Abstract
In this article, we review results on primitive elements of free algebras of main types of Schreier varieties of algebras. A variety of linear algebras over a field is Schreier if any subalgebra of a free algebra of this variety is free in the same variety of algebras. A system of elements of a free algebra is primitive if it is a subset of some set of free generators of this algebra. We consider free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. We present matrix criteria for systems of elements of elements. Primitive elements distinguish automorphisms: endomorphisms sending primitive elements to primitive elements are automorphisms. We give a series of examples of almost primitive elements (an element of a free algebra is almost primitive if it is not a primitive element of the whole algebra, but it is a primitive element of any proper subalgebra which contains it). We also consider generic elements and Δ-primitive elements.
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