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.