An effective criterion for Nielsen-Schreier varieties

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.

arXiv comment35 pages, to appear in IMRN

References in corpus (7)

• Gröbner bases for operads
• Lectures on Noncommutative Geometry
• Character formulas for the operad of two compatible brackets and for the bihamiltonian operad
• Operads of compatible structures and weighted partitions
• Special and exceptional mock-Lie algebras
• Endofunctors and Poincaré-Birkhoff-Witt theorems
• Word operads and admissible orderings

Cited by in corpus (4)

• Polynomial identities in Novikov algebras
• Non-additive derived functors via chain resolutions
• A characterisation of Lie algebras using ideals and subalgebras
• Distributive lattices of varieties of Novikov algebras