  
  [1X2 [33X[0;0YObstruction for the existence of compact Clifford-Klein form[133X[101X
  
  [33X[0;0YIn this chapter we describe functions for algorithm from [BJS+].[133X
  
  
  [1X2.1 [33X[0;0YTechnical functions[133X[101X
  
  [1X2.1-1 NonCompactDimension[101X
  
  [33X[1;0Y[29X[2XNonCompactDimension[102X( [3XG[103X ) [32X function[133X
  
  [33X[0;0YFor  a  real  Lie  algebra  [22XG[122X constructed by the function [3XRealFormById[103X (from
  [DFdG14]),  this  function returns the non-compact dimension of [22XG[122X (dimension
  of a non-compact part in Cartan decomposition of [22XG[122X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:=RealFormById("E",6,2); # E6(6)[127X[104X
    [4X[28X<Lie algebra of dimension 78 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27XdG:=NonCompactDimension(G);[127X[104X
    [4X[28X42[128X[104X
  [4X[32X[104X
  
  [1X2.1-2 PCoefficients[101X
  
  [33X[1;0Y[29X[2XPCoefficients[102X( [3Xtype[103X, [3Xrank[103X ) [32X function[133X
  
  [33X[0;0YLet  [22XG[122X  be a compact connected Lie group of the type [3Xtype[103X and the rank [3Xrank[103X.
  Let  [22XΛP_G=Λ (y_1,...,y_l)[122X be the exterior algebra over the spaces [22XP_G[122X of the
  primitive    elements   in   [22XH^*(G)[122X.   Denote   the   degrees   as   follows
  [22X|y_j|=2p_j-1,j=1,...,l[122X. This function returns coefficients [22Xp_1,...,p_l[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPCoefficients("D",5);[127X[104X
    [4X[28X[ 2, 4, 6, 8, 5 ][128X[104X
  [4X[32X[104X
  
  [1X2.1-3 PCalculate[101X
  
  [33X[1;0Y[29X[2XPCalculate[102X( [3Xpi[103X, [3Xqi[103X ) [32X function[133X
  
  [33X[0;0YHere  [22Xpi={  p_1,...,p_l}[122X  and [22Xqi={ q_1,...,q_m}[122X are sets of coefficients ([22Xl≥
  m[122X).        This        function        returns        the        polynomial:
  [22XP(t)=∏_j=m+1^l(1+t^2p_j-1)∏_i=1^m(1-t^2p_i)/(1-t^2q_i)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPCalculate([4,2,3],[2,2]);   [127X[104X
    [4X[28Xt^9+t^5+t^4+1[128X[104X
  [4X[32X[104X
  
  [1X2.1-4 AllZeroDH[101X
  
  [33X[1;0Y[29X[2XAllZeroDH[102X( [3Xtype[103X, [3Xrank[103X, [3Xid[103X ) [32X function[133X
  
  [33X[0;0YLet  [22XG^C[122X  be a complex Lie algebra of the type [3Xtype[103X and the rank [3Xrank[103X. Let [22XG[122X
  be a real form of [22XG^C[122X with the index [3Xid[103X (see [3XRealFormsInformation[103X,[DFdG14]).
  This function returns the set of degrees of [22XP(t)[122X that have zero coefficients
  over all permutation (see Section 7 in [BJS+]).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAllZeroDH("F",4,2); [127X[104X
    [4X[28X[ 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27 ][128X[104X
  [4X[32X[104X
  
