The algebraic and geometric classification of nilpotent anticommutative algebras

Abstract

We give algebraic and geometric classifications of $6$-dimensional complex nilpotent anticommutative algebras. Specifically, we find that, up to isomorphism, there are $14$ one-parameter families of $6$-dimensional nilpotent anticommutative algebras, complemented by $130$ additional isomorphism classes. The corresponding geometric variety is irreducible and determined by the Zariski closure of a one-parameter family of algebras. In particular, there are no rigid $6$-dimensional complex nilpotent anticommutative algebras.

arXiv admin note: text overlap with arXiv:1905.05361; text overlap with arXiv:1912.02691 by other authors

References in corpus (21)

• Six-dimensional nilpotent Lie algebras
• The variety of $2$-dimensional algebras over an algebraically closed field
• Classification of five-dimensional nilpotent Jordan algebras
• Degenerations of Zinbiel and nilpotent Leibniz algebras
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• The geometric classification of Leibniz algebras
• Complete classification of algebras of level two
• The classification of $2$-dimensional rigid algebras
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• Classification of bilinear maps with radical of codimension 2
• Degenerations of Jordan Superalgebras
• The algebraic classification of nilpotent Tortkara algebras
• Degenerations of Jordan Algebras and ''Marginal'' Algebras
• The geometric classification of nilpotent Tortkara algebras
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Cited by in corpus (26)

• The algebraic and geometric classification of nilpotent Novikov algebras
• Degenerations of nilpotent algebras
• The algebraic and geometric classification of nilpotent terminal algebras
• Non-associative central extensions of null-filiform associative algebras
• The variety of dual mock-Lie algebras
• The algebraic classification of nilpotent algebras
• The algebraic classification of nilpotent Tortkara algebras
• The algebraic and geometric classification of nilpotent noncommutative Jordan algebras
• The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras
• Non-associative algebraic structures: classification and structure
• Central extensions of filiform Zinbiel algebras
• The geometric classification of nilpotent algebras
• One-generated nilpotent terminal algebras
• One-generated nilpotent bicommutative algebras
• The algebraic classification of nilpotent $\mathfrak{CD}$-algebras
• Central extensions of 3-dimensional Zinbiel algebras
• Noncommutative Algebra and Representation Theory: Symmetry, Structure & Invariants
• The algebraic and geometric classification of nilpotent left-symmetric algebras
• The algebraic and geometric classification of nilpotent right alternative algebras
• The geometric classification of nilpotent $\mathfrak{CD}$-algebras
• The algebraic and geometric classification of antiassociative algebras
• The algebraic classification of nilpotent commutative $\mathfrak{CD}$-algebras
• A classification of nilpotent compatible Lie algebras
• One-generated nilpotent assosymmetric algebras
• The algebraic classification of nilpotent commutative algebras
• The geometric classification of nilpotent commutative $\mathfrak{CD}$-algebras