  
  [1X64 [33X[0;0YLie Algebras[133X[101X
  
  [33X[0;0YA  Lie algebra [22XL[122X is an algebra such that [22Xx x = 0[122X and [22Xx(yz) + y(zx) + z(xy) =
  0[122X  for  all [22Xx, y, z ∈ L[122X. A common way of creating a Lie algebra is by taking
  an  associative  algebra  together with the commutator as product. Therefore
  the  product  of  two  elements  [22Xx, y[122X of a Lie algebra is usually denoted by
  [22X[x,y][122X,  but  in [5XGAP[105X this denotes the list of the elements [22Xx[122X and [22Xy[122X; hence the
  product  of  elements  is  made  by the usual [10X*[110X. This gives no problems when
  dealing  with Lie algebras given by a table of structure constants. However,
  for  matrix  Lie  algebras  the  situation  is  not so easy as [10X*[110X denotes the
  ordinary  (associative) matrix multiplication. In [5XGAP[105X this problem is solved
  by  wrapping  elements  of  a  matrix Lie algebra up as [10XLieObject[110Xs, and then
  define the [10X*[110X for [10XLieObject[110Xs to be the commutator (see [14X64.1[114X).[133X
  
  
  [1X64.1 [33X[0;0YLie Objects[133X[101X
  
  [33X[0;0YLet  [10Xx[110X be a ring element, then [10XLieObject(x)[110X (see [2XLieObject[102X ([14X64.1-1[114X)) wraps [10Xx[110X
  up into an object that contains the same data (namely [10Xx[110X). The multiplication
  [10X*[110X  for  Lie  objects is formed by taking the commutator. More exactly, if [10Xl1[110X
  and  [10Xl2[110X  are  the  Lie objects corresponding to the ring elements [10Xr1[110X and [10Xr2[110X,
  then  [10Xl1 * l2[110X is equal to the Lie object corresponding to [10Xr1 * r2 - r2 * r1[110X.
  Two rules for Lie objects are worth noting:[133X
  
  [30X    [33X[0;6YAn element is [13Xnot[113X equal to its Lie element.[133X
  
  [30X    [33X[0;6YIf  we  take  the  Lie object of an ordinary (associative) matrix then
        this is again a matrix; it is therefore a collection (of its rows) and
        a  list. But it is [13Xnot[113X a collection of collections of its entries, and
        its family is [13Xnot[113X a collections family.[133X
  
  [33X[0;0YGiven  a  family  [10XF[110X  of  ring  elements,  we  can form its Lie family [10XL[110X. The
  elements  of [10XF[110X and [10XL[110X are in bijection, only the multiplications via [10X*[110X differ
  for  both  families.  More  exactly,  if  [10Xl1[110X  and  [10Xl2[110X  are  the Lie elements
  corresponding  to  the elements [10Xf1[110X and [10Xf2[110X in [10XF[110X, we have [10Xl1 * l2[110X equal to the
  Lie  element corresponding to [10Xf1 * f2 - f2 * f1[110X. Furthermore, the product of
  Lie  elements  [10Xl1[110X,  [10Xl2[110X  and  [10Xl3[110X is left-normed, that is [10Xl1*l2*l3[110X is equal to
  [10X(l1*l2)*l3[110X.[133X
  
  [33X[0;0YThe main reason to distinguish elements and Lie elements on the family level
  is  that  this  helps to avoid forming domains that contain elements of both
  types. For example, if we could form vector spaces of matrices then at first
  glance  it would be no problem to have both ordinary and Lie matrices in it,
  but  as  soon  as  we  find  out that the space is in fact an algebra (e.g.,
  because its dimension is that of the full matrix algebra), we would run into
  strange problems.[133X
  
  [33X[0;0YNote that the family situation with Lie families may be not familiar.[133X
  
  [30X    [33X[0;6YWe  have  to  be  careful when installing methods for certain types of
        domains  that  may involve Lie elements. For example, the zero element
        of  a  matrix  space  is either an ordinary matrix or its Lie element,
        depending  on  the  space.  So either the method must be aware of both
        cases,  or the method selection must distinguish the two cases. In the
        latter situation, only one method may be applicable to each case; this
        means  that  it  is  not  sufficient  to  treat  the Lie case with the
        additional   requirement   [10XIsLieObjectCollection[110X   but  that  we  must
        explicitly require non-Lie elements for the non-Lie case.[133X
  
  [30X    [33X[0;6YBeing  a  full matrix space is a property that may hold for a space of
        ordinary  matrices  or  a  space  of Lie matrices. So methods for full
        matrix spaces must also be aware of Lie matrices.[133X
  
  [1X64.1-1 LieObject[101X
  
  [29X[2XLieObject[102X( [3Xobj[103X ) [32X attribute
  
  [33X[0;0YLet  [3Xobj[103X  be  a ring element. Then [10XLieObject( [3Xobj[103X[10X )[110X is the corresponding Lie
  object.  If  [3Xobj[103X  lies  in  the  family [10XF[110X, then [10XLieObject( [3Xobj[103X[10X )[110X lies in the
  family [10XLieFamily( F )[110X (see [2XLieFamily[102X ([14X64.1-3[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= [ [ 1, 0 ], [ 0, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xlo:= LieObject( m );[127X[104X
    [4X[28XLieObject( [ [ 1, 0 ], [ 0, 1 ] ] )[128X[104X
    [4X[25Xgap>[125X [27Xm*m;[127X[104X
    [4X[28X[ [ 1, 0 ], [ 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xlo*lo;[127X[104X
    [4X[28XLieObject( [ [ 0, 0 ], [ 0, 0 ] ] )[128X[104X
  [4X[32X[104X
  
  [1X64.1-2 IsLieObject[101X
  
  [29X[2XIsLieObject[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsLieObjectCollection[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsRestrictedLieObject[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsRestrictedLieObjectCollection[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YAn object lies in [2XIsLieObject[102X if and only if it lies in a family constructed
  by [2XLieFamily[102X ([14X64.1-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= [ [ 1, 0 ], [ 0, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xlo:= LieObject( m );[127X[104X
    [4X[28XLieObject( [ [ 1, 0 ], [ 0, 1 ] ] )[128X[104X
    [4X[25Xgap>[125X [27XIsLieObject( m );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsLieObject( lo );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X64.1-3 LieFamily[101X
  
  [29X[2XLieFamily[102X( [3XFam[103X ) [32X attribute
  
  [33X[0;0Yis  a family [10XF[110X in bijection with the family [3XFam[103X, but with the Lie bracket as
  infix multiplication. That is, for [10Xx[110X, [10Xy[110X in [3XFam[103X, the product of the images in
  [10XF[110X will be the image of [10Xx * y - y * x[110X.[133X
  
  [33X[0;0YThe standard type of objects in a Lie family [10XF[110X is [10X[3XF[103X[10X!.packedType[110X.[133X
  
  [33X[0;0YThe  bijection  from [3XFam[103X to [10XF[110X is given by [10XEmbedding( [3XFam[103X[10X, F )[110X (see [2XEmbedding[102X
  ([14X32.2-10[114X)); this bijection respects addition and additive inverses.[133X
  
  [1X64.1-4 UnderlyingFamily[101X
  
  [29X[2XUnderlyingFamily[102X( [3XFam[103X ) [32X attribute
  
  [33X[0;0YIf  [3XFam[103X is a Lie family then [10XUnderlyingFamily( [3XFam[103X[10X )[110X is a family [10XF[110X such that
  [10X[3XFam[103X[10X = LieFamily( F )[110X.[133X
  
  [1X64.1-5 UnderlyingRingElement[101X
  
  [29X[2XUnderlyingRingElement[102X( [3Xobj[103X ) [32X attribute
  
  [33X[0;0YLet  [3Xobj[103X  be  a  Lie  object  constructed  from  a ring element [10Xr[110X by calling
  [10XLieObject( r )[110X. Then [10XUnderlyingRingElement( [3Xobj[103X[10X )[110X returns the ring element [10Xr[110X
  used  to  construct  [3Xobj[103X.  If  [10Xr[110X  lies in the family [10XF[110X, then [3Xobj[103X lies in the
  family [10XLieFamily( F )[110X (see [2XLieFamily[102X ([14X64.1-3[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xlo:= LieObject( [ [ 1, 0 ], [ 0, 1 ] ] );[127X[104X
    [4X[28XLieObject( [ [ 1, 0 ], [ 0, 1 ] ] )[128X[104X
    [4X[25Xgap>[125X [27Xm:=UnderlyingRingElement(lo);[127X[104X
    [4X[28X[ [ 1, 0 ], [ 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xlo*lo;[127X[104X
    [4X[28XLieObject( [ [ 0, 0 ], [ 0, 0 ] ] )[128X[104X
    [4X[25Xgap>[125X [27Xm*m;[127X[104X
    [4X[28X[ [ 1, 0 ], [ 0, 1 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X64.2 [33X[0;0YConstructing Lie algebras[133X[101X
  
