GAP (Sophus) - Chapter 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 Append( L4, Descendants( i, 1 )); 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)> ]

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

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( 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 ] ], 
  agOrder := [ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 ], field := GF(2), 
  glAutos := [  ], glOper := [  ], glOrder := 1, 
  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 ], prime := 2, size := 4096 
 )
gap> 
gap> AutomorphismGroup( L7[100] );
<group with 12 generators>

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

2 A sample calculation with Sophus