On lower central series quotients of finitely generated algebras over Z

Abstract

Let A be an associative unital algebra, B k its successive quotients of lower central series and N k the successive quotients of ideals generated by lower central series. The geometric and algebraic aspects of B k 's and N k 's have been of great interest since the pioneering work of [11]. In this paper, we will concentrate on the case where A is a noncommutative polynomial algebra over Z modulo a single homogeneous relation. Both the torsion part and the free part of B k 's and N k 's are explored. Many examples are demonstrated in detail, and several general theorems are proved. Finally we end up with Appendix A about the torsion subgroups of N k (A n (Z)) and some open problems.

The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3

“…3.2]. Some further results concerning the quotients T (i) /T (i+1) for various associative rings A were obtained by Cordwell, Fei and Zhou in [7].…”

“…Interesting computational data about the torsion subgroup of T (i) /T (i+1) for various i was presented in [7]. In particular, this data suggests that the additive group of Z X /T (5) may have no torsion.…”

An Algebro-geometric Construction of Lower Central Series of Associative Algebras

“…Then the set {[x l , y], l ≥ 1} is linearly independent in N 2 . See [CF13] for an elaboration of this example, and related examples.…”

Os ideais de uma álgebra associativa gerados por comutadores e tópicos relacionados

“…Desde então, álgebras associativas Lie-nilpotente tem sido investigadas em vários trabalhos sob diversos pontos de vista; veja, por exemplo, [2], [21], [22], [24], [29], [31], [33] e [36]. por exemplo, em [1], [3], [4], [6], [10], [14], [16], [25] e [27].…”

“…Quanto aos polinômios(10) temos que s [a 1 , a 2 ][a 3 , a 4 ] + [a 1 , a 3 ][a 2 , a 4 ] = −s [a 1 , a 2 ][a 4 , a 3 ] + [a 1 , a 3 ][a 4 ,a 2 ] = Para verificar que W ⊂ I é suficiente mostrar que [bsc, a 1 , a 2 ] ∈ I para todos s ∈ S e b, c, a 1 , a 2 ∈ K X . Temos [bsc, a 1 , a 2…”