  
  [1X3 [33X[0;0YReal Lie Algebras[133X[101X
  
  
  [1X3.1 [33X[0;0YConstruction of simple real Lie algebras[133X[101X
  
  [33X[0;0YA  few  functions  print some information on what they are doing to the info
  class [3XInfoCorelg[103X.[133X
  
  [1X3.1-1 RealFormsInformation[101X
  
  [33X[1;0Y[29X[2XRealFormsInformation[102X( [3Xtype[103X, [3Xrank[103X ) [32X function[133X
  
  [33X[0;0YThis  function  displays  information regarding the simple real Lie algebras
  that  can be constructed from the complex Lie algebra of type [3Xtype[103X (which is
  a  string)  and rank [3Xrank[103X (a positive integer). Each Lie algebra is given an
  index  which  is an integer, and for each index some information is given on
  the  Lie  algebra,  such  as  a commonly used name. In all cases the index 0
  refers to the realification of the complex Lie algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRealFormsInformation( "A", 4 );[127X[104X
    [4X[28X[128X[104X
    [4X[28X  There are 4 simple real forms with complexification A4[128X[104X
    [4X[28X    1 is of type su(5), compact form[128X[104X
    [4X[28X    2 - 3 are of type su(p,5-p) with 1 <= p <= 2[128X[104X
    [4X[28X    4 is of type sl(5,R)[128X[104X
    [4X[28X  Index '0' returns the realification of A4[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XRealFormsInformation( "E", 6 );[127X[104X
    [4X[28X [128X[104X
    [4X[28X  There are 5 simple real forms with complexification E6[128X[104X
    [4X[28X    1 is the compact form[128X[104X
    [4X[28X    2 is EI   = E6(6), with k_0 of type sp(4) (C4)[128X[104X
    [4X[28X    3 is EII  = E6(2), with k_0 of type su(6)+su(2) (A5+A1)[128X[104X
    [4X[28X    4 is EIII = E6(-14), with k_0 of type so(10)+R (D5+R)[128X[104X
    [4X[28X    5 is EIV  = E6(-26), with k_0 of type f_4 (F4)[128X[104X
    [4X[28X  Index '0' returns the realification of E6[128X[104X
    [4X[28X[128X[104X
    [4X[25Xgap>[125X [27XNumberRealForms("D",10);[127X[104X
    [4X[28X12[128X[104X
  [4X[32X[104X
  
  [1X3.1-2 NumberRealForms[101X
  
  [33X[1;0Y[29X[2XNumberRealForms[102X( [3Xtype[103X, [3Xrank[103X ) [32X function[133X
  
  [33X[0;0YThis function returns the number of (isomorphism types of) all real forms of
  the simple complex Lie algebras of type [3Xtype[103X and rank [3Xrank[103X.[133X
  
  [1X3.1-3 RealFormById[101X
  
  [33X[1;0Y[29X[2XRealFormById[102X( [3Xtype[103X, [3Xrank[103X, [3Xid[103X ) [32X function[133X
  [33X[1;0Y[29X[2XRealFormById[102X( [3Xtype[103X, [3Xrank[103X, [3Xid[103X, [3XF[103X ) [32X function[133X
  [33X[1;0Y[29X[2XRealFormById[102X( [[3Xtype[103X, [3Xrank[103X, [3Xid[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XRealFormById[102X( [[3Xtype[103X, [3Xrank[103X, [3Xid[103X, ][3XF[103X ) [32X function[133X
  [33X[1;0Y[29X[2XRealFormById[102X( [3Xlist[103X ) [32X function[133X
  [33X[1;0Y[29X[2XRealFormById[102X( [3Xlist[103X, [3XF[103X ) [32X function[133X
  
  [33X[0;0YLet  [22XL[122X  be the complex Lie algebra of type [3Xtype[103X and rank [3Xrank[103X. This function
  constructs  the  real  form  of  [22XL[122X  with  index [3Xid[103X (see [2XRealFormsInformation[102X
  ([14X3.1-1[114X)).  By  default  this  Lie  algebra  is  constructed  over  the field
  [3XSqrtField[103X. However, by adding as an optional fourth argument the field [3XF[103X, it
  is  possible to construct the Lie algebra output by this function over [3XF[103X. It
  is  required that the complex unit [3XE(4)[103X is contained in [3XF[103X. The function also
  accepts  [3Xtype[103X, [3Xrank[103X, and [3Xid[103X in a list as an argument. Moreover, if [3Xlist[103X is a
  list  of  such  triples,  then the function constructs the direct sum of the
  simple  real  forms specified by the individual triples. If the index [3Xind[103X is
  0,  then  the realification of [22XL[122X is constructed, which, strictly speaking is
  not a real form of [22XL[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRealFormById( "A", 4, 2 );[127X[104X
    [4X[28X<Lie algebra of dimension 24 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27XRealFormById( "A", 4, 2, CF(4) );[127X[104X
    [4X[28X<Lie algebra of dimension 24 over GaussianRationals>[128X[104X
    [4X[25Xgap>[125X [27XRealFormById( [ ["A", 4, 2], ["D", 5, 2] ], SqrtField);[127X[104X
    [4X[28X<Lie algebra of dimension 69 over SqrtField>[128X[104X
  [4X[32X[104X
  
