Varieties of right-commutative algebras defined by identities (aa)a=[a,b]a=0

Abstract

An algebra R is called a {\it right-commutative} algebra if it satisfies the right-commutative identity

(x_1x_2)x_3-(x_1x_3)x_2=0,

for all elements x_1,x_2,x_3\in R.

This algebra is known by other name as well, for example, {\it NAP} algebra or a {\it non-associative permutative} algebra. Varieties of right-commutative algebras contain important subvarieties such as Novikov and bicommutative algebras.

In this work, we consider two subclasses \mathfrak{M} and \mathfrak{N} of the class of right-commutative algebras. The subclass \mathfrak{M} defined by the identities (x_1x_1)x_1=0, [x_1,x_2]x_1=0 and x_1(x_2x_1)-x_2(x_1x_1)=0. The subclass \mathfrak{N} defined by the identities (x_1x_1)x_1=0, [x_1,x_2]x_1=0 and x_1(x_1x_2)-2x_1(x_2x_1)+x_2(x_1x_1)=0.

We study the free algebras in the varieties \mathfrak{M} and \mathfrak{N} of right-commutative algebras. We investigate the S_n-module structures of the free algebras F_n(\mathfrak{M}) and F_n(\mathfrak{N}) in the varieties \mathfrak{M} and \mathfrak{N}, respectively. As a consequence, we prove that the subvarieties \mathfrak{M} and \mathfrak{M} of the class of right-commutative algebras satisfy the distributivity conditions. Moreover, we describe all subclasses of the classes \mathfrak{M} and \mathfrak{N}.