<p>The <strong class="pkg">Sophus</strong> package was originally designed to aid the author to classify some small-dimensional nilpotent Lie algebras over small fields. The classification follows the ideas that were used to classify small <span class="Math">p</span>-groups by O'Brien [O'B90]. The theory developed by O'Brien could easily be adopted to Lie algebras, and the details of this new theory can be found in [Sch]. Here we only summarise the main ideas, so that the user can understand the procedures implemented in this package. In this section L denotes a finitely generated, and hence finite-dimensional, nilpotent Lie algebra. Suppose that L has nilpotency class c, and hence the lower central series is as follows:
<p>We say that a basis <span class="Math">\mathcal B=\{b_1,\ldots,b_n\}</span> for <span class="Math">L</span> is <em>compatible with the lower central series</em> if there are indices <span class="Math">1=i_1<i_2<\cdots<i_{c}<n</span> such that <span class="Math">\{b_{i_k},\ldots,b_n\}</span> is a basis of <span class="Math">\gamma_k(L)</span> for <span class="Math">k\in\{1,\ldots,c\}</span>. We compute the structure constants table with respect to this basis, that is, we compute coefficients <span class="Math">\alpha_{i,j}^k</span> for <span class="Math">1\leq i<j<k\leq n</span> such that</p>
<p>Suppose that <span class="Math">b_i\in\gamma_j(L)\setminus\gamma_{j+1}(L)</span>. Then we say that the number <span class="Math">j</span> is the <em>weight</em> of the basis element <span class="Math">b_i</span>.</p>
<p>Note that in the nilpotent Lie algebra <span class="Math">L</span> minimal generating sets have the same size, namely the dimension of <span class="Math">L/L'. If \dim L/L'=d</span> then we call <span class="Math">L</span> a <span class="Math">d</span><em>-generator algebra</em>. We call a basis <span class="Math">\mathcal B</span> a <em>nilpotent basis</em> if the following hold.</p>
<ul>
<li><p>The basis <span class="Math">\mathcal B</span> is compatible with the lower central series.</p>
</li>
<li><p>For each <span class="Math">b_i\in \mathcal B</span> with weight <span class="Math">w\geq 2</span> there are <span class="Math">b_{j_1},\ b_{j_2}\in\mathcal B</span> with weight 1 and <span class="Math">w-1</span>, respectively such that <span class="Math">b_i=[b_{j_1},b_{j_2}]</span>. The product <span class="Math">[b_{j_1},b_{j_2}]</span> is called the definition of <span class="Math">b_i</span>.</p>
</li>
</ul>
<p>A Lie algebra <span class="Math">K</span> is said to be a <em>central extension</em> of <span class="Math">L</span> if <span class="Math">L\cong K/I</span> for some ideal <span class="Math">I</span> such that <span class="Math">I\leq Z(K)\cap K'. Suppose that c denotes the nilpotency class of L. Then a Lie algebra K is an immediate descendant of L if K has class c+1 and K/\gamma_{c+1}(K)\cong L. As in this case \gamma_{c+1}(K)\leq Z(K)\cap K'</span>, it follows that an immediate descendant <span class="Math">K</span> is a central extension of <span class="Math">L</span>. If <span class="Math">s=\dim \gamma_{c+1}(K)</span> then <span class="Math">K</span> is said to be a <em>step-<span class="Math">s</span></em> immediate descendant of <span class="Math">L</span>.</p>
<p>Let <span class="Math">L</span> be a <span class="Math">d</span>-generator nilpotent Lie algebra with class <span class="Math">c</span>, and let <span class="Math">F</span> be a free Lie algebra of rank <span class="Math">d</span>. Choose an ideal <span class="Math">I</span> of <span class="Math">F</span> such that <span class="Math">L\cong F/I</span>. Then the Lie algebra <span class="Math">L^*=F/[I,F]</span> is called the <em>Lie cover</em> of <span class="Math">L</span>. The <em>Lie multiplicator</em> in <span class="Math">L^*</span> is the subspace <span class="Math">I/[I,F]</span> and the <em>Lie nucleus</em> is <span class="Math">\gamma_c(L^*)</span>. It clear from the definition that <span class="Math">L^*/M\cong L</span>. It is verified in <a href="chapBib.html#biBSch">[Sch]</a> that, up to isomorphism, the Lie cover, the Lie multiplicator and the Lie nucleus are determined by the isomorphism type of <span class="Math">L</span>. Further, each central extension of the nilpotent Lie algebra <span class="Math">L</span> is a quotient of the Lie cover <span class="Math">L^*</span>. Thus it is possible to obtain all such descendants by first computing the Lie cover; this procedure is explained in <a href="chapBib.html#biBSch">[Sch]</a>. Similar ideas can be used to compute the automorphism group of a nilpotent Lie algebra, and to verify isomorphism between two nilpotent Lie algebras; see <a href="chapBib.html#biBSch">[Sch]</a> for details.</p>
<p>The main functions in <strong class="pkg">Sophus</strong> are thus able to compute</p>
<ul>
<li><p>a nilpotent basis for a nilpotent Lie algebra;</p>
</li>
<li><p>the cover of a nilpotent Lie algebra;</p>
</li>
<li><p>the immediate descendants of a nilpotent Lie algebra;</p>
</li>
<li><p>the full automorphism group of a nilpotent Lie algebra.</p>
</li>
</ul>
<p>There is also a function in the package to check if two nilpotent Lie algebras are isomorphic. After repeated applications of the immediate descendants algorithm, it is, in theory, possible to list all nilpotent Lie algebras of a given dimension over a prime field <span class="Math">\mathbb F_p</span>. Of course, this computation requires relatively large computational resources, and quickly becomes unfeasible as the dimension or the characteristic <span class="Math">p</span> grows.</p>
<p>The <strong class="pkg">Sophus</strong> package was written for the GAP~4 computer algebra system. In many procedures it is very important that we can compute the stabiliser of a subspace under some matrix group action. This is carried out using the procedures implemented in the <var class="Arg">autpgrp</var> package <a href="chapBib.html#biBautpgrp">[EO]</a>. Hence this package is required to run <strong class="pkg">Sophus</strong>.</p>
<p>The current version of <strong class="pkg">Sophus</strong> deals with general nilpotent Lie algebras over finite prime fields. If you are to compute with Lie algebras obtained from group algebras via the bracket operation, then another GAP package LAGUNA <a href="chapBib.html#biBLaguna">[RS]</a> may also offer some very efficient methods.</p>
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