<Chapter Label="Intro">
<Heading>The families of Lie algebras included in the database</Heading>  

<Section Label="nonsolv">
<Heading>Non-solvable Lie algebras</Heading>

The package contains the list of non-solvable Lie algebras over finite fields up to dimension 6. The classification follows the one in <Cite Key="Strade"/>.



<ManSection>
<Meth Name="NonSolvableLieAlgebra" 
         Arg="F, pars"/>
   <Description>
<A>F</A> is a finite field, <A>pars</A> is a list of parameters with 
length between 1 and 4. The first entry of <A>pars</A>
is the dimension of the algebra, and the possible additional entries of 
<A>pars</A> describe the algebra if there are more algebras with  
dimension <A>pars[1]</A>.
   </Description>
</ManSection>

The possible values of pars are as follows.

<Subsection>
<Heading>Dimension 1 and 2</Heading>
There are no non-solvable Lie algebras with dimension less than 3, and so if
<A>pars[1]</A> is less than 3 then <A>NonSolvableLieAlgebra</A> returns an error message. 

</Subsection>

<Subsection>
<Heading>Dimension 3</Heading>
There is just 1 non-solvable Lie algebra over an arbitrary finite field 
<A>F</A> (see Section~<Ref Sect="appdim3"/>) which is returned by <A>NonSolvableLieAlgebra( F, [3] )</A>.
</Subsection>

<Subsection>
<Heading>Dimension 4</Heading>
If <A>F</A> has odd characteristic then there is a unique non-solvable 
Lie algebra with dimension 4 over <A>F</A> and this algebra is returned by <A>NonSolvableLieAlgebra( F, [4] )</A>. 
If <A>F</A> has characteristic 2, then there are two distinct Lie algebras and they are returned by <A>NonSolvableLieAlgebra( F, [4,i] )</A> for <A>i=1, 2</A>.
See Section~<Ref Sect="appdim4"/> for a description of the algebras.
</Subsection>

<Subsection>
<Heading>Dimension 5</Heading>

If <A>F</A> has characteristic 2 then there are 5 isomorphism classes of 
non-solvable Lie 
algebras over <A>F</A> and they are described in
Sections~<Ref Sect="appdim5char2"/>. The possible values of <A>pars</A> are as 
follows.
<List>
<Item>
<A>[5,1]</A>: the Lie algebra in <Ref Sect="appdim5char2"/>(1).</Item>
<Item>
<A>[5,2,a]</A>: <A>a=0, 1</A>; the Lie algebras in <Ref Sect="appdim5char2"/>(2).</Item>
<Item>
<A>[5,3,a]</A>:  <A>a=0, 1</A>; the Lie algebras in <Ref Sect="appdim5char2"/>(3).</Item>
</List>

If the characteristic of <A>F</A> is odd, then the list of Lie algebras is as follows (see Section~<Ref Sect="appdim5charodd"/>). 
<List>
<Item>
<A>[5,1,a]</A>: <A>a=1, 0</A>; 
the Lie algebras that occur in <Ref Sect="appdim5charodd"/>(1). </Item>
<Item>
<A>[5,2]</A>: the Lie algebra in <Ref Sect="appdim5charodd"/>(2).
</Item>
<Item>
<A>[5,3]</A>: 
this algebra only exists if the characteristic of <A>F</A> is 3 or 5. 
In the former case the algebra is the one in <Ref Sect="appdim5charodd"/>(3), while in the latter it is in <Ref Sect="appdim5charodd"/>(4).
</Item>
</List>
</Subsection>

