The geometric classification of nilpotent ℭ𝔇-algebras

Abstract

We give a geometric classification of complex 4-dimensional nilpotent [Formula: see text]-algebras. The corresponding geometric variety has dimension 18 and decomposes into 2 irreducible components determined by the Zariski closures of a two-parameter family of algebras and a four-parameter family of algebras. In particular, there are no rigid 4-dimensional complex nilpotent [Formula: see text]-algebras.

Non-associative algebraic structures: classification and structure

“…-assosymmetric algebras [120]; -bicommutative algebras [163]; -CD-algebras [144]; -commutative algebras [89]; -left-symmetric algebras [3]; -Leibniz algebras [170]; -noncommutative Jordan algebras [128]; -Novikov algebras [137]; -right commutative algebras [4]; -right alternative algebras [121]; -terminal algebras [153]; -weakly associative algebras [9]; II. 5-dimensional nilpotent:…”

“…In [119,152] we completely solve the geometric classification problem for nilpotent and 2-step nilpotent, commutative nilpotent and anticommutative nilpotent algebras of arbitrary dimension.…”

“…Theorem 3 (Theorem C, [152]). For any n ≥ 2, the variety of all n-dimensional anticommutative nilpotent algebras is irreducible and has dimension (n−2)(n 2 +2n+3) 6 .…”

The algebraic classification and degenerations of nilpotent Poisson algebras

“…Ignatyev, Kaygorodov, and Popov determined the dimensions of the irreducible components of n-dimensional 2-step nilpotent algebras (commutative and anticommutative) [36]. On the other hand, Kaygorodov, Khrypchenko, and Lopes studied the dimensions of the irreducible components of n-dimensional (all, commutative, anticommutative) nilpotent algebras [40]. Degenerations have also been used to study the level of complexity of an algebra [56,57].…”

The algebraic classification of nilpotent commutative CD-algebras

“…There are many results related to the algebraic and geometric classification of low-dimensional algebras in the varieties of Jordan, Lie, Leibniz and Zinbiel algebras; for algebraic classifications see, for example, [1, 6, 8, 9, 11, 14-16, 25, 28, 30, 36, 38, 41]; for geometric classifications and descriptions of degenerations see, for example, [33]. In the present paper, we give an algebraic classification of nilpotent commutative CD-algebras.…”