  [1X3.1-4 IdRealForm[101X
  
  [33X[1;0Y[29X[2XIdRealForm[102X( [3XL[103X ) [32X function[133X
  
  [33X[0;0YLet  [22XL[122X be a semisimple real Lie algebra, where each simple summand is a real
  form  of a simple complex Lie algebra. This function returns (a list of) the
  id(s) of the simple real form(s).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := RealFormById( [ ["A", 4, 2], ["D", 5, 2] ], SqrtField);;[127X[104X
    [4X[25Xgap>[125X [27XIdRealForm( L );[127X[104X
    [4X[28X[ [ "A", 4, 2 ], [ "D", 5, 2 ] ][128X[104X
    [4X[25Xgap>[125X [27XK := RealFormById("A",5,2);;[127X[104X
    [4X[25Xgap>[125X [27XIdRealForm( K );[127X[104X
    [4X[28X[ "A", 5, 2 ][128X[104X
  [4X[32X[104X
  
  [1X3.1-5 NameRealForm[101X
  
  [33X[1;0Y[29X[2XNameRealForm[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X  is  a  reductive  real Lie algebra whose centre is stable under the
  Cartan  involution  of [3XL[103X. This function returns a string giving the names of
  the  real  forms of the simple components of the derived subalgebra of [3XL[103X, as
  well as the number of compact and non-compact dimensions of the centre of [3XL[103X.
  For  the  simple  real  Lie  algebras  we  use  the naming conventions as in
  [Kna02].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL := RealFormById( [ ["A", 4, 2], ["D", 5, 2] ], SqrtField);;[127X[104X
    [4X[25Xgap>[125X [27XNameRealForm( L );[127X[104X
    [4X[28X"su(1,4)+so(2,8)"[128X[104X
  [4X[32X[104X
  
  [1X3.1-6 AllRealForms[101X
  
  [33X[1;0Y[29X[2XAllRealForms[102X( [3Xtype[103X, [3Xrank[103X ) [32X function[133X
  
  [33X[0;0YThis  function  returns all real forms of the simple complex Lie algebras of
  type  [3Xtype[103X  and  rank  [3Xrank[103X  up  to  isomorphism.  In  the  same way as with
  [2XRealFormById[102X  ([14X3.1-3[114X)  it  is  possible to add the base field as an optional
  third argument.[133X
  
  [1X3.1-7 RealFormParameters[101X
  
  [33X[1;0Y[29X[2XRealFormParameters[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YFor  a  real Lie algebra [3XL[103X constructed by the function [2XRealFormById[102X ([14X3.1-3[114X),
  this  function returns a list of the parameters defining [3XL[103X as a real form of
  its  complexification. The first element of the list is the type of [3XL[103X (given
  by  a string), the second element is its rank, the third and fourth elements
  are  the  list  of  signs and the permutation defining the Cartan involution
  (see Section [14X1.1[114X).[133X
  
  [1X3.1-8 IsRealFormOfInnerType[101X
  
  [33X[1;0Y[29X[2XIsRealFormOfInnerType[102X( [3XL[103X ) [32X property[133X
  
  [33X[0;0YReturns  [3Xtrue[103X  if  and  only  if  the  real  form [3XL[103X is a defined by an inner
  involutive automorphism.[133X
  
  [1X3.1-9 IsRealification[101X
  
  [33X[1;0Y[29X[2XIsRealification[102X( [3XL[103X ) [32X property[133X
  
  [33X[0;0YReturns  [3Xtrue[103X  if  and  only  if  the  real form [3XL[103X is the realification of a
  complex simple Lie algebra.[133X
  
