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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