  
  [1X4 [33X[0;0YReal nilpotent orbits[133X[101X
  
  
  [1X4.1 [33X[0;0YNilpotent orbits in real Lie algebras[133X[101X
  
  [33X[0;0Y[5XCoReLG[105X  has  a  database  of  the  nilpotent orbits of the real forms of the
  simple  Lie  algebras  of ranks up to 8. When called the first time in a GAP
  session, [5XCoReLG[105X will first read the database of nilpotent orbits.[133X
  
  [1X4.1-1 NilpotentOrbitsOfRealForm[101X
  
  [29X[2XNilpotentOrbitsOfRealForm[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YHere  [3XL[103X is a real form of a complex simple Lie algebra of rank up to 8. This
  function  returns  the  list  of  nilpotent  orbits (under the action of the
  adjoint  group)  of  [3XL[103X.  For  this  function to work, [3XL[103X must be defined over
  [3XSqrtField[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= RealFormById( "F", 4, 3 );;[127X[104X
    [4X[25Xgap>[125X [27Xno:= NilpotentOrbitsOfRealForm( L );;[127X[104X
    [4X[28X#I CoReLG: read database of real triples ... done[128X[104X
    [4X[25Xgap>[125X [27Xno[1];[127X[104X
    [4X[28X<nilpotent orbit in Lie algebra>[128X[104X
  [4X[32X[104X
  
  [1X4.1-2 RealCayleyTriple[101X
  
  [29X[2XRealCayleyTriple[102X( [3Xo[103X ) [32X attribute
  
  [33X[0;0YHere [3Xo[103X is a nilpotent orbit constructed by [2XNilpotentOrbitsOfRealForm[102X ([14X4.1-1[114X)
  of  a  simple real Lie algebra. This function returns a real Cayley triple [3X[
  f,  h,  e  ][103X  corresponding  to  the  orbit  [3Xo[103X.  The  third  element  [3Xe[103X is a
  representative of the orbit.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= RealFormById( "F", 4, 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xno:= NilpotentOrbitsOfRealForm( L );;[127X[104X
    [4X[25Xgap>[125X [27Xo:= no[10];[127X[104X
    [4X[28X<nilpotent orbit in Lie algebra>[128X[104X
    [4X[25Xgap>[125X [27Xt:=RealCayleyTriple(o);;[127X[104X
    [4X[25Xgap>[125X [27Xtheta:= CartanDecomposition(L).CartanInv;[127X[104X
    [4X[28Xfunction( v ) ... end[128X[104X
    [4X[25Xgap>[125X [27Xtheta(t[1]) = -t[3];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xtheta(t[2]) = -t[2];[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xt[3]*t[1] = t[2];[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X4.1-3 WeightedDynkinDiagram[101X
  
  [29X[2XWeightedDynkinDiagram[102X( [3Xo[103X ) [32X attribute
  
  [33X[0;0YHere [3Xo[103X is a nilpotent orbit constructed by [2XNilpotentOrbitsOfRealForm[102X ([14X4.1-1[114X)
  of  a  simple  real  Lie  algebra. This function returns the weighted Dynkin
  diagram  of the orbit, which identifies its orbit in the complexification of
  the real Lie algebra in which [3Xo[103X lies.[133X
  
  
  [1X4.2 [33X[0;0YNilpotent orbits in real Theta groups[133X[101X
  
  [33X[0;0YLet [22XL=⨁_i∈ Z_m L_i[122X be a real semisimple Lie algebra, where [22XZ_m=Z/mZ[122X if [22Xm≥ 1[122X,
  and  [22XZ_0=Z[122X. We provide some functions which help to determine the [22XG_0[122X-orbits
  of  nilpotent  elements  in  [22XL_1[122X,  where [22XG_0[122X is the adjoint group of [22XL_0[122X. An
  approach  to compute these orbits is described in [DFG15]; the first step is
  to  construct,  up to [22XG_0[122X-conjugacy, all carrier subalgebras of [22XL[122X. Functions
  [2XCarrierAlgsForNilpOrbsInZGrading[102X                 ([14X4.2-2[114X)                 and
  [2XCarrierAlgsForNilpOrbsInZmGrading[102X  ([14X4.2-3[114X)  do  this in the case that [22XL[122X is a
  split real form of a complex simple Lie algebra.[133X
  
  [1X4.2-1 RealWeylGroup[101X
  
  [29X[2XRealWeylGroup[102X( [3XL[103X ) [32X function
  [29X[2XRealWeylGroup[102X( [3XL[103X, [3XH[103X ) [32X function
  
  [33X[0;0YHere  [3XL[103X  is a real semisimple Lie algebra with Cartan subalgebra [3XH[103X. (If [3XH[103X is
  not  given,  then  [3XCartanSubalgebra(L)[103X will be taken.) This function returns
  the  real Weyl group [22XN_G(H)/C_G(H)[122X associated with [3XH[103X, where [22XG[122X is the adjoint
  group  of [3XL[103X. The real Weyl group will be stored in the Cartan subalgebra, so
  that  a new call to this function, with the same input, will return the real
  Weyl group immediately.[133X
  
  [1X4.2-2 CarrierAlgsForNilpOrbsInZGrading[101X
  
  [29X[2XCarrierAlgsForNilpOrbsInZGrading[102X( [3Xtype[103X, [3Xrank[103X, [3Xd[103X ) [32X attribute
  
  [33X[0;0YGives a record containing (up to conjugacy) the carrier algebras of the real
  theta group specified by the input: here [3Xtype[103X and [3Xrank[103X are the type and rank
  of the simple real Lie algebra (split real form) where everything happens; [3Xd[103X
  is  a  list  of degrees of the simple roots, defining a [22XZ[122X grading on the Lie
  algebra.  The  output  is  a  record  with  the following entries: [3XL[103X the Lie
  algebra,  [3Xgrad[103X the grading that was used (different format for Z-grading, [3XL0[103X
  the  0-component  of  the  grading, [3XHs[103X the Cartan subalgebras of [3XL0[103X that are
  used,  [3Xcars[103X  the  carrier  algebras.  Here [3Xcars[103X is a list of lists; for each
  Cartan  subalgebra  of  [3XL0[103X  there  is one list: the first corresponds to the
  split Cartan subalgebra, and so has just the complex carrier algebras (which
  are  also  real),  the  other  lists contain lists as well, for each complex
  carrier  algebra  (i.e., for each element of the first list) there is a list
  containing  the  real  carrier  algebras which are strongly [22XH_i[122X-regular, and
  over  the  complex  numbers  conjugate to the given complex carrier algebra.
  Furthermore,  a  carrier algebra is given by a record, containing the fields
  [3Xg0[103X, [3Xgp[103X (positive degree), and [3Xgn[103X (negative degree).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xca:=CarrierAlgsForNilpOrbsInZGrading("G",2,[1,0]);[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X Hs  := [ <Lie algebra of dimension 2 over SqrtField>, [128X[104X
    [4X[28X          <Lie algebra of dimension 2 over SqrtField> ], [128X[104X
    [4X[28X  L  :=   <Lie algebra of dimension 14 over SqrtField>, [128X[104X
    [4X[28X  L0 :=   <Lie algebra of dimension 4 over SqrtField>, [128X[104X
    [4X[28Xcars :=  [ [128X[104X
    [4X[28X          [rec( g0 := [ v.13+(3)*v.14 ], gn := [ [ v.9 ] ], gp := [ [ v.3 ] ] )][128X[104X
    [4X[28X            ,[ [  ] ] [128X[104X
    [4X[28X         ], [128X[104X
    [4X[28Xgrad := rec( g0 := [ v.2, v.8, v.13, v.14 ], [128X[104X
    [4X[28X               gn := [ [ v.7, v.9 ], [ v.10 ], [ v.11, v.12 ] ], [128X[104X
    [4X[28X               gp := [ [ v.1, v.3 ], [ v.4 ], [ v.5, v.6 ] ] ) )[128X[104X
  [4X[32X[104X
  
