Computing Nilpotent Quotients in Finitely Presented Lie Rings

Abstract

A nilpotent quotient algorithm for finitely presented Lie rings over Z (LieNQ) is described. The paper studies graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. The nilpotent presentation consists of generators for the abelian group and the products---expressed as linear combinations---for pairs formed by generators. Using that presentation the word problem is decidable in $L$. Provided that the Lie ring $L$ is graded, it is possible to determine the canonical presentation for a lower central factor of $L$. LieNQ's complexity is studied and it is shown that optimizing the presentation is NP-hard. Computational details are provided with examples, timing and some structure theorems obtained from computations. Implementation in C and GAP 3.5 interface is available.