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<title>GAP (HAP commands) - Chapter 16:  Lie commutators and nonabelian Lie tensors</title>
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<p><a id="X7A3DC9327EE1BE6C" name="X7A3DC9327EE1BE6C"></a></p>
<div class="ChapSects"><a href="chap16.html#X7A3DC9327EE1BE6C">16 <span class="Heading"> Lie commutators and nonabelian Lie tensors</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap16.html#X7CFDEEC07F15CF82">16.1 <span class="Heading">  </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X80BBA6247ED4DCCF">16.1-1 LieCoveringHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7BEEE3D380CF22F1">16.1-2 LeibnizQuasiCoveringHomomorphism</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X7B384F9486A7C92B">16.1-3 LieEpiCentre</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X849324D680C0EE5E">16.1-4 LieExteriorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X809B166C835516EB">16.1-5 LieTensorSquare</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap16.html#X802F2E417D872042">16.1-6 LieTensorCentre</a></span>
</div></div>
</div>

<h3>16 <span class="Heading"> Lie commutators and nonabelian Lie tensors</span></h3>

<p>Functions on this page are joint work with <strong class="button">Hamid Mohammadzadeh</strong>, and implemented by him.</p>

<p><a id="X7CFDEEC07F15CF82" name="X7CFDEEC07F15CF82"></a></p>

<h4>16.1 <span class="Heading">  </span></h4>

<p><a id="X80BBA6247ED4DCCF" name="X80BBA6247ED4DCCF"></a></p>

<h5>16.1-1 LieCoveringHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieCoveringHomomorphism</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">L</span> over a field, and returns a surjective Lie homomorphism <span class="SimpleMath">phi : C→ L</span> where:</p>


<ul>
<li><p>the kernel of <span class="SimpleMath">phi</span> lies in both the centre of <span class="SimpleMath">C</span> and the derived subalgebra of <span class="SimpleMath">C</span>,</p>

</li>
<li><p>the kernel of <span class="SimpleMath">phi</span> is a vector space of rank equal to the rank of the second Chevalley-Eilenberg homology of <span class="SimpleMath">L</span>.</p>

</li>
</ul>
<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLieCovers.html">2</a></span> </p>

<p><a id="X7BEEE3D380CF22F1" name="X7BEEE3D380CF22F1"></a></p>

<h5>16.1-2 LeibnizQuasiCoveringHomomorphism</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeibnizQuasiCoveringHomomorphism</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">L</span> over a field, and returns a surjective homomorphism <span class="SimpleMath">phi : C→ L</span> of Leibniz algebras where:</p>


<ul>
<li><p>the kernel of <span class="SimpleMath">phi</span> lies in both the centre of <span class="SimpleMath">C</span> and the derived subalgebra of <span class="SimpleMath">C</span>,</p>

</li>
<li><p>the kernel of <span class="SimpleMath">phi</span> is a vector space of rank equal to the rank of the kernel <span class="SimpleMath">J</span> of the homomorphism <span class="SimpleMath">L ⊗ L → L</span> from the tensor square to <span class="SimpleMath">L</span>. (We note that, in general, <span class="SimpleMath">J</span> is NOT equal to the second Leibniz homology of <span class="SimpleMath">L</span>.)</p>

</li>
</ul>
<p><strong class="button">Examples:</strong></p>

<p><a id="X7B384F9486A7C92B" name="X7B384F9486A7C92B"></a></p>

<h5>16.1-3 LieEpiCentre</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieEpiCentre</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">L</span> over a field, and returns an ideal <span class="SimpleMath">Z^∗(L)</span> of the centre of <span class="SimpleMath">L</span>. The ideal <span class="SimpleMath">Z^∗(L)</span> is trivial if and only if <span class="SimpleMath">L</span> is isomorphic to a quotient <span class="SimpleMath">L=E/Z(E)</span> of some Lie algebra <span class="SimpleMath">E</span> by the centre of <span class="SimpleMath">E</span>.</p>

<p><strong class="button">Examples:</strong> <span class="URL"><a href="../tutorial/chap7.html">1</a></span> , <span class="URL"><a href="../www/SideLinks/About/aboutLieCovers.html">2</a></span> </p>

<p><a id="X849324D680C0EE5E" name="X849324D680C0EE5E"></a></p>

<h5>16.1-4 LieExteriorSquare</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieExteriorSquare</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">L</span> over a field. It returns a record <span class="SimpleMath">E</span> with the following components.</p>


<ul>
<li><p><span class="SimpleMath">E.homomorphism</span> is a Lie homomorphism <span class="SimpleMath">µ : (L ∧ L) ⟶ L</span> from the nonabelian exterior square <span class="SimpleMath">(L ∧ L)</span> to <span class="SimpleMath">L</span>. The kernel of <span class="SimpleMath">µ</span> is the Lie multiplier.</p>

</li>
<li><p><span class="SimpleMath">E.pairing(x,y)</span> is a function which inputs elements <span class="SimpleMath">x, y</span> in <span class="SimpleMath">L</span> and returns <span class="SimpleMath">(x ∧ y)</span> in the exterior square <span class="SimpleMath">(L ∧ L)</span> .</p>

</li>
</ul>
<p><strong class="button">Examples:</strong></p>

<p><a id="X809B166C835516EB" name="X809B166C835516EB"></a></p>

<h5>16.1-5 LieTensorSquare</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieTensorSquare</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">L</span> over a field and returns a record <span class="SimpleMath">T</span> with the following components.</p>


<ul>
<li><p><span class="SimpleMath">T.homomorphism</span> is a Lie homomorphism <span class="SimpleMath">µ : (L ⊗ L) ⟶ L</span> from the nonabelian tensor square of <span class="SimpleMath">L</span> to <span class="SimpleMath">L</span>.</p>

</li>
<li><p><span class="SimpleMath">T.pairing(x,y)</span> is a function which inputs two elements <span class="SimpleMath">x, y</span> in <span class="SimpleMath">L</span> and returns the tensor <span class="SimpleMath">(x ⊗ y)</span> in the tensor square <span class="SimpleMath">(L ⊗ L)</span> .</p>

</li>
</ul>
<p><strong class="button">Examples:</strong></p>

<p><a id="X802F2E417D872042" name="X802F2E417D872042"></a></p>

<h5>16.1-6 LieTensorCentre</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieTensorCentre</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite dimensional Lie algebra <span class="SimpleMath">L</span> over a field and returns the largest ideal <span class="SimpleMath">N</span> such that the induced homomorphism of nonabelian tensor squares <span class="SimpleMath">(L ⊗ L) ⟶ (L/N ⊗ L/N)</span> is an isomorphism.</p>

<p><strong class="button">Examples:</strong></p>


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