2 The families of Lie algebras included in the database
  
  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 NonSolvableLieAlgebra,
  SolvableLieAlgebra,   or   NilpotentLieAlgebra  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.
  
  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 $L$ is such a Lie
  algebra,  then  we  fix  a  basis  $B=\{b_1,\ldots,b_n\}$  and  consider the
  coefficients  of  the products $[b_i,b_j]$ where $j>i$. We concatenate these
  coefficient sequences and consider the long sequence so obtained as a number
  written  in  base  $p$.  Then we convert this number to base 62 and write it
  down  using the digits $0,\ldots,9,a\ldots,z,A\ldots,Z$. Then this string is
  stored  in  the  files  gap/nilpotent/nilpotent_data*.gi.  See  the function
  ReadStringToNilpotentLieAlgebra  in  the file gap/nilpotent/nilpotent.gi for
  the precise details.
  
  
  2.1 Non-solvable Lie algebras
  
  The  package  contains  the  list  of  non-solvable Lie algebras over finite
  fields up to dimension 6. The classification follows the one in [Str].
  
  2.1-1 NonSolvableLieAlgebra
  
  > NonSolvableLieAlgebra( F, pars ) ___________________________________method
  
  F  is  an  arbitrary  finite field, pars 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  [Str].  The  first entry of pars is the dimension of the algebra,
  and  the  possible  additional entries of pars describe the algebra if there
  are more algebras with dimension pars[1].
  
  The possible values of pars are as follows.
  
  
  2.1-2 Dimension 1 and 2
  
  There are no non-solvable Lie algebras with dimension less than 3, and so if
  pars[1] is less than 3 then NonSolvableLieAlgebra returns an error message.
  
  
  2.1-3 Dimension 3
  
  There  is just one non-solvable Lie algebra over an arbitrary finite field F
  (see Section 3.2) which is returned by NonSolvableLieAlgebra( F, [3] ).
  
  
  2.1-4 Dimension 4
  
  If  F has odd characteristic then there is a unique non-solvable Lie algebra
  with    dimension   4   over   F   and   this   algebra   is   returned   by
  NonSolvableLieAlgebra(  F,  [4] ). If F has characteristic 2, then there are
  two distinct Lie algebras and they are returned by NonSolvableLieAlgebra( F,
  [4,i] ) for i=1, 2. See Section 3.3 for a description of the algebras.
  
  
  2.1-5 Dimension 5
  
  If  F  has  characteristic  2  then  there  are  5  isomorphism  classes  of
  non-solvable  Lie  algebras  over F and they are described in Section 3.4-1.
  The possible values of pars are as follows.
  
  --    [5,1]: the Lie algebra in 3.4-1(1).
  
  --    [5,2,i]: i=0, 1; the Lie algebras in 3.4-1(2).
  
  --    [5,3,i]: i=0, 1; the Lie algebras in 3.4-1(3).
  
  If  the  characteristic  of  F  is  odd, then the list of Lie algebras is as
  follows (see Section 3.4-2).
  
  --    [5,1,i]: i=1, 0; the Lie algebras that occur in 3.4-2(1).
  
  --    [5,2]: the Lie algebra in 3.4-2(2).
  
  --    [5,3]:  this algebra only exists if the characteristic of F is 3 or 5.
        In  the  former  case the algebra is the one in 3.4-2(3), while in the
        latter it is in 3.4-2(4).
  
  
  2.1-6 Dimension 6
  
  The 6-dimensional non-solvable Lie algebras are described in Section 3.5. If
  F has characteristic 2, then the possible values of pars is as follows.
  
  --    [6,1]: the Lie algebra in 3.5-1(1).
  
  --    [6,2]: the Lie algebra in 3.5-1(2).
  
  --    [6,3,i]: i=0, 1; the two Lie algebras 3.5-1(3).
  
  --    [6,4,x]:  x=0,  1,  2,  3  or  x  is  a  field  element.  In this case
        AllNonSolvableLieAlgebras returns one of the Lie algebras in 3.5-1(4).
        If  x=0,  1,  2,  3 then the Lie algebra corresponding to the (x+1)-th
        matrix  of  3.5-1(4)  is returned. If x is a field element, then a Lie
        algebra is returned which corresponds to the 5th matrix in 3.5-1(4).
  
  --    [6,5]: the Lie algebra in 3.5-1(5).
  