  [33X[0;0YIn this section we describe functions that create Lie algebras. Creating and
  working  with  subalgebras  goes  exactly  in  the  same  way as for general
  algebras; so for that we refer to Chapter [14X62[114X.[133X
  
  [1X64.2-1 LieAlgebraByStructureConstants[101X
  
  [29X[2XLieAlgebraByStructureConstants[102X( [3XR[103X, [3Xsct[103X[, [3Xnameinfo[103X] ) [32X function
  
  [33X[0;0Y[2XLieAlgebraByStructureConstants[102X  does the same as [2XAlgebraByStructureConstants[102X
  ([14X62.4-1[114X),  and  has the same meaning of arguments, except that the result is
  assumed  to  be a Lie algebra. Note that the function does not check whether
  [3Xsct[103X satisfies the Jacobi identity. (So if one creates a Lie algebra this way
  with  a  table  that  does not satisfy the Jacobi identity, errors may occur
  later on.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 2, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 2, [ 1/2, 1 ] );[127X[104X
    [4X[25Xgap>[125X [27XL:= LieAlgebraByStructureConstants( Rationals, T );[127X[104X
    [4X[28X<Lie algebra of dimension 2 over Rationals>[128X[104X
  [4X[32X[104X
  
  [1X64.2-2 RestrictedLieAlgebraByStructureConstants[101X
  
  [29X[2XRestrictedLieAlgebraByStructureConstants[102X( [3XR[103X, [3Xsct[103X[, [3Xnameinfo[103X], [3Xpmapping[103X ) [32X function
  
  [33X[0;0Y[2XRestrictedLieAlgebraByStructureConstants[102X      does      the      same     as
  [2XLieAlgebraByStructureConstants[102X  ([14X64.2-1[114X),  and  has  the same meaning of all
  arguments,  except that the result is assumed to be a restricted Lie algebra
  (see  [14X64.8[114X)  with  the [22Xp[122X-map given by the additional argument [3Xpmapping[103X. This
  last argument is a list of the length equal to the dimension of the algebra;
  its [22Xi[122X-th entry specifies the [22Xp[122X-th power of the [22Xi[122X-th basis vector in the same
  format  [10X[  coeff1,  position1,  coeff2,  position2, ... ][110X as [2XSetEntrySCTable[102X
  ([14X62.4-4[114X) uses to specify entries of the structure constants table.[133X
  
  [33X[0;0YNote  that  the  function  does  not  check whether [3Xsct[103X satisfies the Jacobi
  identity, of whether [3Xpmapping[103X specifies a legitimate [22Xp[122X-mapping.[133X
  
  [33X[0;0YThe  following  example  creates  a  commutative  restricted  Lie algebra of
  dimension 3, in which the [22Xp[122X-th power of the [22Xi[122X-th basis element is the [22Xi+1[122X-th
  basis element (except for the 3rd basis element which goes to zero).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 3, Zero(GF(5)), "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XL:= RestrictedLieAlgebraByStructureConstants( [127X[104X
    [4X[25X>[125X [27X                                    GF(5), T, [[1,2],[1,3],[]] );[127X[104X
    [4X[28X<Lie algebra of dimension 3 over GF(5)>[128X[104X
    [4X[25Xgap>[125X [27XList(Basis(L),PthPowerImage);[127X[104X
    [4X[28X[ v.2, v.3, 0*v.1 ][128X[104X
    [4X[25Xgap>[125X [27XPthPowerImage(L.1+L.2);[127X[104X
    [4X[28Xv.2+v.3[128X[104X
  [4X[32X[104X
  
  [1X64.2-3 LieAlgebra[101X
  
  [29X[2XLieAlgebra[102X( [3XL[103X ) [32X function
  [29X[2XLieAlgebra[102X( [3XF[103X, [3Xgens[103X[, [3Xzero[103X][, [3X"basis"[103X] ) [32X function
  
  [33X[0;0YFor  an associative algebra [3XL[103X, [10XLieAlgebra( [3XL[103X[10X )[110X is the Lie algebra isomorphic
  to [3XL[103X as a vector space but with the Lie bracket as product.[133X
  
  [33X[0;0Y[10XLieAlgebra( [3XF[103X[10X, [3Xgens[103X[10X )[110X is the Lie algebra over the division ring [3XF[103X, generated
  [13Xas  Lie  algebra[113X by the Lie objects corresponding to the vectors in the list
  [3Xgens[103X.[133X
  
  [33X[0;0Y[13XNote[113X that the algebra returned by [2XLieAlgebra[102X does not contain the vectors in
  [3Xgens[103X.  The  elements  in [3Xgens[103X are wrapped up as Lie objects (see [14X64.1[114X). This
  allows one to create Lie algebras from ring elements with respect to the Lie
  bracket  as  product.  But  of  course the product in the Lie algebra is the
  usual [10X*[110X.[133X
  
  [33X[0;0YIf  there  are  three  arguments,  a  division ring [3XF[103X and a list [3Xgens[103X and an
  element  [3Xzero[103X,  then  [10XLieAlgebra( [3XF[103X[10X, [3Xgens[103X[10X, [3Xzero[103X[10X )[110X is the corresponding [3XF[103X-Lie
  algebra with zero element the Lie object corresponding to [3Xzero[103X.[133X
  
  [33X[0;0YIf  the  last  argument  is  the string [10X"basis"[110X then the vectors in [3Xgens[103X are
  known to form a basis of the algebra (as an [3XF[103X-vector space).[133X
  
  [33X[0;0Y[13XNote[113X that even if each element in [3Xgens[103X is already a Lie element, i.e., is of
  the form [10XLieElement( [3Xelm[103X[10X )[110X for an object [3Xelm[103X, the elements of the result lie
  in the Lie family of the family that contains [3Xgens[103X as a subset.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:= FullMatrixAlgebra( GF( 7 ), 4 );;[127X[104X
    [4X[25Xgap>[125X [27XL:= LieAlgebra( A );[127X[104X
    [4X[28X<Lie algebra of dimension 16 over GF(7)>[128X[104X
    [4X[25Xgap>[125X [27Xmats:= [ [ [ 1, 0 ], [ 0, -1 ] ], [ [ 0, 1 ], [ 0, 0 ] ], [127X[104X
    [4X[25X>[125X [27X            [ [ 0, 0 ], [ 1, 0] ] ];;[127X[104X
    [4X[25Xgap>[125X [27XL:= LieAlgebra( Rationals, mats );[127X[104X
    [4X[28X<Lie algebra over Rationals, with 3 generators>[128X[104X
  [4X[32X[104X
  
  [1X64.2-4 FreeLieAlgebra[101X
  
  [29X[2XFreeLieAlgebra[102X( [3XR[103X, [3Xrank[103X[, [3Xname[103X] ) [32X function
  [29X[2XFreeLieAlgebra[102X( [3XR[103X, [3Xname1[103X, [3Xname2[103X, [3X...[103X ) [32X function
  
  [33X[0;0YReturns  a free Lie algebra of rank [3Xrank[103X over the ring [3XR[103X. [10XFreeLieAlgebra( [3XR[103X[10X,
  [3Xname1[103X[10X,  [3Xname2[103X[10X,...)[110X  returns  a free Lie algebra over [3XR[103X with generators named
  [3Xname1[103X,  [3Xname2[103X,  and so on. The elements of a free Lie algebra are written on
  the Hall-Lyndon basis.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieAlgebra( Rationals, "x", "y", "z" );[127X[104X
    [4X[28X<Lie algebra over Rationals, with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27Xg:= GeneratorsOfAlgebra( L );; x:= g[1];; y:=g[2];; z:= g[3];;[127X[104X
    [4X[25Xgap>[125X [27Xz*(y*(x*(z*y)));[127X[104X
    [4X[28X(-1)*((x*(y*z))*(y*z))+(-1)*((x*((y*z)*z))*y)+(-1)*(((x*z)*(y*z))*y)[128X[104X
  [4X[32X[104X
  
