The algebraic and geometric classification of nilpotent noncommutative Jordan algebras

Abstract

We give algebraic and geometric classifications of complex four-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 three rigid algebras and two one-parametric families of algebras.

One-generated nilpotent bicommutative algebras

“…The description of one-generated, or cyclic, groups is well known: there exists a unique one-generated group of order n, up to isomorphism. In the case of algebras, similar results have been obtained in varieties such as associative [16], noncommutative Jordan [27], Leibniz and Zinbiel [38], since it was proved that there is only one ndimensional one-generated nilpotent algebra from each of them. However, this circumstance does not hold for every variety of non-associative algebras.…”

Unified computational approach to nilpotent algebra classification problems

“…6}. Moreover, from [17] [5] respectively. As such, it is clear that these algebras have the same central extensions.…”

One-generated nilpotent terminal algebras

“…Another interesting direction is a study of one-generated objects. The description of one-generated finite 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 that the description of n-dimensional one-generated nilpotent associative [12], noncommutative Jordan [22], Leibniz and Zinbiel algebras [38]. It was proven that there is only one n-dimensional one-generated nilpotent algebra in these varieties.…”