An effective criterion for Nielsen-Schreier varieties

TL;DR

This paper introduces a new combinatorial criterion using operads to identify Nielsen-Schreier varieties of algebras, revealing infinitely many such varieties with one binary operation and degree four identities.

Contribution

It provides an effective criterion for Nielsen-Schreier varieties over zero characteristic fields, expanding known classes to include pre-Lie and Lie-admissible algebras and infinitely many new varieties.

Findings

Pre-Lie and Lie-admissible algebras are Nielsen-Schreier.

Infinitely many non-equivalent Nielsen-Schreier varieties with degree four identities.

Operads are crucial in establishing the criterion.

Abstract

All algebras of a certain type are said to form a Nielsen-Schreier variety if every subalgebra of a free algebra is free. This property has been perceived as extremely rare; in particular, only six Nielsen-Schreier varieties of algebras with one binary operation have been discovered in prior work on this topic. We propose an effective combinatorial criterion for the Nielsen-Schreier property in the case of algebras over a field of zero characteristic; in our approach, operads play a crucial role. Using this criterion, we show that the well known varieties of all pre-Lie algebras and of all Lie-admissible algebras are Nielsen-Schreier, and, quite surprisingly, that there are already infinitely many non-equivalent Nielsen-Schreier varieties of algebras with one binary operation and identities of degree four.