<p>In this package we demonstrate the algorithmic usefulness of the so-called Mal'cev correspondence for computations with infinite polycyclic groups; it is a correspondence that associates to every \(\Q\)-powered nilpotent group \(H\) a unique rational nilpotent Lie algebra \(L_H\) and vice-versa. The Mal'cev correspondence was discovered by Anatoly Mal'cev in 1951 [Mal51].
<h4>1.2 <span class="Heading">Setup for computing the correspondence</span></h4>
<p>Let <span class="SimpleMath">\(G\)</span> be a finitely generated torsion-free nilpotent group, i.e.\ a <span class="SimpleMath">\(T\)</span>-group. Then <span class="SimpleMath">\(G\)</span> can be embedded in a <span class="SimpleMath">\(\Q\)</span>-powered hull <span class="SimpleMath">\(G^\)</span>. The group <span class="SimpleMath">\(G^\)</span> is a <span class="SimpleMath">\(\Q\)</span>-powered nilpotent group and is unique up to isomorphism. We denote the Lie algebra which corresponds to <span class="SimpleMath">\(G^\)</span> under the Mal'cev correspondence by \(L(G)= L_{G^}\). We provide an algorithm for setting up the Mal'cev correspondence between <span class="SimpleMath">\(G^\)</span> and the Lie algebra <span class="SimpleMath">\(L(G)\)</span>. That is, if <span class="SimpleMath">\(G\)</span> is given by a polycyclic presentation with respect to a Mal'cev basis, then we can compute a structure constants table of \(L(G)\). Furthermore for a given \(g\in G\) we can compute the corresponding element in \(L(G)\) and vice versa.
<p>Every element of a polycyclically presented group has a unique normal form. An algorithm for computing this normal form is called a collection algorithm. Such an algorithm lies at the heart of most methods dealing with polycyclically presented groups. The current state of the art is collection from the left <a href="chapBib_mj.html#biBGeb02">[Geb02]</a><a href="chapBib_mj.html#biBLGS90">[LGS90]</a><a href="chapBib_mj.html#biBVLe90">[VL90]</a> }. This package contains a new collection algorithm for polycyclically presented groups, which we call Mal'cev collection [AL07]. Mal'cev collection is in some cases dramatically faster than collection from the left, while using less memory.</p>
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