A Nielsen--Schreier variety of algebras without the PBW property

Abstract

We prove the Nielsen--Schreier property for the variety of algebras defined by the identity $x(x^2)^2=(x^2)^2x$: every subalgebra of every free algebra in this variety is itself free. We also show that this variety of algebras does not have the Poincaré--Birkhoff--Witt property for universal multiplicative enveloping algebras. Our strategy of proof actually leads to infinitely many new varieties of non-associative algebras with the same behaviour. This offers new evidence supporting a conjecture of the first author and Umirbaev suggesting that the Nielsen--Schreier property over a field of zero characteristic is equivalent to freeness of universal multiplicative enveloping algebras of free algebras.