Free commutative moufang loops and anticommutative graded rings

Abstract

(1.1) This paper is mainly designed to answer the problems raised by Bruck in 1959 (see [7]). Let L, be the free commutative moufang loop (CML) on n generators. Only L,, the first nonassociative one, was previously known. We shall give explicit descriptions of the two following ones, L, and L, . Besides, for any k > 1, define L(n,k) as the free object on n generators in the category of CML’s whose central nilpotency class is at most k. Bruck [ 9 J had given a concrete description of Len,*), We shall describe similarly Lcn,3). Furthermore, denote as usual by &(x, y, Z; U, , a2 ,..., Ui, Ui + , ,..., u,) the (p + 1 )-associator: