CONSTANTS OF PARTIAL DERIVATIONS AND PRIMITIVE OPERATIONS

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 Sigma = {[ x, y], p(m,n) vertical bar m, n >= 1}. We point out bases of operations of the set S in the classes of all algebras, all commutative algebras, right alternative and Jordan algebras.