<Subsection>
<Heading>Dimension 6</Heading>
The 6-dimensional non-solvable Lie algebras are described in Section~<Ref Sect="appdim6"/>.
If <A>F</A> has characteristic 2, then 
the possible values of <A>pars</A> is as 
follows.
<List>
<Item>
<A>[6,1]</A>: the Lie algebra in <Ref Sect="appdim6char2"/>(1).
</Item>
<Item>
<A>[6,2]</A>: the Lie algebra in <Ref Sect="appdim6char2"/>(2).
</Item>
<Item>
<A>[6,3,a]</A>: <A>a=0, 1</A>; 
the two Lie algebras <Ref Sect="appdim6char2"/>(3).
</Item>
<Item>
<A>[6,4,a]</A>: <A>a=0, 1, 2, 3</A> or <A>a</A> is a field element.
In this case <A>NonSolvableLieAlgebras</A>
returns one of the Lie algebras in <Ref Sect="appdim6char2"/>(4).
If <A>a=0, 1, 2, 3</A>
then the Lie algebra corresponding to the 
<A>a+1</A>-th matrix of <Ref Sect="appdim6char2"/>(4) is returned. If
<A>a</A> is a field element, then a Lie algebra is returned which corresponds
to the 4th matrix in <Ref Sect="appdim6char2"/>(4).</Item>
<Item>
<A>[6,5]</A>: the Lie algebra in <Ref Sect="appdim6char2"/>(5). 
</Item>
<Item><A>[6,6,1], [6,6,2], [6,6,3,a], [6,6,4,a]</A>: <A>a</A> is a field element; the Lie algebras
in <Ref Sect="appdim6char2"/>(6).
The third and fourth entries of <A>pars</A>
determine the isomorphism type of the radical as a solvable Lie algebra.
More precisely, if the third argument <A>pars[3]</A> 
is 1 or 2 then the radical is isomorphic
to <A>SolvableLieAlgebra( F,[3,pars[3]] )</A>. If the third argument <A>pars[3]</A> 
is 3 or 4 then the radical is isomorphic
to <A>SolvableLieAlgebra( F,[3,pars[3],pars[4]] )</A>; see <Ref Meth="SolvableLieAlgebra"/>.
</Item>
<Item>
<A>[6,7]</A>: the Lie algebra in <Ref Sect="appdim6char2"/>(7).
</Item>
<Item>
<A>[6,8]</A>: the  Lie algebra in <Ref Sect="appdim6char2"/>(8).</Item>
</List>

If the characteristic of <A>F</A> is odd, then the possible values of <A>pars</A> are the following (see Sections~<Ref Sect="appdim6charodd"/>, <Ref Sect="appdim6char3"/>, and <Ref Sect="appdim6char5"/>). 

<List>
<Item>
<A>[6,1]</A>: the Lie algebra in <Ref Sect="appdim6charodd"/>(1).</Item>
<Item>
<A>[6,2]</A>: the Lie algebra in <Ref Sect="appdim6charodd"/>(2).</Item>
<Item><A>[6,3,1], [6,3,2], [6,3,3,a], [6,3,4,a]</A>: <A>a</A> is a field element; the Lie algebras
in <Ref Sect="appdim6charodd"/>(3).
The third and fourth entries of <A>pars</A>
determine the isomorphism type of the radical as a solvable Lie algebra.
More precisely, if the third argument <A>pars[3]</A> 
is 1 or 2 then the radical is isomorphic
to <A>SolvableLieAlgebra( F,[3,pars[3]] )</A>. If the third argument <A>pars[3]</A> 
is 3 or 4 then the radical is isomorphic
to <A>SolvableLieAlgebra( F,[3,pars[3],pars[4]] )</A>; see <Ref Meth="SolvableLieAlgebra"/>.
</Item>
<Item><A>[6,4]</A>: the Lie algebra in <Ref Sect="appdim6charodd"/>(4).</Item>
<Item><A>[6,5]</A>: the Lie algebra in <Ref Sect="appdim6charodd"/>(5).</Item>
<Item><A>[6,6]</A>: the Lie algebra in <Ref Sect="appdim6charodd"/>(6).</Item>
<Item><A>[6,7]</A>: the Lie algebra in <Ref Sect="appdim6charodd"/>(7).</Item>
</List>

If the characteristic is 3 or 5 then there are additional families. In 
characteristic 3, these families are as follows.
<List>
<Item>
<A>[6,8,a]</A>: <A>a</A> is a field element; returns one of the 
Lie algebras in <Ref Sect="appdim6char3"/>(1).</Item>
<Item><A>[6,9]</A>: the Lie algebra in <Ref Sect="appdim6char3"/>(2).</Item>
<Item><A>[6,10]</A>: the Lie algebra in <Ref Sect="appdim6char3"/>(3).</Item>
<Item><A>[6,11,a]</A>: <A>a=0, 1</A>; one of the two Lie algebras in 
<Ref Sect="appdim6char3"/>(4). 
</Item>
<Item><A>[6,12]</A>: the first Lie algebra in <Ref Sect="appdim6char3"/>(5).</Item>
<Item><A>[6,13]</A>: the second Lie algebra <Ref Sect="appdim6char3"/>(5).</Item>
</List>

