<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SophusTest</code>( )</td><td class="tdright">( function )</td></tr></table></div>
<p>Tests <strong class="pkg">Sophus</strong> functions, returns true if it finds no mistakes, and returns false otherwise. May take a couple of minutes to complete.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLieNilpotentOverFp</code>( <var class="Arg">L</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns true if <var class="Arg">L</var> is a nilpotent Lie algebra and its underlying field is a finite prime field.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MinimalGeneratorNumber</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Computes the minimal number of generators for <span class="SimpleMath">\(L\)</span>, which is the dimension of <span class="SimpleMath">\(L/L'\).
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AbelianLieAlgebra</code>( <var class="Arg">F</var>, <var class="Arg">d</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the Abelian Lie algebra with dimension <span class="SimpleMath">\(d\)</span> over the field <var class="Arg">F</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieNBWeights</code>( <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Every element of the nilpotent basis <span class="SimpleMath">\(B\)</span> has a weight; See Section <a href="chap1_mj.html#X818ED9677EDCB80E"><span class="RefLink">1</span></a>. This function returns the list of these weights.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieNBDefinitions</code>( <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>This function returns a list. The <var class="Arg">i</var>-th element of this list is 0 if <var class="Arg">B[i]</var> has weight 1. Otherwise the <var class="Arg">i</var>-th element is <var class="Arg">[k,l]</var> if the definition of <var class="Arg">B[i]</var> is <var class="Arg">[B[k],B[l]]</var>. See Section <a href="chap1_mj.html#X818ED9677EDCB80E"><span class="RefLink">1</span></a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNilpotentBasis</code>( <var class="Arg">B</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns <code class="keyw">true</code> if the basis <var class="Arg">B</var> of a Lie algebra was computed with the function <code class="code">NilpotentBasis</code>; <code class="keyw">false</code> otherwise.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLieAlgebraWithNB</code>( <var class="Arg">L</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns <code class="keyw">true</code> if a nilpotent basis for <var class="Arg">L</var> has already been computed using the function <code class="code">NilpotentBasis</code>; <code class="keyw">false</code> otherwise.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieCover</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Computes the cover for the nilpotent Lie algebra <span class="SimpleMath">\(L\)</span> as defined in Section <a href="chap1_mj.html#X818ED9677EDCB80E"><span class="RefLink">1</span></a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoverHomomorphism</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The nilpotent Lie algebra <var class="Arg">C</var> was obtained from a nilpotent Lie algebra <varclass="Arg">L</var> using the <var class="Arg">LieCover( L )</var> function call. This function returns the natural homomorphism from <var class="Arg">C</var> onto <var class="Arg">L</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CoverOf</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The nilpotent Lie algebra <var class="Arg">C</var> was obtained from a nilpotent Lie algebra <varclass="Arg">L</var> using the <var class="Arg">LieCover( L )</var> function call. This function returns <var class="Arg">L</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsLieCover</code>( <var class="Arg">C</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns <code class="keyw">true</code> if the Lie algebra <var class="Arg">C</var> was obtained as the Lie cover of another Lie algebra <var class="Arg">L</var> using the <var class="Arg">LieCover( L )</var> function call.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieMultiplicator</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The nilpotent Lie algebra <var class="Arg">C</var> was obtained from a nilpotent Lie algebra <varclass="Arg">L</var> using the <var class="Arg">LieCover( L )</var> function call. This function returns the central ideal of <var class="Arg">C</var> which is the multiplicator of <var class="Arg">L</var>; see Section <a href="chap1_mj.html#X818ED9677EDCB80E"><span class="RefLink">1</span></a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieNucleus</code>( <var class="Arg">C</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The nilpotent Lie algebra <var class="Arg">C</var> was obtained from a nilpotent Lie algebra <varclass="Arg">L</var> using the <var class="Arg">LieCover( L )</var> function call. This function returns the central ideal of <var class="Arg">C</var> which is the nucleus of <var class="Arg">L</var>; see Section <a href="chap1_mj.html#X818ED9677EDCB80E"><span class="RefLink">1</span></a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NilpotentLieAutomorphism</code>( <var