<Chapter>
<Heading>Computing the Mal'cev correspondence
<Section Label="sec:mainfuncs">
<Heading>The main functions</Heading>
Let <M>G</M> be a <M>T</M>-group
and <M>G^</M> its <M>\Q</M>-powered hull.
In this chapter we describe functionality
for setting up the
Mal'cev correspondence
between <M>G^</M> and the Lie algebra <M>L(G)</M>.
The data structures needed for computations with <M>G^</M> and
<M>L(G)</M> are
stored in a so-called Mal'cev object.
Computational representations of
elements of <M>G^</M>, respectively <M>L(G)</M>,
will be called Mal'cev group elements, respectively
Mal'cev Lie elements.
<ManSection>
<Func Name="MalcevObjectByTGroup" Arg="N"/>
<Description>
If <A>N</A> is a a <A>T</A>-group
(i.e. a finitely generated torsion-free nilpotent group),
given by a polycyclic presentation with respect to a Mal'cev basis,
then this function computes the Mal'cev correspondence for N
and stores the result in a so-called Mal'cev object.
Otherwise this function returns `fail'.
In the moment this function is restricted to groups <A>N</A>
of nilpotency class at most 9.
</Description>
</ManSection>
<ManSection>
<Func Name="UnderlyingGroup" Arg="mo"/>
<Description>
For a Mal'cev object mo this function returns the
<A>T</A>-group, which was used to build <A>mo</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="UnderlyingLieAlgebra" Arg="mo"/>
<Description>
For a Mal'cev object mo this function returns the Lie algebra,
which underlies the correspondence described by <A>mo</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="Dimension" Arg="mo"/>
<Description>
Returns the dimension of the Lie algebra that underlies the
Mal'cev object mo.
</Description>
</ManSection>
<ManSection>
<Func Name="MalcevGrpElementByExponents" Arg="mo, exps"/>
<Description>
For a Mal'cev object mo and an exponent vector exps
with rational entries, this functions returns the Mal'cev group element, which has exponents <A>exps</A> with respect to the
Mal'cev basis of the underlying group of mo.
</Description>
</ManSection>
<ManSection>
<Func Name="MalcevLieElementByCoefficients" Arg="mo, coeffs"/>
<Description>
For a Mal'cev object mo and a coefficient vector
<A>coeffs</A> with rational
entries, this functions returns the Mal'cev Lie element, which
has coefficients <A>coeffs</A> with respect to the basis of
the underlying Lie algebra of <A>mo</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="RandomGrpElm" Arg="mo, range"/>
<Description>
For a Mal'cev object mo this function returns the output of
MalcevGrpElementByExponents( <A>mo</A>, <A>exps</A> ), where
<A>exps</A> is an exponent vector whose entries are randomly
chosen integers between -<A>range</A> and <A>range</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="RandomLieElm" Arg="mo, range"/>
<Description>
For a Mal'cev object mo this function returns the output of
MalcevLieElementByExponents( <A>mo</A>, <A>coeffs</A> ), where
<A>coeffs</A> is a coefficient vector whose entries are randomly
chosen integers between -<A>range</A> and <A>range</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="Log" Arg="g"/>
<Description>
For Mal'cev group element g this function returns the
corresponding Mal'cev Lie element.
</Description>
</ManSection>
<ManSection>
<Func Name="Exp" Arg="x"/>
<Description>
For Mal'cev Lie element x this function returns the
corresponding Mal'cev group element.
</Description>
</ManSection>
<ManSection>
<Func Name="*" Arg="g, h"/>
<Description>
Returns the product of Mal'cev group elements.
</Description>
</ManSection>
<ManSection>
<Func Name="Comm" Arg="x, y"/>
<Description>
If <A>x</A>,<A>y</A> are Mal'cev group elements, then this
function returns the group theoretic commutator of <A>x</A>
and <A>y</A>.
If <A>x</A>,<A>y</A> are Mal'cev Lie elements, then this function
returns the Lie commutator of <A>x</A> and <A>y</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="MalcevSymbolicGrpElementByExponents" Arg="mo, exps"/>
<Description>
For a Mal'cev object mo and an exponent vector exps
with rational indeterminates as entries,
this functions returns the Mal'cev group element, which
has exponents <A>exps</A> with respect to the Mal'cev basis of the
underlying group of <A>mo</A>.
</Description>
</ManSection>
<ManSection>
<Func Name="MalcevLieElementByCoefficients" Arg="mo, coeffs"/>
<Description>
For a Mal'cev object mo and a coefficient vector
<A>coeffs</A> with rational indeterminates as entries,
this functions returns the Mal'cev Lie element, which
has coefficients <A>coeffs</A> with respect to the basis of the
underlying Lie algebra of <A>mo</A>.
</Description>
</ManSection>
</Section>
<Section>
<Heading>An example application</Heading>
<Example>
gap> n := 2;
2
gap> F := FreeGroup( n );
<free group on the generators [ f1, f2 ]>
gap> c := 3;
3
gap> N := NilpotentQuotient( F, c );
Pcp-group with orders [ 0, 0, 0, 0, 0 ]
gap> mo := MalcevObjectByTGroup( N );
<<Malcev object of dimension 5>>
gap> dim := Dimension( mo );
5
gap> UnderlyingGroup( mo );
Pcp-group with orders [ 0, 0, 0, 0, 0 ]
gap> UnderlyingLieAlgebra( mo );
<Lie algebra of dimension 5 over Rationals>
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