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First page of The algebraic and geometric classification of semi-alternative superalgebras

The algebraic and geometric classification of semi-alternative superalgebras

Hani Abdelwahab, Ivan Kaygorodov, Roman Lubkov

math.RA Oct 16, 2026 · v1
formalization algebra
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TL;DR
Lean 4 is used to check the polynomial identities showing that super Sagle cubics follow from the semi-alternative equations, a key lemma.
Abstract
The algebraic and geometric classifications of complex $3$-dimensional semi-alternative superalgebras are given. As corollaries, we obtain the algebraic and geometric classifications of complex $3$-dimensional $\mathfrak{perm}$, associative, and assosymmetric superalgebras.
Problem

The goal is the algebraic and geometric classification, up to isomorphism and degeneration, of complex 3-dimensional semi-alternative superalgebras. As corollaries this also classifies perm, associative, and assosymmetric superalgebras.

Approach

Semi-alternative superalgebra structures are built over Malcev superalgebras using the supercommutator and the Jordan superproduct. A key lemma shows that the supercommutator superalgebra of a 3-dimensional semi-alternative superalgebra of type (1,2) or (2,1) is Malcev. The proof expresses the super Sagle cubic polynomials as combinations of the quadratic semi-alternative equations in the structure constants. The coefficients were found by exact linear algebra and the resulting identities were checked in Lean 4. Degenerations are then studied via orbit closures under the action of GL(V0)×GL(V1).

Results

Complete algebraic and geometric classifications of complex 3-dimensional semi-alternative superalgebras are obtained. Corresponding classifications follow for perm, associative, and assosymmetric superalgebras.