<!-- $Id: intro.xml,v 1.21 2007/02/07 17:15:13 alexk Exp $ --> <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> <!-- ********************************************************* --> <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> <!-- ********************************************************* --> <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> <!-- ********************************************************* --> <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 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\"ubeck and Max Neunh\"offer, which is available from the &GAP; site or from <URL>http://www.math.rwth-aachen.de/˜Frank.Luebeck/GAPDoc/</URL>. <P/> &LAGUNA; is distributed in standard formats (<File>zoo</File>, <File>tar.gz</File>, <File>tar.bz2</File>, <File>-win.zip</File>) and can be obtained from <URL>http://www.cs.st-andrews.ac.uk/˜alexk/laguna.htm</URL>. To unpack the archive <File>laguna-3.4.zoo</File> you need the program <File>unzoo</File>, which can be obtained from the &GAP; homepage <URL>http://www.gap-system.org/</URL> (see section `Distribution'). To install &LAGUNA;, copy this archive into the <File>pkg</File> subdirectory of your &GAP;4.4 installation. The subdirectory <File>laguna</File> will be created in the <File>pkg</File> directory after the following command: <P/> <C>unzoo -x laguna-3.4.zoo</C> <P/> </Section> </Chapter>