<Chapter Label="Intro"> 
<Heading>Introduction</Heading>    

<Section Label="IntroFirst">
<Heading>General aims</Heading>

&LAGUNA; -- <B>L</B>ie <B>A</B>l<B>G</B>ebras and <B>UN</B>its of group
<B>A</B>lgebras -- is the new name of the &GAP;4 package &LAG;. The 
&LAG; package arose as a byproduct of the third author's PhD thesis
<!--[Richard Rossmanith, Centre-by-metabelian group algebras,
Friedrich-Schiller-Universitaet Jena, 1997]--> <Cite Key="Ros97"/>.  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. 
<P/>

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. 
<P/>

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 <M>p</M>-group over the field of <M>p</M> elements.  
<Index Key="SISYPHOS package">&SISYPHOS; package</Index> 
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 
<Cite Key="Wursthorn" />.  
<P/>

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 <M>p</M>-group into the normalized
unit group of its group algebra over the field of <M>p</M> 
elements, but this can easily be done with &LAGUNA;.

</Section>

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<Section Label="IntroFirstAndaHalf">
<Heading>General computations in group rings</Heading>

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 <M>p</M>-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 
<Ref Sect="GenSec"/>--<Ref Sect="Ideals"/> for more details.

</Section>

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<Section Label="IntroSecond">
<Heading>Computations in the normalized unit group</Heading>

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

It is well-known that the normalised unit group <M>V</M> has order
<M>|F|^{|G|-1}</M>, and so <M>V</M> is a finite <M>p</M>-group. Thus
computing <M>V</M> efficiently means to compute a polycyclic
presentation for <M>V</M>. For the theory of polycyclic presentations
refer to  <Cite Key="Sims" Where="Chapter 9"/>.  For this computation we
use an algorithm that was also used in the &SISYPHOS; package.  For a
brief description see Chapter <Ref Chap="Theory"/>.  The functions that
compute the structure of the normalised unit group are described in
Section <Ref Sect="UnitGroup"/>.

</Section>

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<Section Label="IntroThird">
<Heading>Computing Lie properties of the group algebra </Heading>

The functions that are used to compute Lie properties of 
<M>p</M>-modular group algebras were already included in the previous
versions of &LAG;.  The bracket operation <M>[\cdot,\cdot]</M> on a
<M>p</M>-modular  group algebra <M>FG</M> is defined by
<M>[a,b]=ab-ba</M>. It is  well-known and very easy to check that 
<M>(FG, +, [\cdot,\cdot])</M> is a Lie algebra. Then we may ask what 
kind of Lie algebra  properties are satisfied by <M>FG</M>. The results 
in <Cite Key="LR86"/>, <Cite Key="PPS73"/>, and <Cite Key="Ros00"/>  give
fast, practical algorithms to check whether the Lie algebra <M>FG</M> is
abelian, nilpotent, soluble, centre-by-metabelian, etc.  The functions
that implement these algorithms are described in  Section 
<Ref Sect="LieAlgebra"/>.

</Section>

<Section Label="IntroFourth">
<Heading>Installation and system requirements</Heading>

&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 or later and no compatibility with previous releases of &GAP;4 is
guaranteed. 
<P/>

To use the &LAGUNA; online help it is necessary to install the &GAP;4 
package &GAPDoc; by Frank L&#252;beck and Max Neunh&#246;ffer, which is 
available from the &GAP; site or from 
<URL>https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/</URL>.
<P/> 

&LAGUNA; is distributed as a <File>tar.gz</File> archive file and can be
obtained from <URL>https://gap-packages.github.io/laguna/</URL>. To unpack the
archive <File>laguna-X.X.X.tar.gz</File> you need the program
<File>tar</File>. To install &LAGUNA;, copy this archive into the
<File>pkg</File> subdirectory of your &GAP;4 installation.
The subdirectory <File>laguna</File> will be created in the 
<File>pkg</File> directory after the following command:
<P/>
  <C>tar -xf laguna-X.X.X.tar.gz</C>
<P/>

</Section>

</Chapter>
