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



Here we describe the functions that access the classifications of Lie algebras 
that are stored in the package. A function below either returns a single 
Lie algebra, depending on a list of parameters, 
or a collection.
It is important to note that two calls 
of the function <A>NonSolvableLieAlgebra</A>,  <A>SolvableLieAlgebra</A>, or
 <A>NilpotentLieAlgebra</A>  may return 
isomorphic Lie algebras even if the parameters are different (see the 
description of the parameter list for each of the functions). If, however,
the output of a function is a collection, then the members of this collection
are pairwise non-isomorphic.<P/>

The Lie algebras in the database are  stored in the form of
structure constant tables. Usually the size of 
a family of Lie algebras in the database is
small enough so that the entries of the structure constant tables can be
stored without any compression. However the number of 
nilpotent Lie algebras with dimension at least 7 is very large, and so the
structure constant tables are compressed as follows. If <Math>L</Math> is such
a Lie algebra, then we fix a basis <Math>B=\{b_1,\ldots,b_n\}</Math> and
consider the coefficients of the products <Math>[b_i,b_j]</Math> where <Math>j>i</Math>. 
We concatenate these coefficient sequences and consider the long sequence 
so obtained as
a number written in base <Math>p</Math>. Then we convert this number to base
62 and write it down using the digits
<Math>0,\ldots,9,a\ldots,z,A\ldots,Z</Math>. Then this string is stored in the
files <A>gap/nilpotent/nilpotent&uscore;data*.gi</A>. See the function
<A>ReadStringToNilpotentLieAlgebra</A> in the file
<A>gap/nilpotent/nilpotent.gi</A> for the precise details.





<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 an arbitrary finite field, <A>pars</A> is a list of parameters with 
length between 1 and 4. The output is a non-solvable Lie algebra corresponding
to the parameters, which is displayed as a string that describes the
algebra  following <Cite Key="Strade"/>.
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 <A>pars</A> 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 one 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
Section <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,i]</A>: <A>i=0, 1</A>; the Lie algebras in <Ref Sect="appdim5char2"/>(2).</Item>
<Item>
<A>[5,3,i]</A>:  <A>i=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,i]</A>: <A>i=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,i]</A>: <A>i=0, 1</A>; 
the two Lie algebras <Ref Sect="appdim6char2"/>(3).
</Item>
<Item>
<A>[6,4,x]</A>: <A>x=0, 1, 2, 3</A> or <A>x</A> is a field element.
In this case <A>AllNonSolvableLieAlgebras</A>
returns one of the Lie algebras in <Ref Sect="appdim6char2"/>(4).
If <A>x=0, 1, 2, 3</A>
then the Lie algebra corresponding to the 
<A>(x+1)</A>-th matrix of <Ref Sect="appdim6char2"/>(4) is returned. If
<A>x</A> is a field element, then a Lie algebra is returned which corresponds
to the 5th 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,x], [6,6,4,x]</A>: <A>x</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,x], [6,3,4,x]</A>: <A>x</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,x]</A>: <A>x</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,i]</A>: <A>i=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="AllNonSolvableLieAlgebras" 
         Arg="F, dim"/>
   <Description>
Here <A>F</A> is an arbitrary finite field, and <A>dim</A> is at most 6.
A collection is returned whose members form a complete and irredundant list
of  representatives of the isomorphism types of the non-solvable Lie algebras over <A>F</A> with dimension
<A>dim</A>. In order to obtain the algebras contained in the collection, 
one can use the functions <A>AsList</A>, <A>Enumerator</A>, <A>Iterator</A>,
as illustrated by the following example.
<Example>
gap> L := AllNonSolvableLieAlgebras( GF(4), 4 );
&lt;Collection of nonsolvable Lie algebras with dimension 4 over GF(2^2)>
gap>  e := Enumerator( L );
&lt;enumerator>
gap> for i in e do Print( Dimension( LieSolvableRadical( i )), "\n" ); od;
0
1
gap> AsList( L );
[ W(1;2), W(1;2)^{(1)}+GF(4) ]
gap> Dimension( LieCenter( last[2] ));
1
</Example>
As the output of <A>AllNonSolvableLieAlgebras</A> is a collection, the user
can efficiently access the classification of <Math>d</Math>-dimensional 
non-solvable Lie algebras over a given field,  even if the classification
contains a large number of algebras.
For instance, there are
95367431640638 non-solvable Lie algebras over <Math>GF(5^{20})</Math>. Clearly 
one cannot expect to be able to handle a list containing all these algebras;
it is, however, possible to work with the collection of these Lie algebras, as
follows.
<Example>
gap> L := AllNonSolvableLieAlgebras( GF(5^20), 6 );
&lt;Collection of nonsolvable Lie algebras with dimension 6 over GF(5^20)>
gap> e := Enumerator( L );
&lt;enumerator>
gap> Length( last );
95367431640638
gap> Dimension( LieDerivedSubalgebra( e[462468528345] ));
5
</Example>