  --    [6,6,1],  [6,6,2], [6,6,3,x], [6,6,4,x]: x is a field element; the Lie
        algebras  in  3.5-1(6). The third and fourth entries of pars determine
        the  isomorphism  type  of the radical as a solvable Lie algebra. More
        precisely, if the third argument pars[3] is 1 or 2 then the radical is
        isomorphic  to  SolvableLieAlgebra(  F,[3,pars[3]]  ).  If  the  third
        argument  pars[3]  is  3  or  4  then  the  radical  is  isomorphic to
        SolvableLieAlgebra(  F,[3,pars[3],pars[4]]  );  see SolvableLieAlgebra
        (2.2-1).
  
  --    [6,7]: the Lie algebra in 3.5-1(7).
  
  --    [6,8]: the Lie algebra in 3.5-1(8).
  
  If  the characteristic of F is odd, then the possible values of pars are the
  following (see Sections 3.5-2, 3.5-3, and 3.5-4).
  
  --    [6,1]: the Lie algebra in 3.5-2(1).
  
  --    [6,2]: the Lie algebra in 3.5-2(2).
  
  --    [6,3,1],  [6,3,2], [6,3,3,x], [6,3,4,x]: x is a field element; the Lie
        algebras  in  3.5-2(3). The third and fourth entries of pars determine
        the  isomorphism  type  of the radical as a solvable Lie algebra. More
        precisely, if the third argument pars[3] is 1 or 2 then the radical is
        isomorphic  to  SolvableLieAlgebra(  F,[3,pars[3]]  ).  If  the  third
        argument  pars[3]  is  3  or  4  then  the  radical  is  isomorphic to
        SolvableLieAlgebra(  F,[3,pars[3],pars[4]]  );  see SolvableLieAlgebra
        (2.2-1).
  
  --    [6,4]: the Lie algebra in 3.5-2(4).
  
  --    [6,5]: the Lie algebra in 3.5-2(5).
  
  --    [6,6]: the Lie algebra in 3.5-2(6).
  
  --    [6,7]: the Lie algebra in 3.5-2(7).
  
  If  the  characteristic  is  3  or  5 then there are additional families. In
  characteristic 3, these families are as follows.
  
  --    [6,8,x]:  x  is  a  field  element; returns one of the Lie algebras in
        3.5-3(1).
  
  --    [6,9]: the Lie algebra in 3.5-3(2).
  
  --    [6,10]: the Lie algebra in 3.5-3(3).
  
  --    [6,11,i]: i=0, 1; one of the two Lie algebras in 3.5-3(4).
  
  --    [6,12]: the first Lie algebra in 3.5-3(5).
  
  --    [6,13]: the second Lie algebra 3.5-3(5).
  
  If  the  characteristic  is  5,  then  the  additional  Lie algebras are the
  following.
  
  --    [6,8]: the Lie algebra in 3.5-4(1).
  
  --    [6,9]: the Lie algebra in 3.5-4(2).
  
  2.1-7 AllNonSolvableLieAlgebras
  
  > AllNonSolvableLieAlgebras( F, dim ) ________________________________method
  
  Here  F  is an arbitrary finite field, and dim 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  F with dimension dim. In order to obtain the algebras contained in the
  collection,  one  can  use  the  functions  AsList, Enumerator, Iterator, as
  illustrated by the following example.
  
  ---------------------------  Example  ----------------------------
    gap> L := AllNonSolvableLieAlgebras( GF(4), 4 );
    <Collection of nonsolvable Lie algebras with dimension 4 over GF(2^2)>
    gap>  e := Enumerator( L );
    <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
  ------------------------------------------------------------------
  
  As  the  output  of  AllNonSolvableLieAlgebras is a collection, the user can
  efficiently  access  the  classification of $d$-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 $GF(5^{20})$. 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 );
    <Collection of nonsolvable Lie algebras with dimension 6 over GF(5^20)>
    gap> e := Enumerator( L );
    <enumerator>
    gap> Length( last );
    95367431640638
    gap> Dimension( LieDerivedSubalgebra( e[462468528345] ));
    5
  ------------------------------------------------------------------
  
  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
  Enumerator  cannot  be  used  in  that  context.  You  can, however, use the
  functions Iterator and AsList as follows.
  
  ---------------------------  Example  ----------------------------
    gap> L := AllNonSolvableLieAlgebras( GF(3), 6 );
    <Collection of nonsolvable Lie algebras with dimension 6 over GF(3)>
    gap>  e := Iterator( L );
    <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):<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) ]
  ------------------------------------------------------------------
  
  2.1-8 AllSimpleLieAlgebras
  
  > AllSimpleLieAlgebras( F, dim ) _____________________________________method
  
  Here  F  is a finite field, and dim is either an integer not greater than 6,
  or,  if  F=GF(2),  then  dim  is not greater than 9. The output is a list of
  simple  Lie  algebras over F of dimension dim. If dim is at most 6, then the
  classification  by Strade [Str] is used. If F=GF(2) and dim is between 7 and
  9,  then  the  Lie  algebras  in [Vau06] are returned (see Section 3.6). 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 SimpleLieAlgebra.
  