  [1X3.1-10 CartanDecomposition[101X
  
  [33X[1;0Y[29X[2XCartanDecomposition[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YThe  Cartan decomposition of [3XL[103X as a record with entries [3XK[103X, [3XP[103X, and [3XCartanInv[103X,
  such  that  [22XL=K⊕  P[122X  is  the  Cartan decomposition with corresponding Cartan
  involution [3XCartanInv[103X, which is defined as a function on [3XL[103X.[133X
  
  [33X[0;0YThe  Lie  algebras  constructed  by [2XRealFormById[102X ([14X3.1-3[114X) have this attribute
  stored.  For  other semisimple real Lie algebras it is computed. However, we
  do  remark  that  the  in  the  computation the root system is computed with
  respect to a Cartan subalgebra. If the program does not succeed in splitting
  the  Cartan  subalgebra  over the base field of [3XL[103X, then the computation will
  not succeed.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= RealFormById( "A", 5, 3 );[127X[104X
    [4X[28X<Lie algebra of dimension 35 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27XH := CartanSubalgebra(L);;[127X[104X
    [4X[25Xgap>[125X [27XK:= LieCentralizer( L, Subalgebra( L, [Basis( H )[1]] ) );[127X[104X
    [4X[28X<Lie algebra of dimension 17 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27XDK:= LieDerivedSubalgebra( K );[127X[104X
    [4X[28X<Lie algebra of dimension 15 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27XCartanDecomposition( DK );[127X[104X
    [4X[28Xrec( CartanInv := function( v ) ... end, [128X[104X
    [4X[28X  K := <Lie algebra of dimension 15 over SqrtField>, [128X[104X
    [4X[28X  P := <vector space of dimension 0 over SqrtField> )[128X[104X
    [4X[25Xgap>[125X [27X# We see that the semisimple subalgebra DK is compact. [127X[104X
  [4X[32X[104X
  
  [1X3.1-11 RealStructure[101X
  
  [33X[1;0Y[29X[2XRealStructure[102X( [3XL[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XRealStructure[102X( [3XL:[103X [3Xbasis[103X [3X:=[103X [3XB[103X ) [32X attribute[133X
  
  [33X[0;0YThe  real  structure  of  the  real form [3XL[103X is the (complex) conjugation with
  respect  to  [3XL[103X,  that  is,  the  function  which maps an element in [3XL[103X to the
  element  constructed  as  follows:  write  it as a linear combination of the
  basis  elements  of [3XL[103X and replace each coefficient by its complex conjugate.
  If  the  optional argument [3Xbasis:=B[103X is given, then [3XB[103X has to be a basis whose
  span  contains [3XL[103X (which is not checked by the code); in this case the linear
  combination  is done with respect to [3XB[103X. The latter construction is important
  when one considers a subalgebra [3XM[103X of a real form [3XL[103X; here one could either do
  [3XRealstructure(M:basis:=Basis(L))[103X or [3XSetRealStructure(M,RealStructure(L))[103X.[133X
  
  
  [1X3.2 [33X[0;0YMaximal reductive subalgebras[133X[101X
  
  [1X3.2-1 MaximalReductiveSubalgebras[101X
  
  [33X[1;0Y[29X[2XMaximalReductiveSubalgebras[102X( [3Xtype[103X, [3Xrank[103X, [3Xid[103X ) [32X operation[133X
  
  [33X[0;0YHere  the  input  parameters  are  as  in  [2XRealFormById[102X ([14X3.1-3[114X), and [3Xrank[103X is
  between 1 and 8. These parameters correspond to a real simple Lie algebra [22XL[122X.
  This  function  returns  a  list of maximal reductive subalgebras of [22XL[122X. More
  precisely,  it  returns  a  record  with two fields, [3Xliealg[103X and [3Xsubalgs[103X. The
  first field contains the Lie algebra [22XL[122X, and the second field contains a list
  of  its  maximal  reductive  subalgebras.  Isomorphic  copies of the regular
  semisimple Lie algebras can occur more than once. If that happens then those
  copies  are  not  conjugate  under  the  adjoint  group  of  [22XL[122X. There are no
  isomorphisms   between   the   non-reductive   subalgebras.   There  can  be
  non-conjugate  copies  of  those  as well, but the database does not contain
  these.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr:= MaximalReductiveSubalgebras("F",4,3);;[127X[104X
    [4X[25Xgap>[125X [27XNameRealForm( r.liealg );[127X[104X
    [4X[28X"F4(-20)"[128X[104X
    [4X[25Xgap>[125X [27Xfor K in r.subalgs do Print( NameRealForm(K), "\n" ); od;[127X[104X
    [4X[28Xsu(1,2)+su(3)[128X[104X
    [4X[28Xsu(2)+sp(1,2)[128X[104X
    [4X[28Xso(8,1)[128X[104X
    [4X[28Xso(9)[128X[104X
    [4X[28Xsl(2,R)+G2c[128X[104X
  [4X[32X[104X
  