  [1X64.2-5 FullMatrixLieAlgebra[101X
  
  [29X[2XFullMatrixLieAlgebra[102X( [3XR[103X, [3Xn[103X ) [32X function
  [29X[2XMatrixLieAlgebra[102X( [3XR[103X, [3Xn[103X ) [32X function
  [29X[2XMatLieAlgebra[102X( [3XR[103X, [3Xn[103X ) [32X function
  
  [33X[0;0Yis  the  full  matrix  Lie  algebra of [22X[3Xn[103X × [3Xn[103X[122X matrices over the ring [3XR[103X, for a
  nonnegative integer [3Xn[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFullMatrixLieAlgebra( GF(9), 10 );[127X[104X
    [4X[28X<Lie algebra over GF(3^2), with 19 generators>[128X[104X
  [4X[32X[104X
  
  [1X64.2-6 RightDerivations[101X
  
  [29X[2XRightDerivations[102X( [3XB[103X ) [32X attribute
  [29X[2XLeftDerivations[102X( [3XB[103X ) [32X attribute
  [29X[2XDerivations[102X( [3XB[103X ) [32X attribute
  
  [33X[0;0YThese  functions  all  return  the  matrix Lie algebra of derivations of the
  algebra [22XA[122X with basis [3XB[103X.[133X
  
  [33X[0;0Y[10XRightDerivations(  [3XB[103X[10X  )[110X  returns  the  algebra of derivations represented by
  their  right  action  on  the algebra [22XA[122X. This means that with respect to the
  basis  [22XB[122X of [22XA[122X, the derivation [22XD[122X is described by the matrix [22X[ d_{i,j} ][122X which
  means that [22XD[122X maps the [22Xi[122X-th basis element [22Xb_i[122X to [22X∑_{j = 1}^n d_{i,j} b_j[122X.[133X
  
  [33X[0;0Y[10XLeftDerivations(  [3XB[103X[10X  )[110X returns the Lie algebra of derivations represented by
  their left action on the algebra [22XA[122X. So the matrices contained in the algebra
  output  by [10XLeftDerivations( [3XB[103X[10X )[110X are the transposes of the matrices contained
  in the output of [10XRightDerivations( [3XB[103X[10X )[110X.[133X
  
  [33X[0;0Y[2XDerivations[102X is just a synonym for [2XRightDerivations[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:= OctaveAlgebra( Rationals );[127X[104X
    [4X[28X<algebra of dimension 8 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XL:= Derivations( Basis( A ) );[127X[104X
    [4X[28X<Lie algebra of dimension 14 over Rationals>[128X[104X
  [4X[32X[104X
  
  [1X64.2-7 SimpleLieAlgebra[101X
  
  [29X[2XSimpleLieAlgebra[102X( [3Xtype[103X, [3Xn[103X, [3XF[103X ) [32X function
  
  [33X[0;0YThis  function constructs the simple Lie algebra of type given by the string
  [3Xtype[103X  and  rank [3Xn[103X over the field [3XF[103X. The string [3Xtype[103X must be one of [10X"A"[110X, [10X"B"[110X,
  [10X"C"[110X,  [10X"D"[110X, [10X"E"[110X, [10X"F"[110X, [10X"G"[110X, [10X"H"[110X, [10X"K"[110X, [10X"S"[110X, [10X"W"[110X or [10X"M"[110X. For the types [10XA[110X to [10XG[110X, [3Xn[103X
  must  be  a  positive integer. The last five types only exist over fields of
  characteristic  [22Xp>0[122X.  If  the  type  is [10XH[110X, then [3Xn[103X must be a list of positive
  integers of even length. If the type is [10XK[110X, then [3Xn[103X must be a list of positive
  integers  of odd length. For the types [10XS[110X and [10XW[110X, [3Xn[103X must be a list of positive
  integers  of  any  length.  If  the  type is [10XM[110X, then the Melikyan algebra is
  constructed.  In  this  case [3Xn[103X must be a list of two positive integers. This
  Lie algebra only exists over fields of characteristic [22X5[122X. This Lie algebra is
  [22Xℤ  ×  ℤ[122X graded; and the grading can be accessed via the attribute [10XGrading(L)[110X
  (see [2XGrading[102X  ([14X62.9-20[114X)).  In  some  cases  the Lie algebra returned by this
  function  is  not  simple.  Examples are the Lie algebras of type [22XA_n[122X over a
  field  of  characteristic  [22Xp>0[122X  where [22Xp[122X divides [22Xn+1[122X, and the Lie algebras of
  type [22XK_n[122X where [22Xn[122X is a list of length 1.[133X
  
  [33X[0;0YIf  [3Xtype[103X  is  one of [10XA[110X, [10XB[110X, [10XC[110X, [10XD[110X, [10XE[110X, [10XF[110X, [10XG[110X, and [3XF[103X is a field of characteristic
  zero, then the basis of the returned Lie algebra is a Chevalley basis.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSimpleLieAlgebra( "E", 6, Rationals );[127X[104X
    [4X[28X<Lie algebra of dimension 78 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XSimpleLieAlgebra( "A", 6, GF(5) );[127X[104X
    [4X[28X<Lie algebra of dimension 48 over GF(5)>[128X[104X
    [4X[25Xgap>[125X [27XSimpleLieAlgebra( "W", [1,2], GF(5) );[127X[104X
    [4X[28X<Lie algebra of dimension 250 over GF(5)>[128X[104X
    [4X[25Xgap>[125X [27XSimpleLieAlgebra( "H", [1,2], GF(5) );[127X[104X
    [4X[28X<Lie algebra of dimension 123 over GF(5)>[128X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "M", [1,1], GF(5) );[127X[104X
    [4X[28X<Lie algebra of dimension 125 over GF(5)>[128X[104X
  [4X[32X[104X
  
  
  [1X64.3 [33X[0;0YDistinguished Subalgebras[133X[101X
  
  [33X[0;0YHere  we describe functions that calculate well-known subalgebras and ideals
  of  a  Lie  algebra  (such  as  the centre, the centralizer of a subalgebra,
  etc.).[133X
  
  [1X64.3-1 LieCentre[101X
  
  [29X[2XLieCentre[102X( [3XL[103X ) [32X attribute
  [29X[2XLieCenter[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YThe  [13XLie[113X  centre  of the Lie algebra [3XL[103X is the kernel of the adjoint mapping,
  that is, the set [22X{ a ∈ L : ∀ x ∈ L: a x = 0 }[122X.[133X
  
  [33X[0;0YIn  characteristic  [22X2[122X this may differ from the usual centre (that is the set
  of  all  [22Xa ∈ L[122X such that [22Xa x = x a[122X for all [22Xx ∈ L[122X). Therefore, this operation
  is named [2XLieCentre[102X and not [2XCentre[102X ([14X35.4-5[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FullMatrixLieAlgebra( GF(3), 3 );[127X[104X
    [4X[28X<Lie algebra over GF(3), with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XLieCentre( L );[127X[104X
    [4X[28X<two-sided ideal in <Lie algebra of dimension 9 over GF(3)>, [128X[104X
    [4X[28X  (dimension 1)>[128X[104X
  [4X[32X[104X
  
  [1X64.3-2 LieCentralizer[101X
  
  [29X[2XLieCentralizer[102X( [3XL[103X, [3XS[103X ) [32X operation
  
  [33X[0;0Yis the annihilator of [3XS[103X in the Lie algebra [3XL[103X, that is, the set [22X{ a ∈ L : ∀ s
  ∈ S: a*s = 0 }[122X. Here [3XS[103X may be a subspace or a subalgebra of [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "G", 2, Rationals );[127X[104X
    [4X[28X<Lie algebra of dimension 14 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27Xb:= BasisVectors( Basis( L ) );;[127X[104X
    [4X[25Xgap>[125X [27XLieCentralizer( L, Subalgebra( L, [ b[1], b[2] ] ) );[127X[104X
    [4X[28X<Lie algebra of dimension 1 over Rationals>[128X[104X
  [4X[32X[104X
  
