<p><strong class="pkg">LAGUNA</strong> -- <strong class="button">L</strong>ie <strong class="button">A</strong>l<strong class="button">G</strong>ebras and <strong class="button">UN</strong>its of group <strong class="button">A</strong>lgebras -- is the new name of the <strong class="pkg">GAP</strong>4 package <strong class="pkg">LAG</strong>. The <strong class="pkg">LAG</strong> package arose as a byproduct of the third author's PhD thesis <a href="chapBib_mj.html#biBRos97">[Ros97]</a>. Its first version was ported to <strong class="pkg">GAP</strong>4 and was brought into the standard <strong class="pkg">GAP</strong>4 package format during his visit to St Andrews in September 1998.</p>
<p>The main objective of <strong class="pkg">LAG</strong> is to deal with Lie algebras associated with some associative algebras, and, in particular, Lie algebras of group algebras. Using <strong class="pkg">LAG</strong> 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 <strong class="pkg">LAGUNA</strong> the main part of the Lie algebra functionality is heavily built on the previous <strong class="pkg">LAG</strong> releases.</p>
<p>The <strong class="pkg">GAP</strong>4 package <strong class="pkg">LAGUNA</strong> also extends the <strong class="pkg">GAP</strong> 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. <strong class="pkg">LAGUNA</strong> also contains an implementation of an efficient algorithm to calculate the (normalized) unit group of the group algebra of a finite <spanclass="SimpleMath">\(p\)</span>-group over the field of <span class="SimpleMath">\(p\)</span> elements. Thus, the present version of <strong class="pkg">LAGUNA</strong> provides a part of the functionality of the <strong class="pkg">SISYPHOS</strong> program, which was developed by Martin Wursthorn to study the modular isomorphism problem; see <a href="chapBib_mj.html#biBWursthorn">[Wur93]</a>.</p>
<p>The corresponding functions of <strong class="pkg">LAGUNA</strong> use the same algorithmic and theoretical approach as those in <strong class="pkg">SISYPHOS</strong>. The reason why we reimplemented the normalised unit group algorithms in the <strong class="pkg">LAGUNA</strong> package is that <strong class="pkg">SISYPHOS</strong> has no interface to <strong class="pkg">GAP</strong>4, and, even in <strong class="pkg">GAP</strong>3, it is cumbersome to use the <strong class="pkg">SISYPHOS</strong> output for further computation with the normalised unit group. For instance, using <strong class="pkg">SISYPHOS</strong> with its <strong class="pkg">GAP</strong>3 interface, it is difficult to embed a finite <span class="SimpleMath">\(p\)</span>-group into the normalized unit group of its group algebra over the field of <span class="SimpleMath">\(p\)</span> elements, but this can easily be done with <strong class="pkg">LAGUNA</strong>.</p>
<h4>1.2 <span class="Heading">General computations in group rings</span></h4>
<p>The <strong class="pkg">LAGUNA</strong> package provides a set of functions to carry out some basic computations with a group ring and its elements. Among other things, <strong class="pkg">LAGUNA</strong> provides elementary functions to compute such basic notions as support, length, trace and augmentation of an element. For modular group algebras of finite <span class="SimpleMath">\(p\)</span>-groups <strong class="pkg">LAGUNA</strong> is able to calculate the power-structure of the augmentation ideal, which is useful for the construction of the normalised unit group; see Sections <a href="chap4_mj.html#X7B473F157842958E"><span class="RefLink">4.1</span></a>--<a href="chap4_mj.html#X841733AB86D30446"><span class="RefLink">4.3</span></a> for more details.</p>
<h4>1.3 <span class="Heading">Computations in the normalized unit group</span></h4>