  [1X4.2-3 CarrierAlgsForNilpOrbsInZmGrading[101X
  
  [29X[2XCarrierAlgsForNilpOrbsInZmGrading[102X( [3Xtype[103X, [3Xrank[103X, [3Xm0[103X, [3Xstr[103X, [3Xnum[103X ) [32X attribute
  
  [33X[0;0YGives a record (up to conjugacy) containing the carrier algebras of the real
  theta group specified by the input: here [3Xtype[103X and [3Xrank[103X are the type and rank
  of  the  simple real Lie algebra (split real form) where everything happens;
  [3Xm0[103X  is the order of the automorphism defining the grading, [3Xstr[103X is "inner" or
  "outer",  depending  on  whether  the  automorphism is inner or not, [3Xnum[103X the
  [3Xnum[103X-th automorphism in the list [3XFiniteOrderInnerAutomorpisms( type, rank, m0
  )[103X  or  [3XFiniteOrderOuterAutomorphisms(  type,  rank, m0, 2 )[103X which is used to
  define  the  grading.  The output is as for [2XCarrierAlgsForNilpOrbsInZGrading[102X
  ([14X4.2-2[114X),  with  the  exception  that the record entry [3Xgrad[103X is a list with [3Xm0[103X
  entries,  the  [3Xi[103X-th entry containing the basis for the [3Xi[103X-th component of the
  grading.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xgap> ca:=CarrierAlgsForNilpOrbsInZmGrading("G",2,2,"inner",1);[127X[104X
    [4X[28Xrec( [128X[104X
    [4X[28X Hs := [ <Lie algebra of dimension 2 over SqrtField>, [128X[104X
    [4X[28X         <Lie algebra of dimension 2 over SqrtField>, [128X[104X
    [4X[28X         <Lie algebra of dimension 2 over SqrtField>, [128X[104X
    [4X[28X         <Lie algebra of dimension 2 over SqrtField> ], [128X[104X
    [4X[28X  L  :=  <Lie algebra of dimension 14 over SqrtField>, [128X[104X
    [4X[28X  L0 :=  <Lie algebra of dimension 6 over SqrtField>, [128X[104X
    [4X[28Xcars := [ [128X[104X
    [4X[28X         [rec( g0 := [ v.13+v.14 ], gn := [ [ v.11 ] ], gp := [ [ v.5 ] ] ), [128X[104X
    [4X[28X          rec( g0 := [ v.13, v.14 ], gn := [ [ v.2, v.10 ] ], [128X[104X
    [4X[28X               gp := [ [ v.4, v.8 ] ] ), [128X[104X
    [4X[28X          rec( g0 := [ v.13+(3/2)*v.14 ], gn := [ [ v.10 ] ], [128X[104X
    [4X[28X               gp := [ [ v.4 ] ] ), [128X[104X
    [4X[28X          rec( g0 := [ v.1, v.7, v.13, v.14 ], [128X[104X
    [4X[28X               gn := [ [ v.8, v.9, v.10, v.11 ], [ v.12 ] ], [128X[104X
    [4X[28X               gp := [ [ v.2, v.3, v.4, v.5 ], [ v.6 ] ] ), [128X[104X
    [4X[28X          rec( g0 := [ v.13, v.14 ], [128X[104X
    [4X[28X               gn := [ [ v.2, v.9 ], [ v.7 ], [ v.10 ], [ v.12 ], [ v.11 ] ], [128X[104X
    [4X[28X               gp := [ [ v.3, v.8 ], [ v.1 ], [ v.4 ], [ v.6 ], [ v.5 ] ] ) ], [128X[104X
    [4X[28X         [ [  ], [  ], [  ], [  ], [  ] ], [ [  ], [  ], [  ], [  ], [  ] ], [128X[104X
    [4X[28X         [ [  ], [  ], [  ], [  ], [  ] ] ], [128X[104X
    [4X[28X grad := [ [ v.1, v.6, v.7, v.12, v.13, v.14 ], [128X[104X
    [4X[28X         [ v.2, v.3, v.4, v.5, v.8, v.9, v.10, v.11 ] ] )[128X[104X
    [4X[28X[128X[104X
  [4X[32X[104X
  
