Free Subalgebras of Graded Algebras

TL;DR

The paper constructs finitely generated graded nilpotent algebras over any field that surprisingly contain free subalgebras on two generators, challenging previous assumptions about their structure.

Contribution

It introduces a method to produce grading nilpotent algebras and demonstrates their ability to contain free subalgebras, a novel result in algebraic structure theory.

Findings

Existence of finitely generated graded nilpotent algebras with free subalgebras

Method for constructing grading nilpotent algebras

Counterexamples to previous conjectures

Abstract

Let $k$ be a field and let $A=\bigoplus_{n\ge 1}A_n$ be a positively graded $k$-algebra. We recall that $A$ is graded nilpotent if for every $d\ge 1$, the subalgebra of $A$ generated by elements of degree $d$ is nilpotent. We give a method of producing grading nilpotent algebras and use this to prove that over any base field $k$ there exists a finitely generated graded nilpotent algebra that contains a free $k$-subalgebra on two generators.