The algebraic and geometric classification of nilpotent noncommutative Jordan algebras

Abstract

We give algebraic and geometric classifications of complex 4-dimensional nilpotent noncommutative Jordan algebras. Specifically, we find that, up to isomorphism, there are only 18 non-isomorphic nontrivial nilpotent noncommutative Jordan algebras. The corresponding geometric variety is determined by the Zariski closure of 3 rigid algebras and 2 one-parametric families of algebras.

One-generated nilpotent Novikov algebras

“…Another interesting direction is a study of one-generated objects. The description of onegenerated groups is well-known: there is only one one-generated group of order n. In the case of algebras, there are some similar results, such as the description of n-dimensional one-generated nilpotent associative [19], noncommutative Jordan [34], Leibniz and Zinbiel algebras [43]. It was proven that there is only one n-dimensional one-generated nilpotent algebra in these varieties.…”

The algebraic classification of nilpotent ℭ𝔇-algebras

“…Using the same method, all non-Lie central extensions of all 4-dimensional Malcev algebras [21], all non-associative central extensions of all 3dimensional Jordan algebras [20], all anticommutative central extensions of 3-dimensional anticommutative algebras [5], all central extensions of 2-dimensional algebras [7] and some others were described. One can also look at the classification of 3-dimensional nilpotent algebras [16], 4-dimensional nilpotent associative algebras [13], 4-dimensional nilpotent Novikov algebras [27], 4-dimensional nilpotent bicommutative algebras [32], 4-dimensional nilpotent commutative algebras in [16], 4-dimensional nilpotent assosymmetric algebras in [25], 4-dimensional nilpotent noncommutative Jordan algebras in [26], 4-dimensional nilpotent terminal algebras [31], 5-dimensional nilpotent restricted Lie algebras [11], 5-dimensional nilpotent associative commutative algebras [34], 5-dimensional nilpotent Jordan algebras [19], 6-dimensional nilpotent Lie algebras [10,12], 6-dimensional nilpotent Malcev algebras [22], 6-dimensional nilpotent Tortkara algebras [17,18], 6-dimensional nilpotent binary Lie algebras [1], 6-dimensional nilpotent anticommutative CD-algebras [1], 6-dimensional nilpotent anticommutative algebras [29], 8-dimensional dual mock-Lie algebras [8].…”