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<title>GAP (HAP) - Chapter 10:  Lie commutators and nonabelian Lie tensors</title>
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<div class="ChapSects"><a href="chap10.html#X7A3DC9327EE1BE6C">10. <span class="Heading"> Lie commutators and nonabelian Lie tensors</span></a>
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<h3>10. <span class="Heading"> Lie commutators and nonabelian Lie tensors</span></h3>

<div class="pcenter"><table cellspacing="10"  class="GAPDocTable">
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<td class="tdleft">Functions on this page are joint work with Hamid Mohammadzadeh, and implemented by him.</td>
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<tr>
<td class="tdleft"><code class="code"> LieCoveringHomomorphism(L)</code></p>

<p>Inputs a finite dimensional Lie algebra L over a field, and returns a surjective Lie homomorphism phi : C-&gt; L where:</p>


<ul>
<li><p>the kernel of phi lies in both the centre of C and the derived subalgebra of C,</p>

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

</li>
</ul>
</td>
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<td class="tdleft"><code class="code"> LeibnizQuasiCoveringHomomorphism(L)</code></p>

<p>Inputs a finite dimensional Lie algebra L over a field, and returns a surjective homomorphism phi : C-&gt; L of Leibniz algebras where:</p>


<ul>
<li><p>the kernel of phi lies in both the centre of C and the derived subalgebra of C,</p>

</li>
<li><p>the kernel of phi is a vector space of rank equal to the rank of the kernel J of the homomorphism L otimes L -&gt; L from the tensor square to L. (We note that, in general, J is NOT equal to the second Leibniz homology of L.)</p>

</li>
</ul>
</td>
</tr>
<tr>
<td class="tdleft"><code class="code"> LieEpiCentre(L)</code></p>

<p>Inputs a finite dimensional Lie algebra L over a field, and returns an ideal Z^*(L) of the centre of L. The ideal Z^*(L) is trivial if and only if L is isomorphic to a quotient L=E/Z(E) of some Lie algebra E by the centre of E.</td>
</tr>
<tr>
<td class="tdleft"><code class="code"> LieExteriorSquare(L) </code></p>

<p>Inputs a finite dimensional Lie algebra L over a field. It returns a record E with the following components.</p>


<ul>
<li><p>E.homomorphism is a Lie homomorphism µ : (L wedge L) --&gt; L from the nonabelian exterior square (L wedge L) to L. The kernel of µ is the Lie multiplier.</p>

</li>
<li><p>E.pairing(x,y) is a function which inputs elements x, y in L and returns (x wedge y) in the exterior square (L wedge L) .</p>

</li>
</ul>
</td>
</tr>
<tr>
<td class="tdleft"><code class="code"> LieTensorSquare(L) </code></p>

<p>Inputs a finite dimensional Lie algebra L over a field and returns a record T with the following components.</p>


<ul>
<li><p>T.homomorphism is a Lie homomorphism µ : (L otimes L) --&gt; L from the nonabelian tensor square of L to L.</p>

</li>
<li><p>T.pairing(x,y) is a function which inputs two elements x, y in L and returns the tensor (x otimes y) in the tensor square (L otimes L) .</p>

</li>
</ul>
</td>
</tr>
<tr>
<td class="tdleft"><code class="code"> LieTensorCentre(L) </code></p>

<p>Inputs a finite dimensional Lie algebra L over a field and returns the largest ideal N such that the induced homomorphism of nonabelian tensor squares (L otimes L) --&gt; (L/N otimes L/N) is an isomorphism.</td>
</tr>
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