A variety of algebras is said to be a Nielsen-Schreier variety if every subalgebra of every free algebra is free. Using methods of the operad theory, we propose an effective combinatorial criterion for that property in the case of algebras over a field of zero characteristic. Using this criterion, we show, in particular, that the variety of all pre-Lie algebras (also known as right-symmetric algebras) is Nielsen-Schreier, and that, quite surprisingly, there are already infinitely many Nielsen-Schreier varieties of algebras with one binary operation and identities of degree three.
This is joint work with with Ualbai Umirbaev.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