  [1X64.3-3 LieNormalizer[101X
  
  [29X[2XLieNormalizer[102X( [3XL[103X, [3XU[103X ) [32X operation
  
  [33X[0;0Yis  the  normalizer of the subspace [3XU[103X in the Lie algebra [3XL[103X, that is, the set
  [22XN_L(U) = { x ∈ L : [x,U] ⊂ U }[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "G", 2, Rationals );[127X[104X
    [4X[28X<Lie algebra of dimension 14 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27Xb:= BasisVectors( Basis( L ) );;[127X[104X
    [4X[25Xgap>[125X [27XLieNormalizer( L, Subalgebra( L, [ b[1], b[2] ] ) );[127X[104X
    [4X[28X<Lie algebra of dimension 8 over Rationals>[128X[104X
  [4X[32X[104X
  
  [1X64.3-4 LieDerivedSubalgebra[101X
  
  [29X[2XLieDerivedSubalgebra[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0Yis the (Lie) derived subalgebra of the Lie algebra [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X L:= FullMatrixLieAlgebra( GF( 3 ), 3 );[127X[104X
    [4X[28X<Lie algebra over GF(3), with 5 generators>[128X[104X
    [4X[25Xgap>[125X [27XLieDerivedSubalgebra( L );[127X[104X
    [4X[28X<Lie algebra of dimension 8 over GF(3)>[128X[104X
  [4X[32X[104X
  
  [1X64.3-5 LieNilRadical[101X
  
  [29X[2XLieNilRadical[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YThis function calculates the (Lie) nil radical of the Lie algebra [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmats:= [ [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[0,1]] ];;[127X[104X
    [4X[25Xgap>[125X [27XL:= LieAlgebra( Rationals, mats );;[127X[104X
    [4X[25Xgap>[125X [27XLieNilRadical( L );[127X[104X
    [4X[28X<two-sided ideal in <Lie algebra of dimension 3 over Rationals>, [128X[104X
    [4X[28X  (dimension 2)>[128X[104X
  [4X[32X[104X
  
  [1X64.3-6 LieSolvableRadical[101X
  
  [29X[2XLieSolvableRadical[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YReturns the (Lie) solvable radical of the Lie algebra [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FullMatrixLieAlgebra( Rationals, 3 );;[127X[104X
    [4X[25Xgap>[125X [27XLieSolvableRadical( L );[127X[104X
    [4X[28X<two-sided ideal in <Lie algebra of dimension 9 over Rationals>, [128X[104X
    [4X[28X  (dimension 1)>[128X[104X
  [4X[32X[104X
  
  [1X64.3-7 CartanSubalgebra[101X
  
  [29X[2XCartanSubalgebra[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YA  Cartan subalgebra of a Lie algebra [3XL[103X is defined as a nilpotent subalgebra
  of [3XL[103X equal to its own Lie normalizer in [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "G", 2, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XCartanSubalgebra( L );[127X[104X
    [4X[28X<Lie algebra of dimension 2 over Rationals>[128X[104X
  [4X[32X[104X
  
  
  [1X64.4 [33X[0;0YSeries of Ideals[133X[101X
  
  [1X64.4-1 LieDerivedSeries[101X
  
  [29X[2XLieDerivedSeries[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0Yis the (Lie) derived series of the Lie algebra [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmats:= [ [[1,0],[0,0]], [[0,1],[0,0]], [[0,0],[0,1]] ];;[127X[104X
    [4X[25Xgap>[125X [27XL:= LieAlgebra( Rationals, mats );;[127X[104X
    [4X[25Xgap>[125X [27XLieDerivedSeries( L );[127X[104X
    [4X[28X[ <Lie algebra of dimension 3 over Rationals>, [128X[104X
    [4X[28X  <Lie algebra of dimension 1 over Rationals>, [128X[104X
    [4X[28X  <Lie algebra of dimension 0 over Rationals> ][128X[104X
  [4X[32X[104X
  
  [1X64.4-2 LieLowerCentralSeries[101X
  
  [29X[2XLieLowerCentralSeries[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0Yis the (Lie) lower central series of the Lie algebra [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmats:= [ [[ 1, 0 ], [ 0, 0 ]], [[0,1],[0,0]], [[0,0],[0,1]] ];;[127X[104X
    [4X[25Xgap>[125X [27XL:=LieAlgebra( Rationals, mats );;[127X[104X
    [4X[25Xgap>[125X [27XLieLowerCentralSeries( L );[127X[104X
    [4X[28X[ <Lie algebra of dimension 3 over Rationals>, [128X[104X
    [4X[28X  <Lie algebra of dimension 1 over Rationals> ][128X[104X
  [4X[32X[104X
  
  [1X64.4-3 LieUpperCentralSeries[101X
  
  [29X[2XLieUpperCentralSeries[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0Yis the (Lie) upper central series of the Lie algebra [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xmats:= [ [[ 1, 0 ], [ 0, 0 ]], [[0,1],[0,0]], [[0,0],[0,1]] ];;[127X[104X
    [4X[25Xgap>[125X [27XL:=LieAlgebra( Rationals, mats );;[127X[104X
    [4X[25Xgap>[125X [27XLieUpperCentralSeries( L );[127X[104X
    [4X[28X[ <two-sided ideal in <Lie algebra of dimension 3 over Rationals>, [128X[104X
    [4X[28X      (dimension 1)>, <Lie algebra over Rationals, with 0 generators> [128X[104X
    [4X[28X ][128X[104X
  [4X[32X[104X
  
  
  [1X64.5 [33X[0;0YProperties of a Lie Algebra[133X[101X
  
  [1X64.5-1 IsLieAbelian[101X
  
  [29X[2XIsLieAbelian[102X( [3XL[103X ) [32X property
  
  [33X[0;0Yreturns  [9Xtrue[109X  if [3XL[103X is a Lie algebra such that each product of elements in [3XL[103X
  is zero, and [9Xfalse[109X otherwise.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X T:= EmptySCTable( 5, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27X L:= LieAlgebraByStructureConstants( Rationals, T );[127X[104X
    [4X[28X<Lie algebra of dimension 5 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XIsLieAbelian( L );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X64.5-2 IsLieNilpotent[101X
  
  [29X[2XIsLieNilpotent[102X( [3XL[103X ) [32X property
  
  [33X[0;0YA  Lie  algebra  [3XL[103X  is  defined  to  be (Lie) [13Xnilpotent[113X when its (Lie) lower
  central series reaches the trivial subalgebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 5, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XL:= LieAlgebraByStructureConstants( Rationals, T );[127X[104X
    [4X[28X<Lie algebra of dimension 5 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XIsLieNilpotent( L );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X64.5-3 IsLieSolvable[101X
  
  [29X[2XIsLieSolvable[102X( [3XL[103X ) [32X property
  
  [33X[0;0YA  Lie  algebra  [3XL[103X  is  defined  to be (Lie) [13Xsolvable[113X when its (Lie) derived
  series reaches the trivial subalgebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 5, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XL:= LieAlgebraByStructureConstants( Rationals, T );[127X[104X
    [4X[28X<Lie algebra of dimension 5 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XIsLieSolvable( L );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  
  [1X64.6 [33X[0;0YSemisimple Lie Algebras and Root Systems[133X[101X
  
  [33X[0;0YThis  section  contains  some  functions  for  dealing  with  semisimple Lie
  algebras and their root systems.[133X
  
  [1X64.6-1 SemiSimpleType[101X
  
  [29X[2XSemiSimpleType[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YLet  [3XL[103X  be  a  semisimple  Lie  algebra,  i.e.,  a  direct sum of simple Lie
  algebras.  Then  [2XSemiSimpleType[102X  returns  the  type  of  [3XL[103X,  i.e.,  a string
  containing the types of the simple summands of [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "E", 8, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27Xb:= BasisVectors( Basis( L ) );;[127X[104X
    [4X[25Xgap>[125X [27XK:= LieCentralizer(L, Subalgebra(L, [ b[61]+b[79]+b[101]+b[102] ]));[127X[104X
    [4X[28X<Lie algebra of dimension 102 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27Xlev:= LeviMalcevDecomposition(K);;[127X[104X
    [4X[25Xgap>[125X [27XSemiSimpleType( lev[1] );[127X[104X
    [4X[28X"B3 A1"[128X[104X
  [4X[32X[104X
  
