Lower central series of a free associative algebra over the integers and finite fields
Name
Etingof_Lower central series.pdf
Size
270.1 KB
Format
Adobe PDF
Checksum (MD5)
30f5b10eea01d033089ba0eb343cdd4b
Author(s)
Bhupatiraju, Surya
Jordan, David
Kuszmaul, William
Li, Jason
Etingof, Pavel I.
Date Issued
September 2012
Journal
Journal of Algebra
Publisher
Elsevier
Citation
Bhupatiraju, Surya, Pavel Etingof, David Jordan, William Kuszmaul, and Jason Li. “Lower Central Series of a Free Associative Algebra over the Integers and Finite Fields.” Journal of Algebra 372 (December 2012): 251–274.
Version
Author's final manuscript
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-NoDerivatives
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1016/j.jalgebra.2012.07.052
