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### 2. The families of Lie algebras included in the database

#### 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 [S].

##### 2.1-1 NonSolvableLieAlgebra

`> NonSolvableLieAlgebra`( F, pars ) | ( method )

F is a finite field, pars is a list of parameters with length between 1 and 4. 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 1 non-solvable Lie algebra over an arbitrary finite field F (see Section~B.1) 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~B.2 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 Sections~B.3-1. The possible values of pars are as follows.

- [5,1]: the Lie algebra in B.3-1(1).

- [5,2,a]: a=0, 1; the Lie algebras in B.3-1(2).

- [5,3,a]: a=0, 1; the Lie algebras in B.3-1(3).

If the characteristic of F is odd, then the list of Lie algebras is as follows (see Section~B.3-2).

- [5,1,a]: a=1, 0; the Lie algebras that occur in B.3-2(1).

- [5,2]: the Lie algebra in B.3-2(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 B.3-2(3), while in the latter it is in B.3-2(4).

##### 2.1-6 Dimension 6

The 6-dimensional non-solvable Lie algebras are described in Section~B.4. If F has characteristic 2, then the possible values of pars is as follows.

- [6,1]: the Lie algebra in B.4-1(1).

- [6,2]: the Lie algebra in B.4-1(2).

- [6,3,a]: a=0, 1; the two Lie algebras B.4-1(3).

- [6,4,a]: a=0, 1, 2, 3 or a is a field element. In this case NonSolvableLieAlgebras returns one of the Lie algebras in B.4-1(4). If a=0, 1, 2, 3 then the Lie algebra corresponding to the a+1-th matrix of B.4-1(4) is returned. If a is a field element, then a Lie algebra is returned which corresponds to the 4th matrix in B.4-1(4).

- [6,5]: the Lie algebra in B.4-1(5).

- [6,6,1], [6,6,2], [6,6,3,a], [6,6,4,a]: a is a field element; the Lie algebras in B.4-1(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 B.4-1(7).

- [6,8]: the Lie algebra in B.4-1(8).

If the characteristic of F is odd, then the possible values of pars are the following (see Sections~B.4-2, B.4-3, and B.4-4).

- [6,1]: the Lie algebra in B.4-2(1).

- [6,2]: the Lie algebra in B.4-2(2).

- [6,3,1], [6,3,2], [6,3,3,a], [6,3,4,a]: a is a field element; the Lie algebras in B.4-2(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 B.4-2(4).

- [6,5]: the Lie algebra in B.4-2(5).

- [6,6]: the Lie algebra in B.4-2(6).

- [6,7]: the Lie algebra in B.4-2(7).

If the characteristic is 3 or 5 then there are additional families. In characteristic 3, these families are as follows.

- [6,8,a]: a is a field element; returns one of the Lie algebras in B.4-3(1).

- [6,9]: the Lie algebra in B.4-3(2).

- [6,10]: the Lie algebra in B.4-3(3).

- [6,11,a]: a=0, 1; one of the two Lie algebras in B.4-3(4).

- [6,12]: the first Lie algebra in B.4-3(5).

- [6,13]: the second Lie algebra B.4-3(5).

If the characteristic is 5, then the additional Lie algebras are the following.

- [6,8]: the Lie algebra in B.4-4(1).

- [6,9]: the Lie algebra in B.4-4(2).

##### 2.1-7 NonSolvableLieAlgebras

`> NonSolvableLieAlgebras`( F, dim ) | ( method )

Here F is a finite field, and dim is at least 6. The list of non-solvable Lie algebras over F of dimension dim is returned.

##### 2.1-8 SimpleLieAlgebras

`> SimpleLieAlgebras`( 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 the list of simple Lie algebras over F of dimension dim. If dim is at most 6, then the classification by Strade~[S] is used. If F=GF(2) and dim is between 7 and 9, then the Lie algebras in [V06] are returned.

#### 2.2 Solvable and nilpotent Lie algebras

The package contains the classification of solvable Lie algebras of dimensions 2,3, and 4 (taken from [G05]), and the classification of nilpotent Lie algebras of dimensions 5 and 6 (from [G07]). 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 A. 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 a 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 [G05]. 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 [G05] can have one or two parameters. In that case the list pars also has to contain the parameters.

    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 a 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 [G07]. A Lie algebra in the list of [G07] can have one parameter. In that case the list pars also has to contain the parameter.

    gap> NilpotentLieAlgebra( GF(3^7), [ 6, 24, Z(3^7)^101 ] );
    <Lie algebra of dimension 6 over GF(3^7)>

##### 2.2-3 SolvableLieAlgebras

`> SolvableLieAlgebras`( F, dim ) | ( method )

Here F is a finite field, and dim is one of 2,3,4. The list of all solvable Lie algebras over F of dimension dim is returned.

##### 2.2-4 NilpotentLieAlgebras

`> NilpotentLieAlgebras`( F, dim ) | ( method )

Here F is a finite field, and dim not greater than 9 if F=GF(2), dim is not greater than 7 if F=GF(3) or F=GF(5), and dim is not greater than 6 otherwise. The list of all nilpotent Lie algebras over F of dimension dim is returned. If dim is not greater than 6, then the list of nilpotent Lie algebras is determined by [G07], otherwise the classification can be found in [S05].

##### 2.2-5 NumberOfNilpotentLieAlgebras

`> NumberOfNilpotentLieAlgebras`( F, dim ) | ( method )

Here F is a finite field, and dim not greater than 9 if F=GF(2), dim is not greater than 7 if F=GF(3) or F=GF(5), and dim is not greater than 6 otherwise. 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.

    gap> L:= SolvableLieAlgebra( Rationals, [4,7,1/2,4/9] );
    <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( <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 ] )

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.

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