The algebraic and geometric classification of nilpotent terminal algebras

Abstract

We give algebraic and geometric classifications of $4$-dimensional complex nilpotent terminal algebras. Specifically, we find that, up to isomorphism, there are $41$ one-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $18$ two-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, $2$ three-parameter families of $4$-dimensional nilpotent terminal (non-Leibniz) algebras, complemented by $21$ additional isomorphism classes. The corresponding geometric variety has dimension 17 and decomposes into 3 irreducible components determined by the Zariski closures of a one-parameter family of algebras, a two-parameter family of algebras and a three-parameter family of algebras. In particular, there are no rigid $4$-dimensional complex nilpotent terminal algebras.

References in corpus (6)

• The algebraic and geometric classification of nilpotent binary Lie algebras
• The algebraic and geometric classification of nilpotent anticommutative algebras
• Degenerations of nilpotent algebras
• The algebraic and geometric classification of nilpotent bicommutative algebras
• Non-associative central extensions of null-filiform associative algebras
• Degenerations of Filippov algebras

Cited by in corpus (19)

• The algebraic and geometric classification of nilpotent bicommutative algebras
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• The algebraic and geometric classification of Zinbiel algebras
• The geometric classification of nilpotent algebras
• Central extensions of 3-dimensional Zinbiel algebras
• The algebraic classification of nilpotent $\mathfrak{CD}$-algebras
• One-generated nilpotent bicommutative algebras
• The algebraic and geometric classification of nilpotent left-symmetric algebras
• The algebraic and geometric classification of antiassociative algebras
• On anticommutative algebras for which $[R_a,R_b]$ is a derivation
• The algebraic classification of nilpotent commutative $\mathfrak{CD}$-algebras
• The algebraic classification of nilpotent Novikov algebras
• One-generated nilpotent assosymmetric algebras
• The algebraic classification of nilpotent commutative algebras
• The geometric classification of $2$-step nilpotent algebras and applications
• Conservative algebras of $2$-dimensional algebras, IV
• The algebraic and geometric classification of derived Jordan and bicommutative algebras