Engel's Theorem for Alternative and Special Jordan Superalgebras

TL;DR

The paper extends Engel-type nilpotency criteria to nonassociative structures by proving that a finite dimensional alternative superalgebra $A$ with $R_{a}$ nilpotent for every homogeneous $a$ is nilpotent, establishing that Engelian alternative superalgebras are nilpotent. It then shows that finite dimensional Engelian special Jordan superalgebras over any field with $ ext{char} eq 2$ are nilpotent, using their realization as subsuperalgebras of $A^{+}$ and a weakly closed set argument. A graded-nil but non-nilpotent example in characteristic $3$ demonstrates the necessity of the hypotheses. Collectively, these results generalize Engel’s theorem to broader superalgebraic contexts and clarify the boundary between graded-nil and nilpotent behavior, while addressing finite-field questions in the Jordan case.

Abstract

In this paper, a nilpotency criterion is given for finite dimensional alternative superalgebras inspired by the celebrated Engel's Theorem for Lie algebras. As a consequence, a similar result is proved for finite-dimensional special Jordan superalgebras over a field $\mathbb{F}$ of characteristic not $2$, without restrictions on the cardinality of $\mathbb{F}$. In that case, the latter extends Engel's Theorem for Jordan superalgebras constructed by Okunev and Shestakov and it gives a partial positive answer to an open problem announced by Murakami et al. for Jordan superalgebras over finite fields. We also establish some connections between the concepts of graded-nil and nilpotent alternative superalgebras.

Theorems & Definitions (17)

• Definition 2.1
• Proposition 2.2
• Example 3.1
• Proposition 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• proof
• Theorem 3.5