GAP (Guarana) - Chapter 1: Introduction

Goto Chapter: Top 1 2 3 4 Bib Ind
[Top of Book] [Contents] [Previous Chapter] [Next Chapter]

[MathJax on]

1 Introduction
1.1 About Guarana
1.2 Setup for computing the correspondence
1.3 Collection

### 1 Introduction

#### 1.1 About Guarana

In this package we demonstrate the algorithmic usefulness of the so-called Mal'cev correspondence for computations with infinite polycyclic groups; it is a correspondence that associates to every Q-powered nilpotent group H a unique rational nilpotent Lie algebra L_H and vice-versa. The Mal'cev correspondence was discovered by Anatoly Mal'cev in 1951 [Mal51].

#### 1.2 Setup for computing the correspondence

Let G be a finitely generated torsion-free nilpotent group, i.e.\ a T-group. Then G can be embedded in a Q-powered hull G^. The group G^ is a Q-powered nilpotent group and is unique up to isomorphism. We denote the Lie algebra which corresponds to G^ under the Mal'cev correspondence by L(G)= L_G^. We provide an algorithm for setting up the Mal'cev correspondence between G^ and the Lie algebra L(G). That is, if G is given by a polycyclic presentation with respect to a Mal'cev basis, then we can compute a structure constants table of L(G). Furthermore for a given g∈ G we can compute the corresponding element in L(G) and vice versa.

#### 1.3 Collection

Every element of a polycyclically presented group has a unique normal form. An algorithm for computing this normal form is called a collection algorithm. Such an algorithm lies at the heart of most methods dealing with polycyclically presented groups. The current state of the art is collection from the left [Geb02][LGS90][VL90] }. This package contains a new collection algorithm for polycyclically presented groups, which we call Mal'cev collection [AL07]. Mal'cev collection is in some cases dramatically faster than collection from the left, while using less memory.

[Top of Book] [Contents] [Previous Chapter] [Next Chapter]
Goto Chapter: Top 1 2 3 4 Bib Ind

generated by GAPDoc2HTML