If the characteristic is 5, then the additional Lie algebras are the following.
<List>
<Item><A>[6,8]</A>: the Lie algebra in <Ref Sect="appdim6char5"/>(1).</Item>
<Item><A>[6,9]</A>: the Lie algebra in <Ref Sect="appdim6char5"/>(2).</Item>
</List>
</Subsection>

<ManSection>
<Meth Name="NonSolvableLieAlgebras" 
         Arg="F, dim"/>
   <Description>
Here <A>F</A> is a finite field, and <A>dim</A> is at least 6.
The list of non-solvable Lie algebras over <A>F</A> of dimension
<A>dim</A> is returned.
   </Description>
</ManSection>

<ManSection>
<Meth Name="SimpleLieAlgebras" 
         Arg="F, dim"/>
   <Description>
Here <A>F</A> is a finite field, and <A>dim</A> is either an integer 
not greater than 6, or, if <A>F=GF(2)</A>, then <A>dim</A> is not greater than 9.
The output is the list of simple Lie algebras over <A>F</A> of dimension
<A>dim</A>. If <A>dim</A> is at most 6, then the classification by
   Strade~<Cite Key="Strade"/> is used. If <A>F=GF(2)</A> and <A>dim</A> is
   between 7 and 9, then the Lie algebras 
 in <Cite Key="VL"/> are returned.
   </Description>
</ManSection>

</Section>


<Section Label="solvnilp">
<Heading>Solvable and nilpotent Lie algebras</Heading>

The package contains the classification of solvable Lie algebras of dimensions 2,3, and 4 
(taken from <Cite Key="wdg05"/>), and the classification of nilpotent Lie algebras of 
dimensions 5 and 6 (from <Cite Key="wdg07"/>). The classification of nilpotent Lie algebras
of dimension 6 is only complete for base fields of characteristic not 2. The classifications
are complemented by a function for identifying a given Lie algebra as a member of the list.
This function also returns an explicit isomorphism.

In Appendix <Ref Chap="listdescr"/> the list is given of the multiplication tables of the solvable
and nilpotent Lie algebras, corresponding to the functions in this section.


<ManSection Label="sectsolv">
<Meth Name="SolvableLieAlgebra" 
         Arg="F, pars"/>
   <Description>
Here <A>F</A> is a field, <A>pars</A> is a list of parameters with 
length between <K>2</K> and <K>4</K>. The first entry of <A>pars</A>
is the dimension of the algebra, which has to be 2, 3, or 4. If the
dimension is 3, or 4, then the second entry of <A>pars</A> is the number 
of the Lie algebra with which it appears in the list of <Cite Key="wdg05"/>.
If the dimension is 2, then there are only two (isomorphism classes of)
solvable Lie algebras. In this case, if the second entry is 1, then the
abelian Lie algebra is returned, if it is 2, then the unique non-abelian
solvable Lie algebra of dimension 2 is returned. A Lie algebra in the list
of <Cite Key="wdg05"/> can have one or two parameters. In that case the
list <A>pars</A> also has to contain the parameters. 
<Example>
gap> SolvableLieAlgebra( Rationals, [4,6,1,2] );
&lt;Lie algebra of dimension 4 over Rationals>
</Example>
   </Description>
</ManSection>


<ManSection>
<Meth Name="NilpotentLieAlgebra" 
         Arg="F, pars"/>
   <Description>
Here <A>F</A> is a field, <A>pars</A> is a list of parameters with 
length between <K>2</K> and <K>3</K>. The first entry of <A>pars</A>
is the dimension of the algebra, which has to be 5 or 6. The second entry of <A>pars</A> 
is the number of the Lie algebra with which it appears in the list of <Cite Key="wdg07"/>.
A Lie algebra in the list of <Cite Key="wdg07"/> can have one parameter. In that case the
list <A>pars</A> also has to contain the parameter. 
<Example>
gap> NilpotentLieAlgebra( GF(3^7), [ 6, 24, Z(3^7)^101 ] );
&lt;Lie algebra of dimension 6 over GF(3^7)>
</Example>
   </Description>
</ManSection>

