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

Abstract

Consider the free algebra An generated over Q by n generators x1,…,xn. Interesting objects attached to A=An are members of its lower central series, Li=Li(A), defined inductively by L1=A, Li+1=[A,Li], and their associated graded components Bi=Bi(A) defined as Bi=Li/Li+1. These quotients Bi for i⩾2, as well as the reduced quotient B¯1=A/(L2+AL3), 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 Fp. New phenomena arise, namely, torsion in Bi over Z, and jumps in dimension over Fp. We describe the torsion in the reduced quotient B¯1 and B2 geometrically in terms of the De Rham cohomology of Zn. As a corollary we obtain a complete description of B¯1(An(Z)) and B¯1(An(Fp)), as well as of B2(An(Z[1/2])) and B2(An(Fp)), p>2. We also give theoretical and experimental results for Bi 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.