class="Arg">L</var>, <var class="Arg">gens</var>, <var class="Arg">imgs</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p><var class="Arg">L</var> is a nilpotent Lie algebra, <var class="Arg">gens</var> is a generating set, and <var class="Arg">imgs</var> is a subset of <var class="Arg">L</var> with the same length as <var class="Arg">gens</var>. Returns the automorphism of <var class="Arg">L</var> which maps the element of <var class="Arg">gens</var> to the elements of <var class="Arg">imgs</var>. It is the responsibility of the user to make sure that the arguments are given so that the automorphism exists. These automorphisms can be compared, multiplied using the <var class="Arg">*</var> sign, and the inverse of such an automorphism can also be computed in the usual manner.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsNilpotentLieAutomorphism</code>( <var class="Arg">A</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns <code class="keyw">true</code> if <var class="Arg">A</var> was obtained using a <var class="Arg">NilpotentLieAutomorphism</var> or an <var class="Arg">IdentityNilpotentLieAutomorphism</var> function call.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroup</code>( <var class="Arg">L</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p><var class="Arg">L</var> is a nilpotent Lie algebra; returns the automorphism group of <var class="Arg">L</var> as a group generated by <strong class="pkg">GAP</strong> algebra automorphisms. The automorphism group is computed as explained in <a href="chapBib_mj.html#biBSch">[Sch]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AutomorphismGroupNilpotentLieAlgebra</code>( <var class="Arg">L</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p><var class="Arg">L</var> is a nilpotent Lie algebra; returns the automorphism group of <var class="Arg">L</var> in the internally used hybrid format. The automorphism group is computed as explained in <a href="chapBib_mj.html#biBSch">[Sch]</a>. The hybrid format, which is very similar to the one used in <a href="chapBib_mj.html#biBautpgrp">[EO]</a>, is a record that contains the following fields.</p>
<ul>
<li><p><code class="code">glAutos</code>: a set of automorphisms which together with <code class="code">agAutos</code> generate the automorphism group;</p>
</li>
<li><p><code class="code">glOrder</code>: an integer whose product with the numbers in <code class="code">agOrder</code> gives the size of the automorphism group;</p>
</li>
<li><p><code class="code">agAutos</code>: a polycyclic generating sequence for a soluble normal subgroup of the automorphism group;</p>
</li>
<li><p><code class="code">agOrder</code>: the relative orders corresponding to <code class="code">agAutos</code>;</p>
</li>
<li><p><code class="code">liealg</code>: The Lie algebra acted upon by the automorphisms.</p>
</li>
<li><p><code class="code">size</code>: the size of the automorphism group.</p>
</li>
<li><p><code class="code">field</code>: the underlying field of the Lie algebra.</p>
</li>
<li><p><code class="code">prime</code>: the characteristic of the underlying field.</p>
</li>
</ul>
<p>We do not return an automorphism group in the standard form because we wish to distinguish between <code class="code">agAutos</code> and <code class="code">glAutos</code>; the latter act non-trivially on the derived quotient of <span class="SimpleMath">\(L\)</span>. This hybrid-group description of the automorphism group permits more efficient computations with it.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ DescendantsOfStep1OfAbelianLieAlgebra</code>( <var class="Arg">d</var>, <var class="Arg">p</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns the <code class="keyw">1</code>-step descendants of the abelian Lie algebra with dimension <var class="Arg">d</var> defined over the field of <var class="Arg">p</var> elements.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">DescendantsOfStep1OfAbelianLieAlgebra(4,3);</span>
[ <Lie algebra of dimension 5 over GF(3)>,
<Lie algebra of dimension 5 over GF(3)> ]
</pre></div>
<h4>3.7 <span class="Heading">Input and output</span></h4>
<p>The package provides with a number of functions that can be used to store lists of Lie algebras. Here we document only the most important ones, see the sourcecode <code class="code">io.gi</code> for the rest.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ReadStringToNilpotentLieAlgebraOverFp</code>( <var class="Arg">string</var>, <var class="Arg">p</var>, <var class="Arg">d</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Decodes <var class="Arg">string</var> into a <var class="Arg">d</var>-dimensional nilpotent Lie algebra defined over the field of <var class="Arg">p</var> elements.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WriteLieAlgebraListToFile</code>( <var class="Arg">list</var>, <var class="Arg">name</var>, <var class="Arg">file</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">list</var> is a list of nilpotent Lie algebras. Encodes each Lie algebra in <var class="Arg">list</var> to a string. The list so obtained is written into <var class="Arg">file</var>. The name of this list will be <var class="Arg">name</var>.</p>