<p>One of the aims of the <strong class="pkg">LAGUNA</strong> package is to carry out efficient computations in the normalised unit group of the group algebra <span class="SimpleMath">\(FG\)</span> of a finite <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span> over the field <span class="SimpleMath">\(F\)</span> of <span class="SimpleMath">\(p\)</span> elements. If <span class="SimpleMath">\(U\)</span> is the unit group of <span class="SimpleMath">\(FG\)</span> then it is easy to see that <span class="SimpleMath">\(U\)</span> is the direct product of <span class="SimpleMath">\(F^*\)</span> and <span class="SimpleMath">\(V(FG)\)</span>, where <span class="SimpleMath">\(F^*\)</span> is the multiplicative group of <span class="SimpleMath">\(F\)</span>, and <span class="SimpleMath">\(V(FG)\)</span> is the group of normalised units. A unit of <span class="SimpleMath">\(FG\)</span> of the form <span class="SimpleMath">\(\alpha_1 \cdot g_1 + \alpha_2 \cdot g_2 + \cdots + \alpha_k \cdot g_k\)</span> with <span class="SimpleMath">\(\alpha_i \in F\)</span> and <span class="SimpleMath">\(g_i \in G\)</span> is said to be normalised if the sum <span class="SimpleMath">\(\alpha_1 + \alpha_2 + \cdots + \alpha_k\)</span> is equal to <span class="SimpleMath">\(1\)</span>.</p>
<p>It is well-known that the normalised unit group <span class="SimpleMath">\(V\)</span> has order <span class="SimpleMath">\(|F|^{|G|-1}\)</span>, and so <span class="SimpleMath">\(V\)</span> is a finite <span class="SimpleMath">\(p\)</span>-group. Thus computing <span class="SimpleMath">\(V\)</span> efficiently means to compute a polycyclic presentation for <span class="SimpleMath">\(V\)</span>. For the theory of polycyclic presentations refer to <a href="chapBib_mj.html#biBSims">[Sim94, Chapter 9]</a>. For this computation we use an algorithm that was also used in the <strong class="pkg">SISYPHOS</strong> package. For a brief description see Chapter <a href="chap3_mj.html#X7D9FCE3A8526ACBE"><span class="RefLink">3</span></a>. The functions that compute the structure of the normalised unit group are described in Section <a href="chap4_mj.html#X863248708784F94C"><span class="RefLink">4.4</span></a>.</p>
<h4>1.4 <span class="Heading">Computing Lie properties of the group algebra </span></h4>
<p>The functions that are used to compute Lie properties of <span class="SimpleMath">\(p\)</span>-modular group algebras were already included in the previous versions of <strong class="pkg">LAG</strong>. The bracket operation <span class="SimpleMath">\([\cdot,\cdot]\)</span> on a <span class="SimpleMath">\(p\)</span>-modular group algebra <span class="SimpleMath">\(FG\)</span> is defined by <span class="SimpleMath">\([a,b]=ab-ba\)</span>. It is well-known and very easy to check that <span class="SimpleMath">\((FG, +, [\cdot,\cdot])\)</span> is a Lie algebra. Then we may ask what kind of Lie algebra properties are satisfied by <span class="SimpleMath">\(FG\)</span>. The results in <a href="chapBib_mj.html#biBLR86">[LR86]</a>, <a href="chapBib_mj.html#biBPPS73">[PPS73]</a>, and <a href="chapBib_mj.html#biBRos00">[Ros00]</a> give fast, practical algorithms to check whether the Lie algebra <span class="SimpleMath">\(FG\)</span> is abelian, nilpotent, soluble, centre-by-metabelian, etc. The functions that implement these algorithms are described in Section <a href="chap4_mj.html#X783C1A3D86A6656B"><span class="RefLink">4.5</span></a>.</p>
<h4>1.5 <span class="Heading">Installation and system requirements</span></h4>
<p><strong class="pkg">LAGUNA</strong> does not use external binaries and, therefore, works without restrictions on the type of the operating system. It is designed for <strong class="pkg">GAP</strong>4.4 or later and no compatibility with previous releases of <strong class="pkg">GAP</strong>4 is guaranteed.</p>
<p>To use the <strong class="pkg">LAGUNA</strong> online help it is necessary to install the <strongclass="pkg">GAP</strong>4 package <strong class="pkg">GAPDoc</strong> by Frank Lübeck and Max Neunhöffer, which is available from the <strong class="pkg">GAP</strong> site or from <span class="URL"><a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/">https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc/</a></span>.</p>
<p><strong class="pkg">LAGUNA</strong> is distributed as a <code class="file">tar.gz</code> archive file and can be obtained from <span class="URL"><a href="https://gap-packages.github.io/laguna/">https://gap-packages.github.io/laguna/</a></span>. To unpack the archive <code class="file">laguna-X.X.X.tar.gz</code> you need the program <code class="file">tar</code>. To install <strong class="pkg">LAGUNA</strong>, copy this archive into the <code class="file">pkg</code> subdirectory of your <strong class="pkg">GAP</strong>4 installation. The subdirectory <code class="file">laguna</code> will be created in the <code class="file">pkg</code> directory after the following command:</p>
