GAP (Sophus) - Chapter 2: A sample calculation with Sophus

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### 2. A sample calculation with Sophus

Before listing the functions of Sophus we present a sample calculation to show the reader what Sophus is able to compute. We list the isomorphism types of the 7-dimensional nilpotent Lie algebras over $\mathbb F_2$.

There is just one nilpotent Lie algebra with dimension 1 and dimension 2. We also set L3 to be a list containing the abelian Lie algebra with dimension 3.

    gap> L1 := [ AbelianLieAlgebra( GF(2), 1 ) ];;
    gap> L2 := [ AbelianLieAlgebra( GF(2), 2 ) ];;
    gap> L3 := [ AbelianLieAlgebra( GF(2), 3 ) ];;

Any 3-dimensional non-abelian nilpotent Lie algebra is an immediate descendant of a 2-dimensional Lie algebra. So we compute the step-1 descendants of L1[1] and append them to L3.

    gap> Append( L3, Descendants( L2[1], 1 ));
    gap> L3;
    [<Lie algebra of dimension 3 over GF(2)>,
    <Lie algebra of dimension 3 over GF(2)> ]

Now we compute the list of 4-dimensional Lie algebras. First we set L4 to contain the 4-dimensional abelian Lie algebra. Then we compute the step-1 descendants of the 3-dimensional algebras and append these descendants to L4.

    gap> L4 := [ AbelianLieAlgebra( GF(2), 4 ) ];;
    gap> for i in L3 do
    gap> Append( L4, Descendants( i, 1 ));
    gap> od;
    gap> L4;
    [ <Lie algebra of dimension 4 over GF(2)>,
    <Lie algebra of dimension 4 over GF(2)>,
    <Lie algebra of dimension 4 over GF(2)> ]

We continue this way up to dimension~7.

    gap> L5 := [ AbelianLieAlgebra( GF(2), 5 ) ];;
    gap> for i in L3 do
    gap> Append( L5, Descendants( i, 2 ));
    gap> od;
    gap> for i in L4 do
    gap> Append( L5, Descendants( i, 1 ));
    gap> od;
    gap> L6 := [ AbelianLieAlgebra( GF(2), 6 ) ];;
    gap> for i in L3 do
    gap> Append( L6, Descendants( i, 3 ));
    gap> od;
    gap> for i in L4 do
    gap> Append( L6, Descendants( i, 2 ));
    gap> od;
    gap> for i in L5 do
    gap> Append( L6, Descendants( i, 1 ));
    gap> od;
    gap> L7 := [ AbelianLieAlgebra( GF(2), 6 ) ];;
    gap> for i in L4 do
    gap> Append( L7, Descendants( i, 3 ));
    gap> od;
    gap> for i in L5 do
    gap> Append( L7, Descendants( i, 2 ));
    gap> od;
    gap> for i in L6 do
    gap> Append( L7, Descendants( i, 1 ));
    gap> od;
    gap> Length( L7 );
    202
    gap>

This computation shows that there are 202 pairwise non-isomorphic nilpotent Lie algebras over $\mathbb F_2$.

Let us compute the automorphism group of a nilpotent Lie algebra from our list. We compute this automorphism group in the hybrid format used by Sophus, then we compute this group as a standard GAP object.

    gap> AutomorphismGroupOfNilpotentLieAlgebra( L7[100] );
    rec( glAutos := [  ],
      agAutos := [ Aut: [ v.1, v.1+v.2, v.3, v.4, v.5, v.5+v.6, v.7 ],
          Aut: [ v.1, v.2+v.3, v.3, v.4, v.5, v.6, v.7 ],
          Aut: [ v.1+v.3, v.2, v.3, v.4+v.5, v.5, v.6+v.7, v.7 ],
          Aut: [ v.1+v.4, v.2, v.3+v.5, v.4+v.6, v.5+v.7, v.6+v.7, v.7 ],
          Aut: [ v.1, v.2+v.4, v.3, v.4+v.5, v.5, v.6+v.7, v.7 ],
          Aut: [ v.1+v.5, v.2, v.3, v.4+v.7, v.5, v.6, v.7 ],
          Aut: [ v.1, v.2+v.5, v.3, v.4, v.5, v.6, v.7 ],
          Aut: [ v.1+v.6, v.2, v.3, v.4+v.7, v.5, v.6, v.7 ],
          Aut: [ v.1, v.2+v.6, v.3, v.4+v.7, v.5, v.6, v.7 ],
          Aut: [ v.1+v.7, v.2, v.3, v.4, v.5, v.6, v.7 ],
          Aut: [ v.1, v.2+v.7, v.3, v.4, v.5, v.6, v.7 ],
          Aut: [ v.1, v.2, v.3+v.7, v.4, v.5, v.6, v.7 ] ], glOrder := 1,
      glOper := [  ], agOrder := [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ],
      liealg := <Lie algebra of dimension 7 over GF(2)>,
      one := Aut: [ v.1, v.2, v.3, v.4, v.5, v.6, v.7 ], size := 4096,
      field := GF(2), prime := 2 )
    gap>
    gap> AutomorphismGroup( L7[100] );
    <group with 12 generators>
    gap>

Finally let us check that two Lie algebras from our list are not isomorphic.

    gap> AreIsomorphicNilpotentLieAlgebras( L7[100], L7[101] );
    false

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