  [1X64.6-2 ChevalleyBasis[101X
  
  [29X[2XChevalleyBasis[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YHere [3XL[103X must be a semisimple Lie algebra with a split Cartan subalgebra. Then
  [10XChevalleyBasis([3XL[103X[10X)[110X  returns  a  list  consisting  of three sublists. Together
  these  sublists  form  a  Chevalley  basis of [3XL[103X. The first list contains the
  positive  root  vectors, the second list contains the negative root vectors,
  and the third list the Cartan elements of the Chevalley basis.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "G", 2, Rationals );[127X[104X
    [4X[28X<Lie algebra of dimension 14 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XChevalleyBasis( L );[127X[104X
    [4X[28X[ [ v.1, v.2, v.3, v.4, v.5, v.6 ], [128X[104X
    [4X[28X  [ v.7, v.8, v.9, v.10, v.11, v.12 ], [ v.13, v.14 ] ][128X[104X
  [4X[32X[104X
  
  [1X64.6-3 IsRootSystem[101X
  
  [29X[2XIsRootSystem[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YCategory of root systems.[133X
  
  [1X64.6-4 IsRootSystemFromLieAlgebra[101X
  
  [29X[2XIsRootSystemFromLieAlgebra[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YCategory  of  root  systems  that  come from (semisimple) Lie algebras. They
  often   have  special  attributes  such  as  [2XUnderlyingLieAlgebra[102X  ([14X64.6-6[114X),
  [2XPositiveRootVectors[102X       ([14X64.6-9[114X),      [2XNegativeRootVectors[102X      ([14X64.6-10[114X),
  [2XCanonicalGenerators[102X ([14X64.6-14[114X).[133X
  
  [1X64.6-5 RootSystem[101X
  
  [29X[2XRootSystem[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0Y[2XRootSystem[102X calculates the root system of the semisimple Lie algebra [3XL[103X with a
  split Cartan subalgebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "G", 2, Rationals );[127X[104X
    [4X[28X<Lie algebra of dimension 14 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XR:= RootSystem( L );[127X[104X
    [4X[28X<root system of rank 2>[128X[104X
    [4X[25Xgap>[125X [27XIsRootSystem( R );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsRootSystemFromLieAlgebra( R );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X64.6-6 UnderlyingLieAlgebra[101X
  
  [29X[2XUnderlyingLieAlgebra[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0YFor  a root system [3XR[103X coming from a semisimple Lie algebra [10XL[110X, returns the Lie
  algebra [10XL[110X.[133X
  
  [1X64.6-7 PositiveRoots[101X
  
  [29X[2XPositiveRoots[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0YThe list of positive roots of the root system [3XR[103X.[133X
  
  [1X64.6-8 NegativeRoots[101X
  
  [29X[2XNegativeRoots[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0YThe list of negative roots of the root system [3XR[103X.[133X
  
  [1X64.6-9 PositiveRootVectors[101X
  
  [29X[2XPositiveRootVectors[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0YA  list  of positive root vectors of the root system [3XR[103X that comes from a Lie
  algebra  [10XL[110X.  This  is  a  list in bijection with the list [10XPositiveRoots( L )[110X
  (see [2XPositiveRoots[102X  ([14X64.6-7[114X)).  The root vector is a non-zero element of the
  root space (in [10XL[110X) of the corresponding root.[133X
  
  [1X64.6-10 NegativeRootVectors[101X
  
  [29X[2XNegativeRootVectors[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0YA  list  of negative root vectors of the root system [3XR[103X that comes from a Lie
  algebra  [10XL[110X.  This  is  a  list in bijection with the list [10XNegativeRoots( L )[110X
  (see [2XNegativeRoots[102X  ([14X64.6-8[114X)).  The root vector is a non-zero element of the
  root space (in [10XL[110X) of the corresponding root.[133X
  
  [1X64.6-11 SimpleSystem[101X
  
  [29X[2XSimpleSystem[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0YA list of simple roots of the root system [3XR[103X.[133X
  
  [1X64.6-12 CartanMatrix[101X
  
  [29X[2XCartanMatrix[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0YThe  Cartan  matrix  of  the  root system [3XR[103X, relative to the simple roots in
  [10XSimpleSystem( [3XR[103X[10X )[110X (see [2XSimpleSystem[102X ([14X64.6-11[114X)).[133X
  
  [1X64.6-13 BilinearFormMat[101X
  
  [29X[2XBilinearFormMat[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0YThe  matrix  of  the  bilinear  form of the root system [3XR[103X. If we denote this
  matrix  by [22XB[122X, then we have [22XB(i,j) = (α_i, α_j)[122X, where the [22Xα_i[122X are the simple
  roots of [3XR[103X.[133X
  
  [1X64.6-14 CanonicalGenerators[101X
  
  [29X[2XCanonicalGenerators[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0YHere  [3XR[103X  must  be a root system coming from a semisimple Lie algebra [10XL[110X. This
  function returns [22X3l[122X generators of [3XL[103X, [22Xx_1, ..., x_l, y_1, ..., y_l, h_1, ...,
  h_l[122X,  where [22Xx_i[122X lies in the root space corresponding to the [22Xi[122X-th simple root
  of  the  root system of [3XL[103X, [22Xy_i[122X lies in the root space corresponding to [22X-[122X the
  [22Xi[122X-th  simple  root, and the [22Xh_i[122X are elements of the Cartan subalgebra. These
  elements  satisfy  the  relations [22Xh_i * h_j = 0[122X, [22Xx_i * y_j = δ_ij h_i[122X, [22Xh_j *
  x_i = c_ij x_i[122X, [22Xh_j * y_i = -c_ij y_i[122X, where [22Xc_ij[122X is the entry of the Cartan
  matrix on position [22Xij[122X.[133X
  
  [33X[0;0YAlso  if  [22Xa[122X is a root of the root system [3XR[103X (so [22Xa[122X is a list of numbers), then
  we  have  the  relation  [22Xh_i  *  x  =  a[i]  x[122X,  where  [22Xx[122X  is  a root vector
  corresponding to [22Xa[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "G", 2, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XR:= RootSystem( L );;[127X[104X
    [4X[25Xgap>[125X [27XUnderlyingLieAlgebra( R );[127X[104X
    [4X[28X<Lie algebra of dimension 14 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XPositiveRoots( R );[127X[104X
    [4X[28X[ [ 2, -1 ], [ -3, 2 ], [ -1, 1 ], [ 1, 0 ], [ 3, -1 ], [ 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xx:= PositiveRootVectors( R );[127X[104X
    [4X[28X[ v.1, v.2, v.3, v.4, v.5, v.6 ][128X[104X
    [4X[25Xgap>[125X [27Xg:=CanonicalGenerators( R );[127X[104X
    [4X[28X[ [ v.1, v.2 ], [ v.7, v.8 ], [ v.13, v.14 ] ][128X[104X
    [4X[25Xgap>[125X [27Xg[3][1]*x[1];[127X[104X
    [4X[28X(2)*v.1[128X[104X
    [4X[25Xgap>[125X [27Xg[3][2]*x[1];[127X[104X
    [4X[28X(-1)*v.1[128X[104X
    [4X[25Xgap>[125X [27X# i.e., x[1] is the root vector belonging to the root [ 2, -1 ][127X[104X
    [4X[25Xgap>[125X [27XBilinearFormMat( R );[127X[104X
    [4X[28X[ [ 1/12, -1/8 ], [ -1/8, 1/4 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X64.7 [33X[0;0YSemisimple Lie Algebras and Weyl Groups of Root Systems[133X[101X
  
  [33X[0;0YThis  section  deals  with  the Weyl group of a root system. A Weyl group is
  represented by its action on the weight lattice. A [13Xweight[113X is by definition a
  linear function [22Xλ: H → F[122X (where [22XF[122X is the ground field), such that the values
  [22Xλ(h_i)[122X  are  all  integers  (where  the  [22Xh_i[122X  are the Cartan elements of the
  [2XCanonicalGenerators[102X  ([14X64.6-14[114X)). On the other hand each weight is determined
  by  these  values.  Therefore we represent a weight by a vector of integers;
  the  [22Xi[122X-th  entry of this vector is the value [22Xλ(h_i)[122X. Now the elements of the
  Weyl  group  are  represented  by matrices, and if [10Xg[110X is an element of a Weyl
  group  and [10Xw[110X a weight, then [10Xw*g[110X gives the result of applying [10Xg[110X to [10Xw[110X. Another
  way  of  applying  the  [22Xi[122X-th  simple  reflection to a weight is by using the
  function [2XApplySimpleReflection[102X ([14X64.7-4[114X).[133X
  
