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<?xml version="1.0" encoding="utf-8"?>
<!DOCTYPE Book SYSTEM "gapdoc.dtd"
[<!ENTITY NoCK "NoCK">] >

<Book Name="NoCK">

<#Include SYSTEM "title.xml">

<TableOfContents/>

<Body>

<Chapter> <Heading>Notation</Heading>

We use the notation and convention for real Lie algebras as is from CoReLG Package, <Cite Key="CoReLG" />.

<Example>
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
</Example>

Classification can be found in Table 9 in <Cite Key="onvin" />, p. 312-317.

</Chapter>

<Chapter> <Heading>Obstruction for the existence of compact Clifford-Klein form</Heading>

In this chapter we describe functions for algorithm from <Cite Key="our"/>.
<Section> <Heading>Technical functions</Heading>
<ManSection>
<Func Name="NonCompactDimension"  Arg="G"
      Comm="real forms"/>
<Description>
For a real Lie algebra <M>G</M> constructed by the function <A>RealFormById</A> (from <Cite Key="CoReLG"/>), this function returns the non-compact dimension of <M>G</M>
(dimension of a non-compact part in Cartan decomposition of <M>G</M>).

<Example>
gap> G:=RealFormById("E",6,2); # E6(6)
<Lie algebra of dimension 78 over SqrtField>
gap> dG:=NonCompactDimension(G);
42
</Example>
</Description>
</ManSection><ManSection>
<Func Name="PCoefficients"  Arg="type, rank"/>
<Description>
Let <M>G</M> be a compact connected Lie group of the type <A>type</A> and the rank <A>rank</A>.  Let
<M>\Lambda\,P_{G}=\Lambda (y_1,...,y_l)</M>
be the exterior algebra over the spaces <M>P_G</M> of the primitive elements in <M>H^*(G)</M>. Denote the degrees as follows
<M>|y_j|=2p_j-1,j=1,...,l</M>. This function returns coefficients <M>p_1,\ldots,p_l</M>.
<Example>
gap> PCoefficients("D",5);
[ 2, 4, 6, 8, 5 ]
</Example>
</Description>
</ManSection><ManSection>
<Func Name="PCalculate"  Arg="pi, qi"/>
<Description>
Here <M>pi=\{ p_1,\ldots,p_l\}</M> and <M>qi=\{ q_1,\ldots,q_m\}</M> are sets of coefficients (<M>l\geq m</M>). This function
returns the polynomial:
<M>P(t)=\prod_{j=m+1}^l(1+t^{2p_j-1})\prod_{i=1}^m(1-t^{2p_i})/(1-t^{2q_i})</M>.
<Example>
gap> PCalculate([4,2,3],[2,2]);
t^9+t^5+t^4+1
</Example>
</Description>
</ManSection><ManSection>
<Func Name="AllZeroDH"  Arg="type, rank, id"/>
<Description>
Let <M>G^C</M> be a complex Lie algebra of the type <A>type</A> and the rank <A>rank</A>. Let <M>G</M> be a real form of <M>G^C</M>
with the index <A>id</A> (see <A>RealFormsInformation</A>,<Cite Key="CoReLG"/>). This function returns the set of degrees of <M>P(t)</M>
that have zero coefficients over all permutation (see Section 7 in <Cite Key="our"/>).
<Example>
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 ]
</Example>
</Description>
</ManSection>
</Section>
</Chapter>
<Chapter> <Heading>Algorithm example</Heading>

In this chapter we use additionaly functions from the following packages: CoReLG <Cite Key="CoReLG"/> and SLA <Cite Key="SLA"/>.

We will show in detail the split case (for a non-split case you should use algoritm to generate regular subalgebras from <Cite Key="DFG"/>).

For example, we take <M>G=\mathfrak{e}_{6(6)}</M> (tuple "E",6,2 in CoReLG notation). We calculate <A>AllZeroDH</A> on it.

<Example>
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 ]
</Example>

We generate all regular subalgebras of complexification.
<Example>
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" ]
</Example>

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

<Example>
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
</Example>

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

<Example>
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
</Example>

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

<Bibliography Databases="NoCKbib.xml" />

<TheIndex/>

</Book>

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