Optimal 5-step nilpotent quadratic algebras

TL;DR

This paper proves the optimality of a known bound for 5-step nilpotent quadratic algebras, showing that for any number of generators, there exists an algebra with a specific number of relations that is 5-step nilpotent.

Contribution

It demonstrates the existence of 5-step nilpotent quadratic algebras matching the Golod--Shafarevich bound, establishing the bound's optimality.

Findings

Existence of 5-step nilpotent quadratic algebras with n generators and n^2/3 relations

Validation of the Golod--Shafarevich bound as optimal for 5-step nilpotency

Construction method for such algebras

Abstract

By the Golod--Shafarevich Theorem, an associative algebra R given by n generators and d<n^2/3 homogeneous quadratic relations is not 5-step nilpotent. We prove that this estimate is optimal. Namely, we show that for every positive integer n, there is an algebra R given by n generators and n^2/3 homogeneous quadratic relations such that R is 5-step nilpotent.