On Schreier varieties of linear algebras
Jacques Lewin · Transactions of the American Mathematical Society · 1968
I. Introduction.A variety ö of algebraic systems (i.e., a class of algebras defined by identical relations) in which the subalgebras of free algebras are again free is called a Schreier variety.Thus the varieties of groups (Schreier [8]), abelian groups, vector spaces, linear (nonassociative) algebras (Kuros [5]), Lie algebras (Sirsov [9], Witt [12]), commutative algebras, anticommutative algebras (Sirsov [10]) are Schreier varieties.Some of the varieties are also what we may call Nielsen varieties, i.e., if gi,..., gn generate the subalgebra G of the free o-algebra P, then there is an effective procedure for obtaining a free set of generators for G. Thus the varieties of abelian groups, vector spaces, groups (Nielsen [7]), Lie algebras (Cohn [2]) are Nielsen varieties.The main purpose of this paper is to show ( §111 and §IV) that a variety of linear algebras over an infinite field is a Schreier variety if and only if it is a Nielsen variety.The main step of the proof is to show that one can "reduce" the elements gl9 ■.., gn of a free algebra in a Schreier variety to a free set of generators by applying a sequence of "elementary transformations" i.e., transformations which (a) replace a sequence Sx, ■ ■ ■, sn by a linear transform of Sx,.. .,snor (b) fix $i,..., 5"_i and replace sn by sn + w, with w in the algebra generated by jlt...,«._}.Our main theorem then enables us to find a set of generators for the automorphism group of a finitely generated free algebra in a Schreier variety (see Cohn [2] for the case of Lie algebras) and to solve the generalized word problem for the free algebras in Schreier varieties defined by finitely many multilinear identical relations ( §V).Our results apply in particular to the varieties of all linear algebras, commutative algebras, anticommutative algebras, and Lie algebras.II.Notation and preliminary results.1.All algebras are not necessarily associative linear algebras with identity 1 over the infinite field .If S is a subset of an algebra A, , Alg (S), Id (5) denote respectively the subspace, subalgebra, ideal of A generated by S.Let X be a set.We denote by F=F(X) the free algebra freely generated by X.Thus if M = M(X) is the free groupoid (see e.g., [1, p. 1]) on X, then M u {1} is a 0-basis for P, and the multiplication in P is an extension by linearity of the multiplication in M u {1}.A monomial in P is an element of M u {1}.The degree d(m) of a monomial is defined as usual by setting d(l)=0, d(x) = l for xeX and d(xm) = d(mx) = d(m) + 1 for x e X and me M. Let Hk be the subspace of