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<p><a id="X809610728132CED7" name="X809610728132CED7"></a></p>
<div class="ChapSects"><a href="chap3_mj.html#X809610728132CED7">3 <span class="Heading"><strong class="pkg">Sophus</strong> functions</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7E043B9C80BF5DDF">3.1 <span class="Heading">Some general functions to compute with Lie algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7831B69779E3E5D6">3.1-1 SophusTest</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F110C75826C1A6D">3.1-2 IsLieNilpotentOverFp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A9FAEC37ACA9285">3.1-3 MinimalGeneratorNumber</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7C7898C07C7711DB">3.1-4 AbelianLieAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7A8E9A01835EDC3C">3.2 <span class="Heading">Functions to compute with nilpotent bases</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X799C8C57797AE5F0">3.2-1 NilpotentBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7E1BDBF08305FC7A">3.2-2 LieNBWeights</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X80CF319778583B3C">3.2-3 LieNBDefinitions</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X79D0807F784E9BEC">3.2-4 IsNilpotentBasis</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X809032BB7ECF5F48">3.2-5 IsLieAlgebraWithNB</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7F149872830B45BA">3.3 <span class="Heading">The cover</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X830028407DC2D80A">3.3-1 LieCover</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F4D3B8B7E9C30F7">3.3-2 CoverHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X783B60268536DD75">3.3-3 CoverOf</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87BB5AEA80EB4E46">3.3-4 IsLieCover</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F9DFA097A920EEF">3.3-5 LieMultiplicator</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X84C5C1A37BCCB7B9">3.3-6 LieNucleus</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7AD387A67CDFF8A9">3.4 <span class="Heading">Automorphisms of nilpotent Lie algebras</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X86E8712281D7A532">3.4-1 NilpotentLieAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X856F9F0B87F8673B">3.4-2 IdentityNilpotentLieAutomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X78778C3087CB7999">3.4-3 IsNilpotentLieAutomorphism</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X823B76A1836A4BD3">3.5 <span class="Heading">Automorphism group and isomorphism testing</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87677B0787B4461A">3.5-1 AutomorphismGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7A9A3C7E7804704B">3.5-2 AutomorphismGroupNilpotentLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X87B4D4C384E9B3DF">3.5-3 AreIsomorphicNilpotentLieAlgebras</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X8396FA6279C8E439">3.6 <span class="Heading">Descendants</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8396FA6279C8E439">3.6-1 Descendants</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X825E625F84273954">3.6-2 DescendantsOfStep1OfAbelianLieAlgebra</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X78DA04477DDD0ACE">3.7 <span class="Heading">Input and output</span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7B4AA5B97D4B53B9">3.7-1 WriteLieAlgebraToString</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X841498AD80B13D34">3.7-2 ReadStringToNilpotentLieAlgebraOverFp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7D54F5838172718B">3.7-3 WriteLieAlgebraListToFile</a></span>
</div></div>
</div>

<h3>3 <span class="Heading"><strong class="pkg">Sophus</strong> functions</span></h3>

<p><a id="X7E043B9C80BF5DDF" name="X7E043B9C80BF5DDF"></a></p>

<h4>3.1 <span class="Heading">Some general functions to compute with Lie algebras</span></h4>

<p><a id="X7831B69779E3E5D6" name="X7831B69779E3E5D6"></a></p>

<h5>3.1-1 SophusTest</h5>

<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>

<p><a id="X7F110C75826C1A6D" name="X7F110C75826C1A6D"></a></p>

<h5>3.1-2 IsLieNilpotentOverFp</h5>

<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>

<p><a id="X7A9FAEC37ACA9285" name="X7A9FAEC37ACA9285"></a></p>

<h5>3.1-3 MinimalGeneratorNumber</h5>

<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">\(L/L'\).



<p><a id="X7C7898C07C7711DB" name="X7C7898C07C7711DB"></a></p>

<h5>3.1-4 AbelianLieAlgebra</h5>

<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>

<p><a id="X7A8E9A01835EDC3C" name="X7A8E9A01835EDC3C"></a></p>

<h4>3.2 <span class="Heading">Functions to compute with nilpotent bases</span></h4>

<p><a id="X799C8C57797AE5F0" name="X799C8C57797AE5F0"></a></p>

<h5>3.2-1 NilpotentBasis</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NilpotentBasis</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Computes a nilpotent basis for <span class="SimpleMath">\(L\)</span>. Nilpotent bases are defined in Section <a href="chap1_mj.html#X818ED9677EDCB80E"><span class="RefLink">1</span></a>.</p>