  [33X[0;0YA  Weyl  group  is generated by the simple reflections. So [2XGeneratorsOfGroup[102X
  ([14X39.2-4[114X)  for  a Weyl group [10XW[110X gives a list of matrices and the [22Xi[122X-th entry of
  this  list is the simple reflection corresponding to the [22Xi[122X-th simple root of
  the corresponding root system.[133X
  
  [1X64.7-1 IsWeylGroup[101X
  
  [29X[2XIsWeylGroup[102X( [3XG[103X ) [32X property
  
  [33X[0;0YA  Weyl  group  is  a  group  generated  by  reflections, with the attribute
  [2XSparseCartanMatrix[102X ([14X64.7-2[114X) set.[133X
  
  [1X64.7-2 SparseCartanMatrix[101X
  
  [29X[2XSparseCartanMatrix[102X( [3XW[103X ) [32X attribute
  
  [33X[0;0YThis is a sparse form of the Cartan matrix of the corresponding root system.
  If we denote the Cartan matrix by [10XC[110X, then the sparse Cartan matrix of [3XW[103X is a
  list  (of  length  equal to the length of the Cartan matrix), where the [10Xi[110X-th
  entry  is a list consisting of elements [10X[ j, C[i][j] ][110X, where [10Xj[110X is such that
  [10XC[i][j][110X is non-zero.[133X
  
  [1X64.7-3 WeylGroup[101X
  
  [29X[2XWeylGroup[102X( [3XR[103X ) [32X attribute
  
  [33X[0;0YThe  Weyl  group  of  the  root  system  [3XR[103X.  It  is  generated by the simple
  reflections.  A simple reflection is represented by a matrix, and the result
  of letting a simple reflection [10Xm[110X act on a weight [10Xw[110X is obtained by [10Xw*m[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "F", 4, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XR:= RootSystem( L );;[127X[104X
    [4X[25Xgap>[125X [27XW:= WeylGroup( R );[127X[104X
    [4X[28X<matrix group with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XIsWeylGroup( W );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XSparseCartanMatrix( W );[127X[104X
    [4X[28X[ [ [ 1, 2 ], [ 3, -1 ] ], [ [ 2, 2 ], [ 4, -1 ] ], [128X[104X
    [4X[28X  [ [ 1, -1 ], [ 3, 2 ], [ 4, -1 ] ], [128X[104X
    [4X[28X  [ [ 2, -1 ], [ 3, -2 ], [ 4, 2 ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xg:= GeneratorsOfGroup( W );;[127X[104X
    [4X[25Xgap>[125X [27X[ 1, 1, 1, 1 ]*g[2];[127X[104X
    [4X[28X[ 1, -1, 1, 2 ][128X[104X
  [4X[32X[104X
  
  [1X64.7-4 ApplySimpleReflection[101X
  
  [29X[2XApplySimpleReflection[102X( [3XSC[103X, [3Xi[103X, [3Xwt[103X ) [32X operation
  
  [33X[0;0YHere  [3XSC[103X  is the sparse Cartan matrix of a Weyl group. This function applies
  the [3Xi[103X-th simple reflection to the weight [3Xwt[103X, thus changing [3Xwt[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "F", 4, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XW:= WeylGroup( RootSystem( L ) );;[127X[104X
    [4X[25Xgap>[125X [27XC:= SparseCartanMatrix( W );;[127X[104X
    [4X[25Xgap>[125X [27Xw:= [ 1, 1, 1, 1 ];;[127X[104X
    [4X[25Xgap>[125X [27XApplySimpleReflection( C, 2, w );[127X[104X
    [4X[25Xgap>[125X [27Xw;[127X[104X
    [4X[28X[ 1, -1, 1, 2 ][128X[104X
  [4X[32X[104X
  
  [1X64.7-5 LongestWeylWordPerm[101X
  
  [29X[2XLongestWeylWordPerm[102X( [3XW[103X ) [32X attribute
  
  [33X[0;0YLet  [22Xg_0[122X be the longest element in the Weyl group [3XW[103X, and let [22X{ α_1, ..., α_l
  }[122X  be a simple system of the corresponding root system. Then [22Xg_0[122X maps [22Xα_i[122X to
  [22X-α_{σ(i)}[122X,  where  [22Xσ[122X  is a permutation of [22X(1, ..., l)[122X. This function returns
  that permutation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "E", 6, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XW:= WeylGroup( RootSystem( L ) );;[127X[104X
    [4X[25Xgap>[125X [27XLongestWeylWordPerm( W );[127X[104X
    [4X[28X(1,6)(3,5)[128X[104X
  [4X[32X[104X
  
  [1X64.7-6 ConjugateDominantWeight[101X
  
  [29X[2XConjugateDominantWeight[102X( [3XW[103X, [3Xwt[103X ) [32X operation
  [29X[2XConjugateDominantWeightWithWord[102X( [3XW[103X, [3Xwt[103X ) [32X operation
  
  [33X[0;0YHere  [3XW[103X  is  a  Weyl  group  and  [3Xwt[103X  a  weight  (i.e., a list of integers).
  [2XConjugateDominantWeight[102X  returns  the unique dominant weight conjugate to [3Xwt[103X
  under [3XW[103X.[133X
  
  [33X[0;0Y[2XConjugateDominantWeightWithWord[102X returns a list of two elements. The first of
  these  is  the dominant weight conjugate to [3Xwt[103X. The second element is a list
  of  indices  of simple reflections that have to be applied to [3Xwt[103X in order to
  get the dominant weight conjugate to it.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "E", 6, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XW:= WeylGroup( RootSystem( L ) );;[127X[104X
    [4X[25Xgap>[125X [27XC:= SparseCartanMatrix( W );;[127X[104X
    [4X[25Xgap>[125X [27Xw:= [ 1, -1, 2, -2, 3, -3 ];;[127X[104X
    [4X[25Xgap>[125X [27XConjugateDominantWeight( W, w );[127X[104X
    [4X[28X[ 2, 1, 0, 0, 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27Xc:= ConjugateDominantWeightWithWord( W, w );[127X[104X
    [4X[28X[ [ 2, 1, 0, 0, 0, 0 ], [ 2, 4, 2, 3, 6, 5, 4, 2, 3, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27Xfor i in [1..Length(c[2])] do[127X[104X
    [4X[25X>[125X [27XApplySimpleReflection( C, c[2][i], w );[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[25Xgap>[125X [27Xw;[127X[104X
    [4X[28X[ 2, 1, 0, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  [1X64.7-7 WeylOrbitIterator[101X
  
  [29X[2XWeylOrbitIterator[102X( [3XW[103X, [3Xwt[103X ) [32X operation
  
  [33X[0;0YReturns  an  iterator for the orbit of the weight [3Xwt[103X under the action of the
  Weyl group [3XW[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "E", 6, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XW:= WeylGroup( RootSystem( L ) );;[127X[104X
    [4X[25Xgap>[125X [27Xorb:= WeylOrbitIterator( W, [ 1, 1, 1, 1, 1, 1 ] );[127X[104X
    [4X[28X<iterator>[128X[104X
    [4X[25Xgap>[125X [27XNextIterator( orb );[127X[104X
    [4X[28X[ 1, 1, 1, 1, 1, 1 ][128X[104X
    [4X[25Xgap>[125X [27XNextIterator( orb );[127X[104X
    [4X[28X[ -1, -1, -1, -1, -1, -1 ][128X[104X
    [4X[25Xgap>[125X [27Xorb:= WeylOrbitIterator( W, [ 1, 1, 1, 1, 1, 1 ] );[127X[104X
    [4X[28X<iterator>[128X[104X
    [4X[25Xgap>[125X [27Xk:= 0;[127X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27Xwhile not IsDoneIterator( orb ) do[127X[104X
    [4X[25X>[125X [27Xw:= NextIterator( orb ); k:= k+1;[127X[104X
    [4X[25X>[125X [27Xod;[127X[104X
    [4X[25Xgap>[125X [27Xk;  # this is the size of the Weyl group of E6[127X[104X
    [4X[28X51840[128X[104X
  [4X[32X[104X
  
