<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">\(phi : C\rightarrow L\)</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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">\(phi : C\rightarrow L\)</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">\(L \otimes L \rightarrow L\)</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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">\(Z^\ast(L)\)</span> of the centre of <span class="SimpleMath">\(L\)</span>. The ideal <span class="SimpleMath">\(Z^\ast(L)\)</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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">\(µ : (L \wedge L) \longrightarrow L\)</span> from the nonabelian exterior square <span class="SimpleMath">\((L \wedge L)\)</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">\((x \wedge y)\)</span> in the exterior square <span class="SimpleMath">\((L \wedge L)\)</span> .</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">\(µ : (L \otimes L) \longrightarrow L\)</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">\((x \otimes y)\)</span> in the tensor square <span class="SimpleMath">\((L \otimes L)\)</span> .</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; 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">\((L \otimes L) \longrightarrow (L/N \otimes L/N)\)</span> is an isomorphism.</p>
