# An uncountable family of almost nilpotent varieties of polynomial growth

## TL;DR

This paper constructs infinite families of almost nilpotent algebra varieties with polynomial growth, expanding the known examples beyond the unique associative or Lie algebra cases.

## Contribution

It introduces the first known uncountable family of almost nilpotent varieties with quadratic growth, and a countable family with linear growth, broadening the scope of such varieties.

## Key findings

- Existence of a countable family of almost nilpotent varieties with linear growth.
- Existence of an uncountable family of almost nilpotent varieties with quadratic growth.
- These families are distinct from previously known unique cases in associative and Lie algebras.

## Abstract

A non-nilpotent variety of algebras is almost nilpotent if any proper subvariety is nilpotent. Let the base field be of characteristic zero. It has been shown that for associative or Lie algebras only one such variety exists. Here we present infinite families of such varieties. More precisely we shall prove the existence of 1) a countable family of almost nilpotent varieties of at most linear growth and 2) an uncountable family of almost nilpotent varieties of at most quadratic growth.