic ≠ 2, 3” presents a complete, elementary, and geometrically flavored classification of all 8‑dimensional 2‑step nilpotent Lie algebras when the base field k is algebraically closed and its characteristic is different from 2 and 3. The authors adopt the viewpoint that a nilpotent Lie algebra g can be identified with its Chevalley‑Eilenberg cochain complex (Λ g*, d), which is a minimal commutative differential graded algebra (CDGA) generated in degree 1. The 2‑step nilpotency condition translates into the statement that the characteristic filtration of the minimal CDGA has length two: W₀ = ker d and W₁ = V, where V is the underlying 8‑dimensional vector space. Consequently, V splits as V = F₀ ⊕ F₁ with dim F₀ = f₀, dim F₁ = f₁, f₀ + f₁ = 8, and the differential restricts to an injective linear map d : F₁ → Λ² W₀. The image Im(d) is a f₁‑dimensional subspace of Λ² W₀, and the classification problem becomes the problem of describing the GL(W₀)‑orbits of such subspaces inside the Grassmannian Gr(Λ² W₀, f₁).

The authors first list the only possible pairs (f₀, f₁) allowed by the inequality f₁ ≤ C(f₀, 2): (8,0), (7,1), (6,2), (5,3), (4,4). The trivial case (8,0) corresponds to the abelian Lie algebra. For (7,1) the image is a single bivector φ; its rank can be 2, 4 or 6, leading to three normal forms:  rank 2 φ = x₁∧x₂,  rank 4 φ = x₁∧x₂ + x₃∧x₄,  rank 6 φ = x₁∧x₂ + x₃∧x₄ + x₅∧x₆. Choosing a basis of V that extends the basis of W₀, the corresponding Lie brackets are simply