Computing nilpotent quotients in finitely presented Lie rings

Csaba Schneider

doi:10.46298/dmtcs.234

Abstract

A nilpotent quotient algorithm for finitely presented Lie rings over \textbfZ (and \textbfQ) is described. The paper studies the graded and non-graded cases separately. The algorithm computes the so-called nilpotent presentation for a finitely presented, nilpotent Lie ring. A 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. Complexity is studied and it is shown that optimising the presentation is NP-hard. Computational details are provided with examples, timing and some structure theorems obtained from computations. Implementation in C and GAP interface are available.

Highlights

The nilpotent quotient algorithm for finitely presented Lie rings – LIENQ as we shall refer to it in the sequel – operates with Lie rings over Z
By efficient and useful we mean that several important pieces of information can be read off immediately (e.g.: nilpotency, nilpotency class etc). By this presentation the so-called word problem is decidable, it is known to be undecidable in the general case
The first part contains the basic properties of the nilpotent presentation and describes an algorithm to compute it in the general case

Summary

Introduction

The nilpotent quotient algorithm for finitely presented Lie rings – LIENQ as we shall refer to it in the sequel – operates with Lie rings over Z. By efficient and useful we mean that several important pieces of information can be read off immediately (e.g.: nilpotency, nilpotency class etc) By this presentation the so-called word problem is decidable, it is known to be undecidable in the general case. In details, (2) is \similar” to the Jacobi identity while (3) and (4) can be interpreted as some \distributive” property of the group commutator This analogy allows us to alter the known group algorithms for Lie rings. The first part contains the basic properties of the nilpotent presentation and describes an algorithm to compute it in the general case. The first lemma is of fundamental importance to our goal, since it essentially shows that it is realistic to build a nilpotent quotient algorithm for finitely presented Lie rings. The lemma is a consequence of the argument presented in [5] on basic commutators

The Nilpotent Presentation of a Lie Ring

Computing a Weighted Nilpotent Presentation

Computing the Abelian Factor

Extending the Presentation

Enforcing Consistency

Enforcing the Defining Relations

Simplifying LieNQ in Graded Lie Rings

Another approach – Canonical Weighted Nilpotent Presentation

Computing the Canonical Weighted Nilpotent Presentation

Implementing LieNQ in C

GAP Interface

Some Sample Computations