Laiz Valim da Rocha | Searcharxiv

arXiv2026

Engel's Theorem for Alternative Superalgebras

In this paper, a nilpotency criterion is given for finite dimensional alternative superalgebras in the spirit of Engel's Theorem for Jordan superalgebras over infinite fields provided by Shestakov and Okunev. For alternative superalgebras, no restrictions on the cardinality of the ground field are required. Furthermore, we establish some connections between the concepts of graded-nil and nilpotent alternative superalgebras, and we also exhibit an example of an Engelian commutative power-associative superalgebra of dimension $4$ which is not nilpotent.

math.RA↗

arXiv2026

Varieties of nilpotent Jordan superalgebras of dimension five

The paper is devoted to the description of the varieties of complex 5-dimensional nilpotent Jordan superalgebras. We find all representatives for the isomorphism classes, using the Jordan normal form, results of simultaneous matrix triangularization, the Jordan-Kronecker theorem for a pair of skew-symmetric bilinear forms and similar arguments developed for $Δ$-modules by Burde and Grunewald. We also provide a complete geometric classification, determining the irreducible components of the corresponding varieties and describing all possible degenerations and non-degenerations between these superalgebras, in particular, applying some $\mathbb{Z}_2$-graded subspaces as invariants.

math.RA↗