<p><a id="X7E1BDBF08305FC7A" name="X7E1BDBF08305FC7A"></a></p>

<h5>3.2-2 LieNBWeights</h5>

<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>

<p><a id="X80CF319778583B3C" name="X80CF319778583B3C"></a></p>

<h5>3.2-3 LieNBDefinitions</h5>

<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>

<p><a id="X79D0807F784E9BEC" name="X79D0807F784E9BEC"></a></p>

<h5>3.2-4 IsNilpotentBasis</h5>

<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>

<p><a id="X809032BB7ECF5F48" name="X809032BB7ECF5F48"></a></p>

<h5>3.2-5 IsLieAlgebraWithNB</h5>

<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>

<p><a id="X7F149872830B45BA" name="X7F149872830B45BA"></a></p>

<h4>3.3 <span class="Heading">The cover</span></h4>

<p><a id="X830028407DC2D80A" name="X830028407DC2D80A"></a></p>

<h5>3.3-1 LieCover</h5>

<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>

<p><a id="X7F4D3B8B7E9C30F7" name="X7F4D3B8B7E9C30F7"></a></p>

<h5>3.3-2 CoverHomomorphism</h5>

<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 <var class="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>

<p><a id="X783B60268536DD75" name="X783B60268536DD75"></a></p>

<h5>3.3-3 CoverOf</h5>

<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 <var class="Arg">L</var> using the <var class="Arg">LieCover( L )</var> function call. This function returns <var class="Arg">L</var>.</p>

<p><a id="X87BB5AEA80EB4E46" name="X87BB5AEA80EB4E46"></a></p>

<h5>3.3-4 IsLieCover</h5>

<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>

<p><a id="X7F9DFA097A920EEF" name="X7F9DFA097A920EEF"></a></p>

<h5>3.3-5 LieMultiplicator</h5>

<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 <var class="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>

<p><a id="X84C5C1A37BCCB7B9" name="X84C5C1A37BCCB7B9"></a></p>

<h5>3.3-6 LieNucleus</h5>

<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 <var class="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>

<p><a id="X7AD387A67CDFF8A9" name="X7AD387A67CDFF8A9"></a></p>

<h4>3.4 <span class="Heading">Automorphisms of nilpotent Lie algebras</span></h4>

<p>We define a special class of automorphisms for our work.</p>

<p><a id="X86E8712281D7A532" name="X86E8712281D7A532"></a></p>

<h5>3.4-1 NilpotentLieAutomorphism</h5>

<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>

<p><a id="X856F9F0B87F8673B" name="X856F9F0B87F8673B"></a></p>

<h5>3.4-2 IdentityNilpotentLieAutomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdentityNilpotentLieAutomorphism</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 identity automorphism of <span class="SimpleMath">\(L\)</span>.</p>

<p><a id="X78778C3087CB7999" name="X78778C3087CB7999"></a></p>

<h5>3.4-3 IsNilpotentLieAutomorphism</h5>

<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>

<p><a id="X823B76A1836A4BD3" name="X823B76A1836A4BD3"></a></p>

<h4>3.5 <span class="Heading">Automorphism group and isomorphism testing</span></h4>

<p><a id="X87677B0787B4461A" name="X87677B0787B4461A"></a></p>

<h5>3.5-1 AutomorphismGroup</h5>

<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>

<p><a id="X7A9A3C7E7804704B" name="X7A9A3C7E7804704B"></a></p>

<h5>3.5-2 AutomorphismGroupNilpotentLieAlgebra</h5>

<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>

<p><a id="X87B4D4C384E9B3DF" name="X87B4D4C384E9B3DF"></a></p>

<h5>3.5-3 AreIsomorphicNilpotentLieAlgebras</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AreIsomorphicNilpotentLieAlgebras</code>( <var class="Arg">L</var>, <var class="Arg">K</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns <code class="keyw">true</code> if <var class="Arg">L</var> and <var class="Arg">K</var> are isomorphic; <code class="keyw">false</code> otherwise.</p>

<p><a id="X8396FA6279C8E439" name="X8396FA6279C8E439"></a></p>

<h4>3.6 <span class="Heading">Descendants</span></h4>

<p><a id="X8396FA6279C8E439" name="X8396FA6279C8E439"></a></p>

<h5>3.6-1 Descendants</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Descendants</code>( <var class="Arg">L</var>, <var class="Arg">step</var> )</td><td class="tdright">( method )</td></tr></table></div>
<p>Returns the <code class="keyw">step</code>-step descendants of a nilpotent Lie algebra <var class="Arg">L</var>.</p>

<p><a id="X825E625F84273954" name="X825E625F84273954"></a></p>

<h5>3.6-2 DescendantsOfStep1OfAbelianLieAlgebra</h5>

<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>

<p><a id="X78DA04477DDD0ACE" name="X78DA04477DDD0ACE"></a></p>

<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 source code <code class="code">io.gi</code> for the rest.</p>

<p><a id="X7B4AA5B97D4B53B9" name="X7B4AA5B97D4B53B9"></a></p>

<h5>3.7-1 WriteLieAlgebraToString</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ WriteLieAlgebraToString</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns a string that encodes the nilpotent Lie algebra <var class="Arg">L</var></p>

<p><a id="X841498AD80B13D34" name="X841498AD80B13D34"></a></p>

<h5>3.7-2 ReadStringToNilpotentLieAlgebraOverFp</h5>

<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>

<p><a id="X7D54F5838172718B" name="X7D54F5838172718B"></a></p>

<h5>3.7-3 WriteLieAlgebraListToFile</h5>

<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>


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