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

Abstract

Consider the free algebra A_n generated over Q by n generators x_1, ..., x_n. Interesting objects attached to A = A_n are members of its lower central series, L_i = L_i(A), defined inductively by L_1 = A, L_{i+1} = [A,L_{i}], and their associated graded components B_i = B_i(A) defined as B_i=L_i/L_{i+1}. These quotients B_i, for i at least 2, as well as the reduced quotient \bar{B}_1=A/(L_2+A L_3), exhibit a rich geometric structure, as shown by Feigin and Shoikhet and later authors, (Dobrovolska-Kim-Ma,Dobrovolska-Etingof,Arbesfeld-Jordan,Bapat-Jordan). We study the same problem over the integers Z and finite fields F_p. New phenomena arise, namely, torsion in B_i over Z, and jumps in dimension over F_p. We describe the torsion in the reduced quotient RB_1 and B_2 geometrically in terms of the De Rham cohomology of Z^n. As a corollary we obtain a complete description of \bar{B}_1(A_n(Z)) and \bar{B}_1(A_n(F_p)), as well as of B_2(A_n(Z[1/2])) and B_2(A_n(F_p)), p>2. We also give theoretical and experimental results for B_i with i>2, formulating a number of conjectures and questions based on them. Finally, we discuss the supercase, when some of the generators are odd (fermionic) and some are even (bosonic), and provide some theoretical results and experimental data in this case.

References in corpus (3)

• The additive group of a Lie nilpotent associative ring
• On $[A,A]/[A,[A,A]]$ and on a $W_n$-action on the consecutive commutators of free associative algebra
• Cohomologie de de Rham entiere (Integral de Rham cohomology)

Cited by in corpus (10)

• The additive group of a Lie nilpotent associative ring
• Hilbert Basis Theorem and Finite Generation of Invariants in Symmetric Fusion Categories in Positive Characteristic
• The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3
• Products of commutators in a Lie nilpotent associative algebra
• Relations in universal Lie nilpotent associative algebras of class 4
• On Lower Central Series Quotients of Finitely Generated Algebras over $\mathbb{Z}$
• On some products of commutators in an associative ring
• An algebro-geometric construction of lower central series of associative algebras
• Products of several commutators in a Lie nilpotent associative algebra
• On Lie nilpotent associative algebras