Engel's Theorem for Alternative Superalgebras

TL;DR

This paper extends Engel's Theorem to finite dimensional alternative superalgebras, providing a nilpotency criterion without restrictions on the ground field, and explores related algebraic concepts.

Contribution

It introduces a nilpotency criterion for alternative superalgebras and connects graded-nil and nilpotent structures, including a counterexample in superalgebra theory.

Findings

Nilpotency criterion for finite dimensional alternative superalgebras.

No restrictions on the ground field's cardinality.

Existence of a non-nilpotent Engelian superalgebra example.

Abstract

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.