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<title>GAP (Guarana) - Chapter 2: Computing the Mal'cev correspondence
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<p><a id="X781C0216819C09A8" name="X781C0216819C09A8"></a></p>
<div class="ChapSects"><a href="chap2_mj.html#X781C0216819C09A8">2 <span class="Heading">Computing the Mal'cev correspondence
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X7D3DC4ED855DC13C">2.1 <span class="Heading">The main functions</span></a>
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<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7A3976DA834C0E90">2.1-1 MalcevObjectByTGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7A6720DD7B4AF466">2.1-2 UnderlyingGroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7CA021E28527763E">2.1-3 UnderlyingLieAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7E6926C6850E7C4E">2.1-4 Dimension</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X860138DA82C4F56D">2.1-5 MalcevGrpElementByExponents</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X849FCF377F5E29C2">2.1-6 MalcevLieElementByCoefficients</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X86CC8994841B1193">2.1-7 RandomGrpElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X876F868B78430678">2.1-8 RandomLieElm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7E7C986487C4EB02">2.1-9 Log</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X86E48BC98197839E">2.1-10 Exp</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X7857704878577048">2.1-11 *</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X80761843831B468E">2.1-12 Comm</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X86C4910279A45173">2.1-13 MalcevSymbolicGrpElementByExponents</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap2_mj.html#X849FCF377F5E29C2">2.1-14 MalcevLieElementByCoefficients</a></span>
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap2_mj.html#X81CAD2F27B2066C4">2.2 <span class="Heading">An example application</span></a>
</span>
</div>
</div>

<h3>2 <span class="Heading">Computing the Mal'cev correspondence

<p><a id="X7D3DC4ED855DC13C" name="X7D3DC4ED855DC13C"></a></p>

<h4>2.1 <span class="Heading">The main functions</span></h4>

<p>Let <span class="SimpleMath">\(G\)</span> be a <span class="SimpleMath">\(T\)</span>-group and <span class="SimpleMath">\(G^\)</span> its <span class="SimpleMath">\(\Q\)</span>-powered hull. In this chapter we describe functionality for setting up the Mal'cev correspondence between \(G^\) and the Lie algebra \(L(G)\). The data structures needed for computations with \(G^\) and \(L(G)\) are stored in a so-called Mal'cev object. Computational representations of elements of <span class="SimpleMath">\(G^\)</span>, respectively <span class="SimpleMath">\(L(G)\)</span>, will be called Mal'cev group elements, respectively Mal'cev Lie elements.</p>

<p><a id="X7A3976DA834C0E90" name="X7A3976DA834C0E90"></a></p>

<h5>2.1-1 MalcevObjectByTGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MalcevObjectByTGroup</code>( <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>If <var class="Arg">N</var> is a a <var class="Arg">T</var>-group (i.e. a finitely generated torsion-free nilpotent group), given by a polycyclic presentation with respect to a Mal'cev basis, then this function computes the Mal'cev correspondence for <var class="Arg">N</var> and stores the result in a so-called Mal'cev object. Otherwise this function returns `fail'. In the moment this function is restricted to groups <var class="Arg">N</var> of nilpotency class at most 9.</p>

<p><a id="X7A6720DD7B4AF466" name="X7A6720DD7B4AF466"></a></p>

<h5>2.1-2 UnderlyingGroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingGroup</code>( <var class="Arg">mo</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev object mo this function returns the T-group, which was used to build mo.



<p><a id="X7CA021E28527763E" name="X7CA021E28527763E"></a></p>

<h5>2.1-3 UnderlyingLieAlgebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ UnderlyingLieAlgebra</code>( <var class="Arg">mo</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev object mo this function returns the Lie algebra, which underlies the correspondence described by mo.



<p><a id="X7E6926C6850E7C4E" name="X7E6926C6850E7C4E"></a></p>

<h5>2.1-4 Dimension</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Dimension</code>( <var class="Arg">mo</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the dimension of the Lie algebra that underlies the Mal'cev object mo.



<p><a id="X860138DA82C4F56D" name="X860138DA82C4F56D"></a></p>

<h5>2.1-5 MalcevGrpElementByExponents</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MalcevGrpElementByExponents</code>( <var class="Arg">mo</var>, <var class="Arg">exps</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev object mo and an exponent vector exps with rational entries, this functions returns the Mal'cev group element, which has exponents <var class="Arg">exps</var> with respect to the Mal'cev basis of the underlying group of mo.



<p><a id="X849FCF377F5E29C2" name="X849FCF377F5E29C2"></a></p>

<h5>2.1-6 MalcevLieElementByCoefficients</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MalcevLieElementByCoefficients</code>( <var class="Arg">mo</var>, <var class="Arg">coeffs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev object mo and a coefficient vector coeffs with rational entries, this functions returns the Mal'cev Lie element, which has coefficients <var class="Arg">coeffs</var> with respect to the basis of the underlying Lie algebra of <var class="Arg">mo</var>.</p>

<p><a id="X86CC8994841B1193" name="X86CC8994841B1193"></a></p>

<h5>2.1-7 RandomGrpElm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomGrpElm</code>( <var class="Arg">mo</var>, <var class="Arg">range</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev object mo this function returns the output of MalcevGrpElementByExponents( mo, exps ), where exps is an exponent vector whose entries are randomly chosen integers between -range and range.