  
  [1X64.8 [33X[0;0YRestricted Lie algebras[133X[101X
  
  [33X[0;0YA  Lie  algebra [22XL[122X over a field of characteristic [22Xp>0[122X is called restricted if
  there  is  a  map  [22Xx ↦ x^p[122X from [22XL[122X into [22XL[122X (called a [22Xp[122X-map) such that ad [22Xx^p =
  ([122Xad[22Xx)^p[122X, [22X(α x)^p = α^p x^p[122X and [22X(x+y)^p = x^p + y^p + ∑_{i=1}^{p-1} s_i(x,y)[122X,
  where  [22Xs_i:  L  ×  L → L[122X are certain Lie polynomials in two variables. Using
  these  relations  we can calculate [22Xy^p[122X for all [22Xy ∈ L[122X, once we know [22Xx^p[122X for [22Xx[122X
  in  a  basis  of  [22XL[122X.  Therefore  a  [22Xp[122X-map  is  represented in [5XGAP[105X  by a list
  containing  the  images  of  the  basis  vectors of a basis [22XB[122X of [22XL[122X. For this
  reason this list is an attribute of the basis [22XB[122X.[133X
  
  [1X64.8-1 IsRestrictedLieAlgebra[101X
  
  [29X[2XIsRestrictedLieAlgebra[102X( [3XL[103X ) [32X property
  
  [33X[0;0YTest whether [3XL[103X is restricted.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "W", [2], GF(5));[127X[104X
    [4X[28X<Lie algebra of dimension 25 over GF(5)>[128X[104X
    [4X[25Xgap>[125X [27XIsRestrictedLieAlgebra( L );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "W", [1], GF(5));[127X[104X
    [4X[28X<Lie algebra of dimension 5 over GF(5)>[128X[104X
    [4X[25Xgap>[125X [27XIsRestrictedLieAlgebra( L );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X64.8-2 PthPowerImages[101X
  
  [29X[2XPthPowerImages[102X( [3XB[103X ) [32X attribute
  
  [33X[0;0YHere  [3XB[103X  is  a  basis of a restricted Lie algebra. This function returns the
  list of the images of the basis vectors of [3XB[103X under the [22Xp[122X-map.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "W", [1], GF(11) );[127X[104X
    [4X[28X<Lie algebra of dimension 11 over GF(11)>[128X[104X
    [4X[25Xgap>[125X [27XB:= Basis( L );[127X[104X
    [4X[28XCanonicalBasis( <Lie algebra of dimension 11 over GF(11)> )[128X[104X
    [4X[25Xgap>[125X [27XPthPowerImages( B );[127X[104X
    [4X[28X[ 0*v.1, v.2, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1, [128X[104X
    [4X[28X  0*v.1 ][128X[104X
  [4X[32X[104X
  
  [1X64.8-3 PthPowerImage[101X
  
  [29X[2XPthPowerImage[102X( [3XB[103X, [3Xx[103X ) [32X operation
  [29X[2XPthPowerImage[102X( [3Xx[103X ) [32X operation
  [29X[2XPthPowerImage[102X( [3Xx[103X, [3Xn[103X ) [32X operation
  
  [33X[0;0YThis function computes the image of an element [3Xx[103X of a restricted Lie algebra
  under its [22Xp[122X-map.[133X
  
  [33X[0;0YIn the first form, a basis of the Lie algebra is provided; this basis stores
  the  [22Xp[122Xth  powers  of  its elements. It is the traditional form, provided for
  backwards compatibility.[133X
  
  [33X[0;0YIn  its second form, only the element [3Xx[103X is provided. It is the only form for
  elements  of  Lie  algebras  with  no  predetermined  basis,  such  as those
  constructed by [2XLieObject[102X ([14X64.1-1[114X).[133X
  
  [33X[0;0YIn  its  third  form,  an  extra  non-negative  integer  [3Xn[103X is specified; the
  [22Xp[122X-mapping is iterated [3Xn[103X times on the element [3Xx[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "W", [1], GF(11) );;[127X[104X
    [4X[25Xgap>[125X [27XB:= Basis( L );;[127X[104X
    [4X[25Xgap>[125X [27Xx:= B[1]+B[11];[127X[104X
    [4X[28Xv.1+v.11[128X[104X
    [4X[25Xgap>[125X [27XPthPowerImage( B, x );[127X[104X
    [4X[28Xv.1+v.11[128X[104X
    [4X[25Xgap>[125X [27XPthPowerImage( x, 2 );[127X[104X
    [4X[28Xv.1+v.11[128X[104X
    [4X[25Xgap>[125X [27Xf := FreeAssociativeAlgebra(GF(2),"x","y");[127X[104X
    [4X[28X<algebra over GF(2), with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xx := LieObject(f.1);; y := LieObject(f.2);;[127X[104X
    [4X[25Xgap>[125X [27Xx*y; x^2; PthPowerImage(x);[127X[104X
    [4X[28XLieObject( (Z(2)^0)*x*y+(Z(2)^0)*y*x )[128X[104X
    [4X[28XLieObject( <zero> of ... )[128X[104X
    [4X[28XLieObject( (Z(2)^0)*x^2 )[128X[104X
  [4X[32X[104X
  
  [1X64.8-4 JenningsLieAlgebra[101X
  
  [29X[2XJenningsLieAlgebra[102X( [3XG[103X ) [32X attribute
  
  [33X[0;0YLet  [3XG[103X  be  a nontrivial [22Xp[122X-group, and let [22X[3XG[103X = G_1 ⊃ G_2 ⊃ ⋯ ⊃ G_m = 1[122X be its
  Jennings  series  (see [2XJenningsSeries[102X  ([14X39.17-14[114X)). Then the quotients [22XG_i /
  G_{i+1}[122X  are elementary abelian [22Xp[122X-groups, i.e., they can be viewed as vector
  spaces  over [10XGF[110X[22X(p)[122X. Now the Jennings-Lie algebra [22XL[122X of [3XG[103X is the direct sum of
  those vector spaces. The Lie bracket on [22XL[122X is induced by the commutator in [3XG[103X.
  Furthermore,  the  map  [22Xg  ↦  g^p[122X  in [3XG[103X induces a [22Xp[122X-map in [22XL[122X making [22XL[122X into a
  restricted  Lie  algebra. In the canonical basis of [22XL[122X this [22Xp[122X-map is added as
  an  attribute.  A  Lie  algebra  created  by [2XJenningsLieAlgebra[102X is naturally
  graded. The attribute [2XGrading[102X ([14X62.9-20[114X) is set.[133X
  
  [1X64.8-5 PCentralLieAlgebra[101X
  
  [29X[2XPCentralLieAlgebra[102X( [3XG[103X ) [32X attribute
  
  [33X[0;0YHere  [3XG[103X  is  a  nontrivial [22Xp[122X-group. [10XPCentralLieAlgebra( [3XG[103X[10X )[110X does the same as
  [2XJenningsLieAlgebra[102X ([14X64.8-4[114X) except that the [22Xp[122X-central series is used instead
  of  the  Jennings series (see [2XPCentralSeries[102X ([14X39.17-13[114X)). This function also
  returns a graded Lie algebra. However, it is not necessarily restricted.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG:= SmallGroup( 3^6, 123 );[127X[104X
    [4X[28X<pc group of size 729 with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27XL:= JenningsLieAlgebra( G );[127X[104X
    [4X[28X<Lie algebra of dimension 6 over GF(3)>[128X[104X
    [4X[25Xgap>[125X [27XHasPthPowerImages( Basis( L ) );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XPthPowerImages( Basis( L ) );[127X[104X
    [4X[28X[ v.6, 0*v.1, 0*v.1, 0*v.1, 0*v.1, 0*v.1 ][128X[104X
    [4X[25Xgap>[125X [27Xg:= Grading( L );[127X[104X
    [4X[28Xrec( hom_components := function( d ) ... end, max_degree := 3, [128X[104X
    [4X[28X  min_degree := 1, source := Integers )[128X[104X
    [4X[25Xgap>[125X [27XList( [1,2,3], g.hom_components );[127X[104X
    [4X[28X[ <vector space over GF(3), with 3 generators>, [128X[104X
    [4X[28X  <vector space over GF(3), with 2 generators>, [128X[104X
    [4X[28X  <vector space over GF(3), with 1 generators> ][128X[104X
  [4X[32X[104X
  