<ManSection>
<Meth Name="SolvableLieAlgebras" 
         Arg="F, dim"/>
   <Description>
Here <A>F</A> is a finite field, and <A>dim</A> is one of 2,3,4.
The list of all solvable Lie algebras over <A>F</A> of dimension
<A>dim</A> is returned.
   </Description>
</ManSection>

<ManSection>
<Meth Name="NilpotentLieAlgebras" 
         Arg="F, dim"/>
   <Description>
Here <A>F</A> is a finite field, and <A>dim</A> not greater than 9 if 
<A>F=GF(2)</A>, <A>dim</A> is not greater than 7 if <A>F=GF(3)</A> or <A>F=GF(5)</A>, and
<A>dim</A> is not greater than 6 otherwise. 
The list of all nilpotent Lie algebras over <A>F</A> of dimension
<A>dim</A> is returned. If <A>dim</A> is not greater than 6, then the list of
   nilpotent Lie algebras is determined by <Cite Key="wdg07"/>, otherwise the 
classification can be found in <Cite Key="sch"/>.
   </Description>
</ManSection>

<ManSection>
<Meth Name="NumberOfNilpotentLieAlgebras" 
         Arg="F, dim"/>
   <Description>
Here <A>F</A> is a finite field, and <A>dim</A> not greater than 9 if 
<A>F=GF(2)</A>, <A>dim</A> is not greater than 7 if <A>F=GF(3)</A> or <A>F=GF(5)</A>, and
<A>dim</A> is not greater than 6 otherwise. 
The number of nilpotent Lie algebras over <A>F</A> of dimension
<A>dim</A> is returned. 
   </Description>
</ManSection>


<ManSection>
<Meth Name="LieAlgebraIdentification" 
         Arg="L"/>
   <Description>
Here <A>L</A> is a solvable Lie algebra of dimension 2,3, or 4, or it is a
nilpotent Lie algebra of dimension 5 or 6 (in the latter case it has to be of
characteristic not 2). This function returns a record with three fields.
<List>
<Item><A>name</A> This is a string containing the name of the Lie algebra. It starts
with a capital L if it is a solvable Lie algebra of dimension 2,3,4. It starts with
a capital N if it is a nilpotent Lie algebra of dimension 5 or 6. A name like 
<Verb>"N6_24( GF(3^7), Z(3^7) )"</Verb> means that the input Lie algebra is isomorphic to
the Lie algebra with number 24 in the list of 6-dimensional nilpotent Lie algebras.
Furthermore the field is given and the value of the parameters (if there are any). </Item>
<Item><A>parameters</A> This contains the parameters that appear in the name of the algebra.</Item>
<Item><A>isomorphism</A> This is an isomorphism of the input Lie algebra to the Lie algebra
from the classification with the given name.</Item>
</List>
<Example>
gap> L:= SolvableLieAlgebra( Rationals, [4,7,1/2,4/9] );
&lt;Lie algebra of dimension 4 over Rationals>
gap> LieAlgebraIdentification( L );
rec( name := "L4_7( Rationals, 256/729, 256/729 )",
  parameters := [ 256/729, 256/729 ],
  isomorphism := CanonicalBasis( &lt;Lie algebra of dimension
    4 over Rationals> ) ->
    [ (10)*v.1+v.2+(4)*v.3, (16/9)*v.1+(848/81)*v.2+(8/9)*v.3,
      (32/81)*v.1+(1408/729)*v.2+(6784/729)*v.3, (8/9)*v.4 ] )
</Example>
In the example we see that the program finds different parameters, than the ones
with which the Lie algebra was constructed. The explanation is that some parametric 
classes of Lie algebras contain isomorphic Lie algebras, for different values of the
parameters. In that case the identification function makes its own choice.
   </Description>
</ManSection>

</Section>

</Chapter>

