Constants of Partial Derivations and Primitive Operations

Abstract

We describe algebras of constants of the set of all partial derivations in free algebras of unitarily closed varieties over a field of characteristic 0. These constants are also called proper polynomials. It is proved that a subalgebra of proper polynomials coincides with the subalgebra generated by values of commutators and Umirbaev–Shestakov primitive elements p_m,n on a set of generators for a free algebra. The space of primitive elements is a linear algebraic system over a signature Σ = {[x, y], p_m,n | m, n ≥ 1}. We point out bases of operations of the set Σ in the classes of all algebras, all commutative algebras, right alternative and Jordan algebras.