<p><a id="X876F868B78430678" name="X876F868B78430678"></a></p>

<h5>2.1-8 RandomLieElm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RandomLieElm</code>( <var class="Arg">mo</var>, <var class="Arg">range</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev object mo this function returns the output of MalcevLieElementByExponents( mo, coeffs ), where coeffs is a coefficient vector whose entries are randomly chosen integers between -range and range.



<p><a id="X7E7C986487C4EB02" name="X7E7C986487C4EB02"></a></p>

<h5>2.1-9 Log</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Log</code>( <var class="Arg">g</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For Mal'cev group element g this function returns the corresponding Mal'cev Lie element.</p>

<p><a id="X86E48BC98197839E" name="X86E48BC98197839E"></a></p>

<h5>2.1-10 Exp</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Exp</code>( <var class="Arg">x</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For Mal'cev Lie element x this function returns the corresponding Mal'cev group element.</p>

<p><a id="X7857704878577048" name="X7857704878577048"></a></p>

<h5>2.1-11 *</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ *</code>( <var class="Arg">g</var>, <var class="Arg">h</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the product of Mal'cev group elements.



<p><a id="X80761843831B468E" name="X80761843831B468E"></a></p>

<h5>2.1-12 Comm</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Comm</code>( <var class="Arg">x</var>, <var class="Arg">y</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>If <var class="Arg">x</var>,<var class="Arg">y</var> are Mal'cev group elements, then this function returns the group theoretic commutator of x and y. If x,y are Mal'cev Lie elements, then this function returns the Lie commutator of <var class="Arg">x</var> and <var class="Arg">y</var>.</p>

<p><a id="X86C4910279A45173" name="X86C4910279A45173"></a></p>

<h5>2.1-13 MalcevSymbolicGrpElementByExponents</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MalcevSymbolicGrpElementByExponents</code>( <var class="Arg">mo</var>, <var class="Arg">exps</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev object mo and an exponent vector exps with rational indeterminates as entries, this functions returns the Mal'cev group element, which has exponents <var class="Arg">exps</var> with respect to the Mal'cev basis of the underlying group of mo.



<p><a id="X849FCF377F5E29C2" name="X849FCF377F5E29C2"></a></p>

<h5>2.1-14 MalcevLieElementByCoefficients</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MalcevLieElementByCoefficients</code>( <var class="Arg">mo</var>, <var class="Arg">coeffs</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>For a Mal'cev object mo and a coefficient vector coeffs with rational indeterminates as entries, this functions returns the Mal'cev Lie element, which has coefficients <var class="Arg">coeffs</var> with respect to the basis of the underlying Lie algebra of <var class="Arg">mo</var>.</p>

<p><a id="X81CAD2F27B2066C4" name="X81CAD2F27B2066C4"></a></p>

<h4>2.2 <span class="Heading">An example application</span></h4>


<div class="example"><pre>
 gap> n := 2;
 2
 gap> F := FreeGroup( n );
 <free group on the generators [ f1, f2 ]>
 gap> c := 3;
 3
 gap> N := NilpotentQuotient( F, c );
 Pcp-group with orders [ 0, 0, 0, 0, 0 ]
 
 gap> mo := MalcevObjectByTGroup( N );
 <<Malcev object of dimension 5>>
 gap> dim := Dimension( mo );
 5
 gap> UnderlyingGroup( mo );
 Pcp-group with orders [ 0, 0, 0, 0, 0 ]
 gap> UnderlyingLieAlgebra( mo );
 <Lie algebra of dimension 5 over Rationals>
  
 gap> g := MalcevGrpElementByExponents( mo, [1,1,0,2,-1/2] );
 [ 1, 1, 0, 2, -1/2 ]
 gap> x := MalcevLieElementByCoefficients( mo, [1/2, 2, -1, 3, 5 ] );
 [ 1/2, 2, -1, 3, 5 ]
 
 gap> h := RandomGrpElm( mo );
 [ 5, -3, 0, -2, 8 ]
 gap> y := RandomLieElm( mo );
 [ 3, 9, 5, 5, 2 ]
 
 gap> z := Log( g );
 [ 1, 1, -1/2, 7/3, -1/3 ]
 gap> Exp( z ) = g;
 true
 gap> k := Exp( y );
 [ 3, 9, 37/2, 77/4, 395/4 ]
 gap> Log( k ) = y;
 true
 
 gap> g*h;
 [ 6, -2, 5, 10, -15/2 ]
 gap> Comm(g,h);
 [ 0, 0, 8, 10, -18 ]
 gap> Comm(x,y);
 [ 0, 0, 3/2, -25/4, -79/4 ]
 
 gap> indets := List( List( [1..dim], i->Concatenation( "a_", String(i) ) ),
 >                   x->Indeterminate( Rationals, x : new ) );
 [ a_1, a_2, a_3, a_4, a_5 ]
 gap> g_sym := MalcevSymbolicGrpElementByExponents( mo, indets );
 [ a_1, a_2, a_3, a_4, a_5 ]
 gap> x_sym := Log( g_sym );
 [ a_1, a_2, -1/2*a_1*a_2+a_3, 1/12*a_1^2*a_2+1/4*a_1*a_2-1/2*a_1*a_3+a_4,
 -1/12*a_1*a_2^2+1/4*a_1*a_2-1/2*a_2*a_3+a_5 ]
 gap> g_sym * g;
 [ a_1+1, a_2+1, a_2+a_3, a_3+a_4+2, 1/2*a_2^2+1/2*a_2+a_3+a_5-1/2 ]
 </pre></div>


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