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 complex 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 complex 3-dimensional anticommutative superalgebras and their principal subvarieties: Lie, Malcev, binary Lie, Tortkara, anticommutative $$\mathfrak {CD}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>CD</mml:mi> </mml:math> -, $$\mathfrak {s}_4$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>s</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:math> -, anticommutative terminal superalgebras, anticommutative conservative and anticommutative quasi-conservative (rigid) superalgebras; and also prove a Grishkov–Shestakov’s conjecture for 3-dimensional binary Lie superalgebras.

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