Six-dimensional nilpotent Lie algebras

Abstract

We give a full classification of 6-dimensional nilpotent Lie algebras over an arbitrary field, including fields that are not algebraically closed and fields of characteristic~2. To achieve the classification we use the action of the automorphism group on the second cohomology space, as isomorphism types of nilpotent Lie algebras correspond to orbits of subspaces under this action. In some cases, these orbits are determined using geometric invariants, such as the Gram determinant or the Arf invariant. As a byproduct, we completely determine, for a 4-dimensional vector space $V$, the orbits of $\GL(V)$ on the set of 2-dimensional subspaces of $V\wedge V$.

Corrected a small error in Theorem 4.4

Cited by in corpus (51)

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• One-generated nilpotent bicommutative algebras
• One-generated nilpotent terminal algebras
• The Nash-Moser Theorem of Hamilton and rigidity of finite dimensional nilpotent Lie algebras
• Central extensions of 3-dimensional Zinbiel algebras
• The algebraic classification of nilpotent $\mathfrak{CD}$-algebras
• Clifford Parallelisms and External Planes to the Klein quadric
• The algebraic and geometric classification of nilpotent left-symmetric algebras
• Some results on the Schur multiplier of nilpotent Lie algebras
• Certain functors of nilpotent Lie algebra with the derived subalgebra of dimension two
• The algebraic and geometric classification of nilpotent right alternative algebras
• The Bogomolov multiplier of Lie algebras
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• Capability of nilpotent Lie algebras of small dimension
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• The structure of nilpotent Lie algebras of class two with derived subalgebra of dimension two
• On a spectral sequence for the cohomology of a nilpotent Lie algebra
• On classification of (n+5)-dimensional nilpotent n-Lie algebra of class two
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