  
  [1X3.3 [33X[0;0YIsomorphisms[133X[101X
  
  [1X3.3-1 IsomorphismOfRealSemisimpleLieAlgebras[101X
  
  [33X[1;0Y[29X[2XIsomorphismOfRealSemisimpleLieAlgebras[102X( [3XK[103X, [3XL[103X ) [32X function[133X
  
  [33X[0;0YHere  [3XK[103X,  [3XL[103X  are  two  real  forms of a semisimple complex Lie algebra. This
  function returns an isomorphism if one exists. Otherwise [3Xfalse[103X is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:=RealFormById("E",6,3);;                            [127X[104X
    [4X[25Xgap>[125X [27XH:=CartanSubalgebra(L);;[127X[104X
    [4X[25Xgap>[125X [27XK:=LieCentralizer(L,Subalgebra(L,Basis(H){[1,2,4]}));;[127X[104X
    [4X[25Xgap>[125X [27XDK:=LieDerivedSubalgebra(K);[127X[104X
    [4X[28X<Lie algebra of dimension 8 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27XIdRealForm(DK);          [127X[104X
    [4X[28X[ "A", 2, 2 ][128X[104X
    [4X[25Xgap>[125X [27XM:=RealFormById("A",2,2);[127X[104X
    [4X[28X<Lie algebra of dimension 8 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27XIsomorphismOfRealSemisimpleLieAlgebras(DK,M);[127X[104X
    [4X[28X<Lie algebra isomorphism between Lie algebras of dimension 8 over SqrtField>[128X[104X
  [4X[32X[104X
  
  
  [1X3.4 [33X[0;0YCartan subalgebras and root systems[133X[101X
  
  [1X3.4-1 MaximallyCompactCartanSubalgebra[101X
  
  [33X[1;0Y[29X[2XMaximallyCompactCartanSubalgebra[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X  is a real semisimple Lie algebra. This function returns a maximally
  compact Cartan subalgebra of [3XL[103X.[133X
  
  [1X3.4-2 MaximallyNonCompactCartanSubalgebra[101X
  
  [33X[1;0Y[29X[2XMaximallyNonCompactCartanSubalgebra[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X  is a real semisimple Lie algebra. This function returns a maximally
  non-compact Cartan subalgebra of [3XL[103X.[133X
  
  [1X3.4-3 CompactDimensionOfCartanSubalgebra[101X
  
  [33X[1;0Y[29X[2XCompactDimensionOfCartanSubalgebra[102X( [3XL[103X ) [32X function[133X
  [33X[1;0Y[29X[2XCompactDimensionOfCartanSubalgebra[102X( [3XL[103X, [3XH[103X ) [32X function[133X
  
  [33X[0;0YHere  [3XL[103X  is a real semisimple Lie algebra. This function returns the compact
  dimension   of   the   Cartan   subalgebra  [3XH[103X.  If  [3XH[103X  is  not  given,  then
  [3XCartanSubalgebra(L)[103X  will  be taken. The compact dimension will be stored in
  the  Cartan  subalgebra,  so that a new call to this function, with the same
  input, will return the compact dimension immediately.[133X
  
  [1X3.4-4 CartanSubalgebrasOfRealForm[101X
  
  [33X[1;0Y[29X[2XCartanSubalgebrasOfRealForm[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X  is  a  real form of a complex semisimple Lie algebra. This function
  returns  a  list of Cartan subalgebras of [3XL[103X. They are representatives of all
  classes of conjugate (by the adjoint group) Cartan subalgebras of [3XL[103X.[133X
  
  [1X3.4-5 CartanSubspace[101X
  
  [33X[1;0Y[29X[2XCartanSubspace[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X  is  a  real  semisimple Lie algebra. This function returns a Cartan
  subspace of [3XL[103X. That is a maximal abelian subspace of the subspace [3XP[103X given in
  the [2XCartanDecomposition[102X ([14X3.1-10[114X) of [3XL[103X.[133X
  
  [1X3.4-6 RootsystemOfCartanSubalgebra[101X
  
  [33X[1;0Y[29X[2XRootsystemOfCartanSubalgebra[102X( [3XL[103X ) [32X operation[133X
  [33X[1;0Y[29X[2XRootsystemOfCartanSubalgebra[102X( [3XL[103X, [3XH[103X ) [32X operation[133X
  
