<Chapter>
<Heading>Introduction</Heading>
<Section Label="sec:about">
<Heading>About Guarana</Heading>
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 <M>\Q</M>-powered nilpotent group <M>H</M> a
unique rational nilpotent Lie algebra <M>L_H</M> and vice-versa.
The Mal'cev correspondence was discovered
by Anatoly Mal'cev in 1951 .
</Section>
<Section Label="sec:setup">
<Heading>Setup for computing the correspondence</Heading>
Let <M>G</M> be a finitely generated torsion-free nilpotent group,
i.e.\ a <M>T</M>-group.
Then <M>G</M> can be embedded in a <M>\Q</M>-powered hull <M>G^</M>.
The group <M>G^</M> is
a <M>\Q</M>-powered nilpotent group and
is unique up to isomorphism.
We denote the Lie algebra
which corresponds to <M>G^</M> under the Mal'cev correspondence by
<M>L(G)= L_{G^}</M>.
We provide an algorithm for setting up the
Mal'cev correspondence
between <M>G^</M> and the Lie algebra <M>L(G)</M>.
That is, if <M>G</M>
is given by a polycyclic presentation with respect to a Mal'cev basis,
then we can compute a structure constants table of <M>L(G)</M>.
Furthermore for a given <M>g\in G</M> we can compute the corresponding element in <M>L(G)</M> and vice versa.
</Section>
<Section Label="sec:collect">
<Heading>Collection</Heading>
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
<Cite Key="Geb02"/><Cite Key="LGS90"/><Cite Key="VLe90"/> }.
This package contains
a new collection algorithm for polycyclically presented groups,
which we call Mal'cev collection .
Mal'cev collection is
in some cases dramatically faster than
collection from the left, while using less memory.
</Section>
</Chapter>
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