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 ∧ V .

The Algebraic and Geometric Classification of Nilpotent Bicommutative Algebras

“…Using this method, all the non-Lie central extensions of all 4-dimensional Malcev algebras were described afterwards [29], and also all the non-associative central extensions of 3-dimensional Jordan algebras [28], all the anticommutative central extensions of the 3-dimensional anticommutative algebras [8], and all the central extensions of the 2-dimensional algebras [10]. Moreover, the method is especially indicated for the classification of nilpotent algebras (see, for example, [26]), and it was used to describe all the 4-dimensional nilpotent associative algebras [15], all the 4-dimensional nilpotent Novikov algebras [33], all the 5-dimensional nilpotent Jordan algebras [27], all the 5-dimensional nilpotent restricted Lie algebras [13], all the 6-dimensional nilpotent Lie algebras [12,14], all the 6-dimensional nilpotent Malcev algebras [30] and some others.…”

The algebraic classification of nilpotent Tortkara algebras

“…After that, the method introduced by Skjelbred and Sund was used to describe all non-Lie central extensions of all 4-dimensional Malcev algebras [21], all non-associative central extensions of 3-dimensional Jordan algebras [20], all anticommutative central extensions of 3-dimensional anticommutative algebras [6], all central extensions of 2-dimensional algebras [7]. The method of central extensions was used to describe all 4-dimensional nilpotent associative algebras [12], all 4-dimensional nilpotent bicommutative algebras [26], all 4-dimensional nilpotent Novikov algebras [24], all 5-dimensional nilpotent Jordan algebras [19], all 5-dimensional nilpotent restricted Lie algebras [11], all 6-dimensional nilpotent Lie algebras [10,13], all 6-dimensional nilpotent Malcev algebras [22], all 6-dimensional nilpotent binary Lie algebras [3], all 6-dimensional nilpotent anticommutative CD-algebras [3] and some other.…”

The algebraic and geometric classification of nilpotent Novikov algebras

“…After that, using the method described by Skjelbred and Sund, all non-Lie central extensions of all 4-dimensional Malcev algebras were described [33], and also all the non-associative central extensions of 3-dimensional Jordan algebras [32], all the anticommutative central extensions of the 3-dimensional anticommutative algebras [10], and all the central extensions of the 2-dimensional algebras [12]. Note that the method of central extensions is an important tool in the classification of nilpotent algebras (see, for example, [30]), which was used to describe all the 4-dimensional nilpotent associative algebras [18], all the 4-dimensional nilpotent bicommutative algebras [38], all the 5-dimensional nilpotent Jordan algebras [31], all the 5-dimensional nilpotent restricted Lie algebras [16], all the 6-dimensional nilpotent Lie algebras [15,17], all the 6-dimensional nilpotent Malcev algebras [34] and some others.…”