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Notation We use the notation and convention for real Lie algebras as is from CoReLG Package, .
gap> G:=RealFormById( "E", 7,3); <Lie algebra of dimension 133 over SqrtField> gap> rankG:=Dimension(CartanSubalgebra(G)); 7 gap> rankRG:=Dimension(CartanSubspace(G)); 3 gap> dimG:=Dimension(G); 133 gap> P:=CartanDecomposition( G ).P; <vector space over SqrtField, with 54 generators> gap> dimPforG:=Dimension(P); 54 gap> K:=CartanDecomposition( G ).K; <Lie algebra of dimension 79 over SqrtField> gap> rankK:= Dimension(CartanSubalgebra(K)); 7 gap> dimK:= Dimension(K); 79
Classification can be found in Table 9 in , p. 312-317.
Obstruction for the existence of compact Clifford-Klein form In this chapter we describe functions for algorithm from .
Technical functions For a real Lie algebra G constructed by the function RealFormById (from ), this function returns the non-compact dimension of G (dimension of a non-compact part in Cartan decomposition of G).
gap> G:=RealFormById("E",6,2); # E6(6) <Lie algebra of dimension 78 over SqrtField> gap> dG:=NonCompactDimension(G); 42
Let G be a compact connected Lie group of the type type and the rank rank. Let \Lambda\,P_{G}=\Lambda (y_1,...,y_l) be the exterior algebra over the spaces P_G of the primitive elements in H^*(G). Denote the degrees as follows |y_j|=2p_j-1,j=1,...,l. This function returns coefficients p_1,\ldots,p_l.
gap> PCoefficients("D",5); [ 2, 4, 6, 8, 5 ]
Here pi=\{ p_1,\ldots,p_l\} and qi=\{ q_1,\ldots,q_m\} are sets of coefficients (l\geq m). This function returns the polynomial: P(t)=\prod_{j=m+1}^l(1+t^{2p_j-1})\prod_{i=1}^m(1-t^{2p_i})/(1-t^{2q_i}).
gap> PCalculate([4,2,3],[2,2]); t^9+t^5+t^4+1
Let G^C be a complex Lie algebra of the type type and the rank rank. Let G be a real form of G^C with the index id (see RealFormsInformation,). This function returns the set of degrees of P(t) that have zero coefficients over all permutation (see Section 7 in ).
gap> AllZeroDH("F",4,2); [ 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27 ]
Algorithm example In this chapter we use additionaly functions from the following packages: CoReLG and SLA . We will show in detail the split case (for a non-split case you should use algoritm to generate regular subalgebras from ). 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.