The algebraic and geometric classification of noncommutative Jordan superalgebras

Abstract

The algebraic and geometric classifications of complex $3$-dimensional noncommutative Jordan superalgebras are given. In particular, we obtain the algebraic and geometric classification of $3$-dimensional Kokoris and standard superalgebras, and, due to one-to-one correspondences between suitable superalgebras, we have classifications for generic Poisson-Jordan and generic Poisson superalgebras. As a byproduct, we have the algebraic and geometric classification of the variety of $3$-dimensional anticommutative superalgebras and its principal subvarieties: Lie, Malcev, binary Lie, Tortkara, anticommutative $\mathfrak{CD}$-, ${\mathfrak{s}}_4$-, anticommutative terminal superalgebras, anticommutative conservative and anticommutative quasi-conservative $($rigid$)$ superalgebras.