We note that we could not enumerate the non-solvable Lie algebras of dimension
6 over finite fields of characteristic 3, and so the function
<A>Enumerator</A> 
cannot be used in that context. You can, however, 
use the functions <A>Iterator</A> and <A>AsList</A> as follows.
<Example>
gap> L := AllNonSolvableLieAlgebras( GF(3), 6 );
&lt;Collection of nonsolvable Lie algebras with dimension 6 over GF(3)>
gap>  e := Iterator( L );
&lt;iterator>
gap> dims := [];;
gap> for i in e do Add( dims, Dimension( LieSolvableRadical( i ))); od;
gap> dims;
[ 0, 0, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 ]
gap> AsList( L );
[ sl(2,3)+sl(2,3), sl(2,GF(9)), sl(2,3)+solv([ 1 ]), sl(2,3)+solv([ 2 ]), 
  sl(2,3)+solv([ 3, 0*Z(3) ]), sl(2,3)+solv([ 3, Z(3)^0 ]), 
  sl(2,3)+solv([ 3, Z(3) ]), sl(2,3)+solv([ 4, 0*Z(3) ]), 
  sl(2,3)+solv([ 4, Z(3) ]), sl(2,3)+solv([ 4, Z(3)^0 ]), sl(2,3):(V(1)+V(0)),
  sl(2,3):V(2), sl(2,3):H, sl(2,3):&lt;x,y,z|[x,y]=y,[x,z]=z>, 
  sl(2,3):V(2,0*Z(3)), sl(2,3):V(2,Z(3)), W(1;1):O(1;1), W(1;1):O(1;1)*, 
  sl(2,3).H(0), sl(2,3).H(1), sl(2,3).(GF(3)+GF(3)+GF(3))(1), 
  sl(2,3).(GF(3)+GF(3)+GF(3))(2) ]
</Example>

   </Description>
</ManSection>

<ManSection>
<Meth Name="AllSimpleLieAlgebras" 
         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 a 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 (see Section <Ref Sect="simple"/>). The algebras in the list are pairwise
   non-isomorphic. Note that the output of this function is a list and not
a collection, 
and the package does not contain a function called <A>SimpleLieAlgebra</A>.
   </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 Section <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 an arbitrary 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 an arbitrary 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="AllSolvableLieAlgebras" 
         Arg="F, dim"/>
   <Description>
Here <A>F</A> is an arbitrary finite field, and <A>dim</A> is at most 4.
A collection of all solvable Lie algebras over <A>F</A> of dimension
<A>dim</A> is returned. The output does not contain isomorphic Lie
algebras. The order in which the Lie algebras appear in the list
is always the same. It is possible to construct an enumerator from the output 
collection for all of the valid choices of the parameters.
See <A>AllNonSolvableLieAlgebra</A> for a more detailed description of
usage.
   </Description>
</ManSection>

<ManSection>
<Meth Name="AllNilpotentLieAlgebras" 
         Arg="F, dim"/>
   <Description>

Here <A>F</A> is a finite field, and <A>dim</A> not greater than 9. 
Further, if <A>dim=9</A> or <A>dim=8</A>, then <A>F</A> must be <A>GF(2)</A>;
if <A>dim=7</A> then <A>F</A> must be one of <A>GF(2)</A>, <A>GF(3)</A>,
<A>GF(5)</A>; if <A>dim=6</A> then either <A>F</A> must be <A>GF(2)</A>
or the size of <A>F</A> must be odd; and if <A>dim&lt;6</A> then <A>F</A> can
be an arbitrary finite field.
A collection of all nilpotent Lie algebras over <A>F</A> of dimension
<A>dim</A> is returned. If <A>dim</A> is not greater than 5 or <A>dim=6</A> 
and the
   characteristic of <A>F</A> is odd, then the collection of
   nilpotent Lie algebras is determined by <Cite Key="wdg07"/>, otherwise the 
classification can be found in <Cite Key="sch"/>. The output does not contain isomorphic Lie
algebras. The order in which the Lie algebras appear in the collection
is always the same. It is possible to construct an enumerator from the output 
collection for all of the valid choices of the parameters.
See <A>AllNonSolvableLieAlgebra</A> for a more detailed description of
usage.
   </Description>
</ManSection>

<ManSection>
<Meth Name="NumberOfNilpotentLieAlgebras" 
         Arg="F, dim"/>
   <Description>
Here <A>F</A> is a finite field, and <A>dim</A> is an integer.
The restrictions for <A>F</A> and <A>dim</A> are the same as in the function
<A>AllNilpotentLieAlgebras</A>.
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, [4,14,3] );
&lt;Lie algebra of dimension 4 over Rationals>
gap>  LieAlgebraIdentification( L );
rec( name := "L4_14( Rationals, 1/3 )", parameters := [ 1/3 ],
  isomorphism := CanonicalBasis( &lt;Lie algebra of dimension
    4 over Rationals> ) -> [ v.3, (-1)*v.2, v.1, (1/3)*v.4 ] )
</Example>
In the example we see that the program finds a different parameter, than the one
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>