The algebraic and geometric classification of nilpotent binary Lie algebras

Abstract

We give a complete algebraic classification of nilpotent binary Lie algebras of dimension at most 6 over an arbitrary base field of characteristic not 2 and a complete geometric classification of nilpotent binary Lie algebras of dimension 6 over C. As an application, we give an algebraic and geometric classification of nilpotent anticommutative CD-algebras of dimension at most 6.Proof. Let θ ∈ Z 2 BL (A, F). Then A θ is a binary Lie algebra. We denote the Jacobian of elements x, y, z in A θ by J A θ (x, y, z). Now consider x, y, z, t ∈ A. (1.3), By the identityt] , x, y) = 0; J([x, y] , z, t) + J([x, t] , z, y) + J([z, y] , x, t) + J([z, t] , x, y) = 0; we deduce that ψ θ ([x, y] , z, t) + ψ θ ([x, t] , z, y) + ψ θ ([z, y] , x, t) + ψ θ ([z, t] , x, y) = 0, as desired.

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 geometric classification of nilpotent Tortkara algebras

“…There are many results related to both the algebraic and geometric classification of small dimensional algebras in the varieties of Jordan, Lie, Leibniz and Zinbiel algebras; for algebraic results see, for example, [1,11,18,19,23,[25][26][27]30]; for geometric results see, for example, [1, 3-6, 8, 10, 11, 19-31, 34]. Here we give a geometric classification of 6-dimensional nilpotent Tortkara algebras over C. Our main result is Theorem 3 which describes the rigid algebras in this variety.…”

“…These algebras are of big interest, since the closures of their orbits under the action of the generalized linear group form irreducible components of the variety under consideration (with respect to the Zariski topology). For example, the rigid algebras in the varieties of all 4-dimensional Leibniz algebras [24], all 4-dimensional nilpotent Novikov algebras [26], all 4-dimensional nilpotent bicommutative algebras [27], all 4-dimensional nilpotent assosymmetric algebras [23], all 6-dimensional nilpotent binary Lie algebras [1], and some other has been classified. There are fewer works in which the full information about degenerations has been found for some variety of algebras.…”

Degenerations of nilpotent associative commutative algebras

“…These algebras are of big interest, since the closures of their orbits under the action of the generalized linear group form irreducible components of the variety under consideration (with respect to the Zariski topology). For example, rigid algebras in the varieties of all 4-dimensional Leibniz algebras [19], all 4-dimensional nilpotent Novikov algebras [21], all 4-dimensional nilpotent assosymmetric algebras [18], all 4-dimensional nilpotent bicommutative algebras [22], all 6-dimensional nilpotent binary Lie algebras [1], and in some other varieties were classified. There are fewer works in which the full information about degenerations was given for some variety of algebras.…”