The variety of $2$-dimensional algebras over an algebraically closed field

Abstract

The work is devoted to the variety of $2$-dimensional algebras over an algebraically closed field. Firstly, we classify such algebras modulo isomorphism. Then we describe the degenerations and the closures of principal algebra series in the variety under consideration. Finally, we apply our results to obtain analogous descriptions for the subvarieties of flexible, and bicommutative algebras. In particular, we describe rigid algebras and irreducible components for these subvarieties.

References in corpus (7)

• Degenerations of Zinbiel and nilpotent Leibniz algebras
• Degenerations of binary Lie and nilpotent Malcev algebras
• The geometric classification of Leibniz algebras
• Complete classification of algebras of level two
• Classification of bilinear maps with radical of codimension 2
• The classification of algebras of level two
• Conservative algebras of $2$-dimensional algebras, II

Cited by in corpus (35)

• Degenerations of Zinbiel and nilpotent Leibniz algebras
• The algebraic and geometric classification of nilpotent binary Lie algebras
• The algebraic and geometric classification of nilpotent anticommutative algebras
• The algebraic and geometric classification of nilpotent Novikov algebras
• Degenerations of nilpotent algebras
• The algebraic and geometric classification of nilpotent bicommutative algebras
• Degenerations of binary Lie and nilpotent Malcev algebras
• The algebraic and geometric classification of nilpotent terminal algebras
• Complete classification of algebras of level two
• The variety of dual mock-Lie algebras
• Degenerations of Jordan Superalgebras
• The algebraic and geometric classification of nilpotent noncommutative Jordan algebras
• The algebraic and geometric classification of nilpotent weakly associative and symmetric Leibniz algebras
• The algebraic classification of nilpotent algebras
• The algebraic classification of nilpotent Tortkara algebras
• Degenerations of Jordan Algebras and ''Marginal'' Algebras
• The geometric classification of nilpotent Tortkara algebras
• Degenerations of Filippov algebras
• The geometric classification of nilpotent algebras
• Degenerations of nilpotent associative commutative algebras
• Central extensions of filiform Zinbiel algebras
• One-generated nilpotent Novikov algebras
• One-generated nilpotent terminal algebras
• The algebraic classification of nilpotent $\mathfrak{CD}$-algebras
• Idempotent geometry in generic algebras
• One-generated nilpotent bicommutative algebras
• The algebraic and geometric classification of nilpotent left-symmetric algebras
• The geometric classification of nilpotent $\mathfrak{CD}$-algebras
• The algebraic and geometric classification of nilpotent right alternative algebras
• The algebraic classification of nilpotent commutative $\mathfrak{CD}$-algebras
• One-generated nilpotent assosymmetric algebras
• The algebraic classification of nilpotent commutative algebras
• Polynomial identities of bicommutative algebras, Lie and Jordan elements
• The automorphism groups and derivation algebras of two-dimensional algebras
• Conservative algebras of $2$-dimensional algebras, IV