Locally finite varieties of nonassociative algebras

TL;DR

This paper investigates the properties and classifications of finite-dimensional nonassociative algebras over finite fields within locally finite varieties, focusing on their structural features and enumeration of algebras with specific properties.

Contribution

It provides new insights into the properties of finite nonassociative algebras and estimates the frequency of various classical features among them.

Findings

Analysis of nilpotence, solvability, simplicity, freeness, projectivity, and injectivity in these algebras.

Numerical estimates of the ratio of algebras with specific properties to total algebras of fixed dimension.

Characterization of algebras with no proper subalgebras or automorphisms.

Abstract

We study locally finite varieties (=primitive classes) of linear algebras over finite fields. We do not assume that our algebras are associative or Lie. We are interested in the basic properties of finite algebras in these varieties such as: nilpotence, solvability, simplicity, freeness, projectivity, and injectivity. We are also interested in the numerical estimates of the ratio of the number of algebras with various classical properties to the total number of all algebras of a fixed dimension $n$. Among these properties are having no proper nontrivial subalgebras or no nontrivial automorphisms, etc.