On Relations in the Universal Lie Nilpotent Associative Algebras of Class 3

Abstract

Let R be a unital associative and commutative ring and let A be a unital associative R -algebra. For arbitrary elements d i ( i = 1, … , n ) of A , define a left-normed commutator [ d 1 , d 2 , … , d n ] recursively, supposing that [ d 1 , d 2 ] = d 1 d 2 – d 2 d 1 , [ d 1 , … , d ( n -1) , d n ] = [[ d 1 , … , d ( n -1) ], d n ] ( n > 2). For any n > 1, let T ( n ) ( A ) denote the two-sided ideal in A generated by all commutators [ d 1 , d 2 , … , d n ] for all d i in A . Recall that an algebra A is called Lie nilpotent of class at most ( n -1) if T ( n ) ( A ) = 0. Now let A = R be the free unital associative R -algebra on a non-empty set X of free generators and let T ( n ) = T ( n ) ( A ). The quotient algebra A / T ( n ) is the universal (or, in another terminology, the relatively free ) unital associative R -algebra of Lie nilpotent of class (n-1) generated by a set X . Such universal algebras and their similar algebras have been intensively studying during the last decade. For their study, information about the relationships between generators of these universal algebras is of importance. Let Z be the ring of integers. The torsion part of the additive group of Z / T (4) has been explicitly described in [4]. This description is based on the following result: Let R be an arbitrary unital associative and commutative ring. Then T (4) is generated as a two-sided ideal in A by the polynomials (1) [ Y 1 , Y 2 , Y 3 , Y 4 ] , (2) [ Y 1 , Y 2 , Y 3 ] [ Y 4 , Y 5 , Y 6 ] , (3) [ Y 1 , Y 2 ] [ Y 3 , Y 4 , Y 5 ] + [ Y 1 , Y 5 ] [ Y 3 , Y 4 , Y 2 ] , (4) [ Y 1 , Y 2 ] [ Y 3 , Y 4 , Y 5 ] + [ Y 1 , Y 4 ] [ Y 3 , Y 2 , Y 5 ] , (5) ([ Y 1 , Y 2 ] [ Y 3 , Y 4 ] + [ Y 1 , Y 3 ] [ Y 2 , Y 4 ]) [ Y 5 , Y 6 ] where, for all i , Y i belongs to X . Let I be the two-sided ideal in A generated by all polynomials (1) – (5). It has been proved in [4] that I = T (4) , that is, i) I is contained in T (4) ; ii) T (4) is contained in I . The proof of the item ii) in [4] is based on a relatively sophisticated simultaneous induction according to degree of non-commutative polynomials that belong to five certain families. The aim of the present article is to give a new proof of ii) that is simpler than that of given in [4]. In our proof we make use of various simplifications and straightforward calculations and do not use induction. More precisely, we prove that the ideal T (4) is generated (as a two-sided ideal in A ) by the commutators of the form [a, Y 1 , b , Y 2 ] where Y 1 , Y 2 are elements of X and a , b are products of elements of X . Then we check that the commutators of such a form belong to I, therefore T (4) is contained in I .