  
  2.2 Solvable and nilpotent Lie algebras
  
  The  package  contains  the  classification  of  solvable  Lie  algebras  of
  dimensions  2,3,  and  4  (taken  from  [dG05]),  and  the classification of
  nilpotent   Lie   algebras   of  dimensions  5  and  6  (from  [dG07]).  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 3.7 the list is
  given  of  the  multiplication  tables  of  the  solvable  and nilpotent Lie
  algebras, corresponding to the functions in this section.
  
  2.2-1 SolvableLieAlgebra
  
  > SolvableLieAlgebra( F, pars ) ______________________________________method
  
  Here  F  is  an  arbitrary  field,  pars is a list of parameters with length
  between  2  and  4. The first entry of pars 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 pars is the number of the Lie algebra with which it appears in the
  list  of [dG05]. 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 [dG05] can have one or two parameters. In that case the list
  pars also has to contain the parameters.
  
  ---------------------------  Example  ----------------------------
    gap> SolvableLieAlgebra( Rationals, [4,6,1,2] );
    <Lie algebra of dimension 4 over Rationals>
  ------------------------------------------------------------------
  
  2.2-2 NilpotentLieAlgebra
  
  > NilpotentLieAlgebra( F, pars ) _____________________________________method
  
  Here  F  is  an  arbitrary  field,  pars is a list of parameters with length
  between  2  and  3. The first entry of pars is the dimension of the algebra,
  which  has  to  be 5 or 6. The second entry of pars is the number of the Lie
  algebra  with  which  it appears in the list of [dG07]. A Lie algebra in the
  list  of  [dG07] can have one parameter. In that case the list pars also has
  to contain the parameter.
  
  ---------------------------  Example  ----------------------------
    gap> NilpotentLieAlgebra( GF(3^7), [ 6, 24, Z(3^7)^101 ] );
    <Lie algebra of dimension 6 over GF(3^7)>
  ------------------------------------------------------------------
  
  2.2-3 AllSolvableLieAlgebras
  
  > AllSolvableLieAlgebras( F, dim ) ___________________________________method
  
  Here  F  is an arbitrary finite field, and dim is at most 4. A collection of
  all  solvable  Lie  algebras over F of dimension dim 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 AllNonSolvableLieAlgebra for a more detailed description of
  usage.
  
  2.2-4 AllNilpotentLieAlgebras
  
  > AllNilpotentLieAlgebras( F, dim ) __________________________________method
  
  Here  F  is a finite field, and dim not greater than 9. Further, if dim=9 or
  dim=8,  then  F  must be GF(2); if dim=7 then F must be one of GF(2), GF(3),
  GF(5);  if  dim=6  then either F must be GF(2) or the size of F must be odd;
  and  if  dim<6  then F can be an arbitrary finite field. A collection of all
  nilpotent  Lie  algebras  over F of dimension dim is returned. If dim is not
  greater  than  5  or  dim=6  and  the  characteristic  of F is odd, then the
  collection  of nilpotent Lie algebras is determined by [dG07], otherwise the
  classification  can  be  found  in  [Sch05].  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 AllNonSolvableLieAlgebra for a more detailed description of usage.
  
  2.2-5 NumberOfNilpotentLieAlgebras
  
  > NumberOfNilpotentLieAlgebras( F, dim ) _____________________________method
  
  Here  F is a finite field, and dim is an integer. The restrictions for F and
  dim  are  the same as in the function AllNilpotentLieAlgebras. The number of
  nilpotent Lie algebras over F of dimension dim is returned.
  
  2.2-6 LieAlgebraIdentification
  
  > LieAlgebraIdentification( L ) ______________________________________method
  
  Here  L  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.
  
  --    name  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
  
  "N6_24( GF(3^7), Z(3^7) )"
  
        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).
  
  --    parameters This contains the parameters that appear in the name of the
        algebra.
  
  --    isomorphism This is an isomorphism of the input Lie algebra to the Lie
        algebra from the classification with the given name.
  
  ---------------------------  Example  ----------------------------
    gap> L:= SolvableLieAlgebra( Rationals, [4,14,3] );
    <Lie algebra of dimension 4 over Rationals>
    gap>  LieAlgebraIdentification( L );
    rec( name := "L4_14( Rationals, 1/3 )", parameters := [ 1/3 ],
      isomorphism := CanonicalBasis( <Lie algebra of dimension
        4 over Rationals> ) -> [ v.3, (-1)*v.2, v.1, (1/3)*v.4 ] )
  ------------------------------------------------------------------
  
  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.