GAP (NoCK) - Chapter 3: Algorithm example

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3 Algorithm example

### 3 Algorithm example

In this chapter we use additionaly functions from the following packages: CoReLG [DFdG14] and SLA [dG]. We will show in detail the split case (for a non-split case you should use algoritm to generate regular subalgebras from [DFdG15]). For example, we take \(G=\mathfrak{e}_{6(6)}\) (tuple "E",6,2 in CoReLG notation). We calculate AllZeroDH on it.

    gap> AllZeroDH("E",6,2);
    [ 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 18, 19, 20, 21, 22, 23, 24, 27,
     28, 29, 30, 31, 32, 35, 36, 37, 38, 39, 40, 41 ]

We generate all regular subalgebras of complexification.

    gap> GC:=SimpleLieAlgebra("E",6,Rationals);;
    gap> REG:=RegularSemisimpleSubalgebras(GC);;
    gap> L0:=List( REG, SemiSimpleType );
    [ "A1", "A1 A1", "A2 A1", "A4", "D5", "A4 A1", "A2 A1 A1", "A2 A1 A2", "A3 A1",
     "A1 A1 A1", "A2", "A3", "A5", "A2 A2", "D4", "A5 A1", "A3 A1 A1", "A1 A1 A1 A1",
     "A2 A2 A2" ]

For each subalgebras we take the split real form and calculate its non-compact dimension.

    gap> L0[4];
    "A4"
    gap> RealFormsInformation( "A", 4 );

      There are 4 simple real forms with complexification A4
        1 is of type su(5), compact form
        2 - 3 are of type su(p,5-p) with 1 <= p <= 2
        4 is of type sl(5,R)
      Index '0' returns the realification of A4

    gap> G:=RealFormById("A",4,4);;
    gap> NonCompactDimension( G );
    14

Number 14 is in output of AllZeroDH function, so for \(\mathfrak{g}=e_{6(6)}\) and \(\mathfrak{h}=\mathfrak{sl}(5,\mathbb{R})\) corresponding homogeneous spaces \(G/H\) do not have compact Clifford–Klein forms.

    gap> L0[5];
    "D5"
    gap> RealFormsInformation( "D", 5 );

      There are 7 simple real forms with complexification D5
        1 is of type so(10), compact form
        2 - 3 are of type so(2p,10-2p) with 1 <= p <= 2
        4 is of type so*(10)
        5 is of type so(9,1)
        6 - 7 are of type so(2p+1,10-2p-1) with 1 <= p <= 2
      Index '0' returns the realification of D5

    gap> G:=RealFormById("D",5,7);;
    gap> NonCompactDimension( G );
    25

Number 25 is not in output of AllZeroDH function, so for \(\mathfrak{g}=e_{6(6)}\) and \(\mathfrak{h}=\mathfrak{so}(5,5)\) our algoritm does not provide a solution to the problem.

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