Lower central series of a free associative algebra over the integers and finite fields

Abstract

Consider the free algebra A[subscript n] generated over Q by n generators x[subscript 1],…,x[subscript n]. Interesting objects attached to A=A[subscript n] are members of its lower central series, L[subscript i] = L[subscript i](A), defined inductively by L[subscript 1] = A, L[subscript i + 1] =[A,L[subscript i]], and their associated graded components B[subscript i] = B[subscript i](A) defined as B[subscript i] = L[subscript i]/L[subscript i + 1]. These quotients B[subscript i] for i ⩾ 2, as well as the reduced quotient [bar over B][subscript 1] = A/(L[subscript 2] + AL[subscript 3]), exhibit a rich geometric structure, as shown by Feigin and Shoikhet (2007) [FS] and later authors (Dobrovolska et al., 1997 [DKM], Dobrovolska and Etingof, 2008 [DE], Arbesfeld and Jordan, 2010 [AJ], Bapat and Jordan, 2010 [BJ]). We study the same problem over the integers Z and finite fields F[subscript p]. New phenomena arise, namely, torsion in B[subscript i] over Z, and jumps in dimension over F[subscript p]. We describe the torsion in the reduced quotient [bar over B][subscript 1] and B[subscript 2] geometrically in terms of the De Rham cohomology of Z[superscript n]. As a corollary we obtain a complete description of [bar over B][subscript 1](A[subscript n](Z)) and [bar over B][subscript 1](A[subscript n](F[subscript p])), as well as of B[subscript 2](A[subscript n](Z[1/2])) and B[subscript 2](A[subscript n](F[subscript p])), p > 2. We also give theoretical and experimental results for B[subscript i] with i > 2, formulating a number of conjectures and questions on their basis. Finally, we discuss the supercase, when some of the generators are odd and some are even, and provide some theoretical results and experimental data in this case.