  [33X[0;0YHere  [3XL[103X  is a semisimple Lie algebra, and [3XH[103X is a Cartan subalgebra. (If [3XH[103X is
  not  given,  then  [3XCartanSubalgebra(L)[103X will be taken.) This function returns
  the root system of [3XL[103X with respect to [3XH[103X. It is necessary that the eigenvalues
  of  the  adjoint  maps  corresponding to all elements of [3XH[103X lie in the ground
  field  of  [3XL[103X.  However,  even  if  they  do,  it is not guaranteed that this
  function succeeds, as it may happen that [5XGAP[105X has no polynomial factorisation
  algorithm over the ground field.[133X
  
  [33X[0;0YThe  root  system  is stored in [3XH[103X, so that a new call to this function, with
  the same input, will return the same root system.[133X
  
  [1X3.4-7 ChevalleyBasis[101X
  
  [33X[1;0Y[29X[2XChevalleyBasis[102X( [3XR[103X ) [32X attribute[133X
  
  [33X[0;0YHere [3XR[103X is a root system of a semisimple Lie algebra [3XL[103X. This function returns
  a Chevalley basis of [3XL[103X, consisting of root vectors of [3XR[103X.[133X
  
  
  [1X3.5 [33X[0;0YDiagrams[133X[101X
  
  [33X[0;0YIn  this  section we document the functionality for computing the Satake and
  Vogan  diagrams of a real semisimple Lie algebra. In both cases the relevant
  function  computes  an  object,  which,  when  printed, does not reveal much
  information.  However,  [3XDisplay[103X  with  as input such an object, displays the
  diagram.  Here  we  use  the convention that every node is represented by an
  integer;  nodes  that  are  painted  black  are  represented  by integers in
  brackets;  and  the  involution  (i.e.,  the  arrows  in  the  diagram)  are
  represented  by  a  permutation  of  the  nodes, printed on a line below the
  diagram.[133X
  
  [1X3.5-1 VoganDiagram[101X
  
  [33X[1;0Y[29X[2XVoganDiagram[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X  is  a  real semisimple Lie algebra. This function returns the Vogan
  diagram of [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= RealFormById( [["E", 6, 3],["A", 3, 2]] );;[127X[104X
    [4X[25Xgap>[125X [27X K:= LieCentralizer( L, Subalgebra( L, Basis( CartanSubalgebra(L) ){[1]} ) );[127X[104X
    [4X[28X<Lie algebra of dimension 51 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27X DK:= LieDerivedSubalgebra( K );[127X[104X
    [4X[28X<Lie algebra of dimension 50 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27Xvd:= VoganDiagram(DK);[127X[104X
    [4X[28X<Vogan diagram in Lie algebra of type A3+A5>[128X[104X
    [4X[25Xgap>[125X [27X Display( vd );[127X[104X
    [4X[28XA3:  (1)---2---3[128X[104X
    [4X[28XA5:  4---(5)---6---7---8[128X[104X
    [4X[28XInvolution: ()[128X[104X
    [4X[28XTypes of direct summands:[128X[104X
    [4X[28X[ [ "A", 3, 2 ], [ "A", 5, 3 ] ][128X[104X
  [4X[32X[104X
  
  [1X3.5-2 SatakeDiagram[101X
  
  [33X[1;0Y[29X[2XSatakeDiagram[102X( [3XL[103X ) [32X attribute[133X
  
  [33X[0;0YHere  [3XL[103X  is  a real semisimple Lie algebra. This function returns the Satake
  diagram of [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= RealFormById( [["E", 6, 3],["A", 3, 2]] );;[127X[104X
    [4X[25Xgap>[125X [27XK:= LieCentralizer( L, Subalgebra( L, Basis( CartanSubalgebra(L) ){[1]} ) );[127X[104X
    [4X[28X<Lie algebra of dimension 51 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27X DK:= LieDerivedSubalgebra( K );[127X[104X
    [4X[28X<Lie algebra of dimension 50 over SqrtField>[128X[104X
    [4X[25Xgap>[125X [27Xsd:= SatakeDiagram( DK );[127X[104X
    [4X[28X<Satake diagram in Lie algebra of type A5xA3>[128X[104X
    [4X[25Xgap>[125X [27XDisplay( sd );[127X[104X
    [4X[28XA5:  1---2---(3)---4---5[128X[104X
    [4X[28XA3:  6---(7)---8[128X[104X
    [4X[28XInvolution:  (1,5)(2,4)(6,8)[128X[104X
  [4X[32X[104X
  
