Introduction
General aims

&LAGUNA; -- Lie AlGebras and UNits of group Algebras -- is the new name of the &GAP;4 package &LAG;. The &LAG; package arose as a byproduct of the third author's PhD thesis . Its first version was ported to &GAP;4 and was brought into the standard &GAP;4 package format during his visit to St Andrews in September 1998.

The main objective of &LAG; is to deal with Lie algebras associated with some associative algebras, and, in particular, Lie algebras of group algebras. Using &LAG; it is possible to verify some properties or calculate certain Lie ideals of such Lie algebras very efficiently, due to their special structure. In the current version of &LAGUNA; the main part of the Lie algebra functionality is heavily built on the previous &LAG; releases.

The &GAP;4 package &LAGUNA; also extends the &GAP; functionality for calculations with units of modular group algebras. In particular, using this package, one can check whether an element of such a group algebra is invertible. &LAGUNA; also contains an implementation of an efficient algorithm to calculate the (normalized) unit group of the group algebra of a finite p-group over the field of p elements. &SISYPHOS; package Thus, the present version of &LAGUNA; provides a part of the functionality of the &SISYPHOS; program, which was developed by Martin Wursthorn to study the modular isomorphism problem; see .

The corresponding functions of &LAGUNA; use the same algorithmic and theoretical approach as those in &SISYPHOS;. The reason why we reimplemented the normalised unit group algorithms in the &LAGUNA; package is that &SISYPHOS; has no interface to &GAP;4, and, even in &GAP;3, it is cumbersome to use the &SISYPHOS; output for further computation with the normalised unit group. For instance, using &SISYPHOS; with its &GAP;3 interface, it is difficult to embed a finite p-group into the normalized unit group of its group algebra over the field of p elements, but this can easily be done with &LAGUNA;.

General computations in group rings

The &LAGUNA; package provides a set of functions to carry out some basic computations with a group ring and its elements. Among other things, &LAGUNA; provides elementary functions to compute such basic notions as support, length, trace and augmentation of an element. For modular group algebras of finite p-groups &LAGUNA; is able to calculate the power-structure of the augmentation ideal, which is useful for the construction of the normalised unit group; see Sections -- for more details.

Computations in the normalized unit group

One of the aims of the &LAGUNA; package is to carry out efficient computations in the normalised unit group of the group algebra FG of a finite p-group G over the field F of p elements. If U is the unit group of FG then it is easy to see that U is the direct product of F^* and V(FG), where F^* is the multiplicative group of F, and V(FG) is the group of normalised units. A unit of FG of the form \alpha_1 \cdot g_1 + \alpha_2 \cdot g_2 + \cdots + \alpha_k \cdot g_k with \alpha_i \in F and g_i \in G is said to be normalised if the sum \alpha_1 + \alpha_2 + \cdots + \alpha_k is equal to 1.

It is well-known that the normalised unit group V has order |F|^{|G|-1}, and so V is a finite p-group. Thus computing V efficiently means to compute a polycyclic presentation for V. For the theory of polycyclic presentations refer to . For this computation we use an algorithm that was also used in the &SISYPHOS; package. For a brief description see Chapter . The functions that compute the structure of the normalised unit group are described in Section .

Computing Lie properties of the group algebra

The functions that are used to compute Lie properties of p-modular group algebras were already included in the previous versions of &LAG;. The bracket operation [\cdot,\cdot] on a p-modular group algebra FG is defined by [a,b]=ab-ba. It is well-known and very easy to check that (FG, +, [\cdot,\cdot]) is a Lie algebra. Then we may ask what kind of Lie algebra properties are satisfied by FG. The results in , , and give fast, practical algorithms to check whether the Lie algebra FG is abelian, nilpotent, soluble, centre-by-metabelian, etc. The functions that implement these algorithms are described in Section .

Installation and system requirements

&LAGUNA; does not use external binaries and, therefore, works without restrictions on the type of the operating system. It is designed for &GAP;4.4 and no compatibility with previous releases of &GAP;4 is guaranteed.

To use the &LAGUNA; online help it is necessary to install the &GAP;4 package &GAPDoc; by Frank L\"ubeck and Max Neunh\"offer, which is available from the &GAP; site or from http://www.math.rwth-aachen.de/˜Frank.Luebeck/GAPDoc/.

&LAGUNA; is distributed in standard formats (zoo, tar.gz, tar.bz2, -win.zip) and can be obtained from http://www.cs.st-andrews.ac.uk/˜alexk/laguna.htm. To unpack the archive laguna-3.4.zoo you need the program unzoo, which can be obtained from the &GAP; homepage http://www.gap-system.org/ (see section `Distribution'). To install &LAGUNA;, copy this archive into the pkg subdirectory of your &GAP;4.4 installation. The subdirectory laguna will be created in the pkg directory after the following command:

unzoo -x laguna-3.4.zoo