  [1X64.8-6 NaturalHomomorphismOfLieAlgebraFromNilpotentGroup[101X
  
  [29X[2XNaturalHomomorphismOfLieAlgebraFromNilpotentGroup[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YThis  is an attribute of Lie algebras created by [2XJenningsLieAlgebra[102X ([14X64.8-4[114X)
  or  [2XPCentralLieAlgebra[102X  ([14X64.8-5[114X).  Then  [3XL[103X is the direct sum of quotients of
  successive terms of the Jennings, or [22Xp[122X-central series of a [22Xp[122X-group G. Let [10XGi[110X
  be     the     [22Xi[122X-th    term    in    this    series,    and    let    [10Xf    =
  NaturalHomomorphismOfLieAlgebraFromNilpotentGroup( [3XL[103X[10X )[110X, then for [10Xg[110X in [10XGi[110X, [10Xf(
  [3Xg[103X[10X,  [3Xi[103X[10X  )[110X  returns the element of [3XL[103X (lying in the [22Xi[122X-th homogeneous component)
  corresponding to [10Xg[110X.[133X
  
  
  [1X64.9 [33X[0;0YThe Adjoint Representation[133X[101X
  
  [33X[0;0YIn  this  section  we  show  functions  for  calculating  with  the  adjoint
  representation  of  a  Lie algebra (and the corresponding trace form, called
  the  Killing form) (see also [2XAdjointBasis[102X ([14X62.9-5[114X) and [2XIndicesOfAdjointBasis[102X
  ([14X62.9-6[114X)).[133X
  
  [1X64.9-1 AdjointMatrix[101X
  
  [29X[2XAdjointMatrix[102X( [3XB[103X, [3Xx[103X ) [32X operation
  
  [33X[0;0Yis  the  matrix  of  the  adjoint representation of the element [3Xx[103X w.r.t. the
  basis [3XB[103X. The adjoint map is the left multiplication by [3Xx[103X. The [22Xi[122X-th column of
  the  resulting  matrix  represents  the  image of the [22Xi[122X-th basis vector of [3XB[103X
  under left multiplication by [3Xx[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "A", 1, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XAdjointMatrix( Basis( L ), Basis( L )[1] );[127X[104X
    [4X[28X[ [ 0, 0, -2 ], [ 0, 0, 0 ], [ 0, 1, 0 ] ][128X[104X
  [4X[32X[104X
  
  [1X64.9-2 AdjointAssociativeAlgebra[101X
  
  [29X[2XAdjointAssociativeAlgebra[102X( [3XL[103X, [3XK[103X ) [32X attribute
  
  [33X[0;0Yis  the associative matrix algebra (with 1) generated by the matrices of the
  adjoint representation of the subalgebra [3XK[103X on the Lie algebra [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "A", 1, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XAdjointAssociativeAlgebra( L, L );[127X[104X
    [4X[28X<algebra of dimension 9 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XAdjointAssociativeAlgebra( L, CartanSubalgebra( L ) );[127X[104X
    [4X[28X<algebra of dimension 3 over Rationals>[128X[104X
  [4X[32X[104X
  
  [1X64.9-3 KillingMatrix[101X
  
  [29X[2XKillingMatrix[102X( [3XB[103X ) [32X attribute
  
  [33X[0;0Yis  the  matrix of the Killing form [22Xκ[122X with respect to the basis [3XB[103X, i.e., the
  matrix [22X( κ( b_i, b_j ) )[122X where [22Xb_1, b_2, ...[122X are the basis vectors of [3XB[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "A", 1, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XKillingMatrix( Basis( L ) );[127X[104X
    [4X[28X[ [ 0, 4, 0 ], [ 4, 0, 0 ], [ 0, 0, 8 ] ][128X[104X
  [4X[32X[104X
  
  [1X64.9-4 KappaPerp[101X
  
  [29X[2XKappaPerp[102X( [3XL[103X, [3XU[103X ) [32X operation
  
  [33X[0;0Yis  the  orthogonal  complement  of the subspace [3XU[103X of the Lie algebra [3XL[103X with
  respect  to the Killing form [22Xκ[122X, that is, the set [22XU^{perp} = { x ∈ L; κ( x, y
  ) = 0 hbox for all y ∈ L }[122X.[133X
  
  [33X[0;0Y[22XU^{perp}[122X  is  a  subspace of [3XL[103X, and if [3XU[103X is an ideal of [3XL[103X then [22XU^{perp}[122X is a
  subalgebra of [3XL[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "A", 1, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27Xb:= BasisVectors( Basis( L ) );;[127X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [b[1],b[2]] );;[127X[104X
    [4X[25Xgap>[125X [27XKappaPerp( L, V );[127X[104X
    [4X[28X<vector space of dimension 1 over Rationals>[128X[104X
  [4X[32X[104X
  
  [1X64.9-5 IsNilpotentElement[101X
  
  [29X[2XIsNilpotentElement[102X( [3XL[103X, [3Xx[103X ) [32X operation
  
  [33X[0;0Y[3Xx[103X is nilpotent in [3XL[103X if its adjoint matrix is a nilpotent matrix.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "A", 1, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XIsNilpotentElement( L, Basis( L )[1] );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X64.9-6 NonNilpotentElement[101X
  
  [29X[2XNonNilpotentElement[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YA  non-nilpotent element of a Lie algebra [3XL[103X is an element [22Xx[122X such that ad[22Xx[122X is
  not  nilpotent. If [3XL[103X is not nilpotent, then by Engel's theorem non-nilpotent
  elements  exist  in  [3XL[103X.  In  this case this function returns a non-nilpotent
  element of [3XL[103X, otherwise (if [3XL[103X is nilpotent) [9Xfail[109X is returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "G", 2, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XNonNilpotentElement( L );[127X[104X
    [4X[28Xv.13[128X[104X
    [4X[25Xgap>[125X [27XIsNilpotentElement( L, last );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X64.9-7 FindSl2[101X
  
  [29X[2XFindSl2[102X( [3XL[103X, [3Xx[103X ) [32X operation
  
  [33X[0;0YThis  function  tries  to  find  a  subalgebra [22XS[122X of the Lie algebra [3XL[103X with [22XS[122X
  isomorphic  to  [22Xsl_2[122X and such that the nilpotent element [3Xx[103X of [3XL[103X is contained
  in  [22XS[122X.  If  such  an  algebra  exists then it is returned, otherwise [9Xfail[109X is
  returned.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "G", 2, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27Xb:= BasisVectors( Basis( L ) );;[127X[104X
    [4X[25Xgap>[125X [27XIsNilpotentElement( L, b[1] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XFindSl2( L, b[1] );[127X[104X
    [4X[28X<Lie algebra of dimension 3 over Rationals>[128X[104X
  [4X[32X[104X
  
  
  [1X64.10 [33X[0;0YUniversal Enveloping Algebras[133X[101X
  
  [1X64.10-1 UniversalEnvelopingAlgebra[101X
  
  [29X[2XUniversalEnvelopingAlgebra[102X( [3XL[103X[, [3XB[103X] ) [32X attribute
  
  [33X[0;0YReturns  the universal enveloping algebra of the Lie algebra [3XL[103X. The elements
  of this algebra are written on a Poincare-Birkhoff-Witt basis.[133X
  
  [33X[0;0YIf  a second argument [3XB[103X is given, it must be a basis of [3XL[103X, and an isomorphic
  copy  of  the  universal  enveloping  algebra  is returned, generated by the
  images (in the universal enveloping algebra) of the elements of [3XB[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "A", 1, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XUL:= UniversalEnvelopingAlgebra( L );[127X[104X
    [4X[28X<algebra-with-one of dimension infinity over Rationals>[128X[104X
    [4X[25Xgap>[125X [27Xg:= GeneratorsOfAlgebraWithOne( UL );[127X[104X
    [4X[28X[ [(1)*x.1], [(1)*x.2], [(1)*x.3] ][128X[104X
    [4X[25Xgap>[125X [27Xg[3]^2*g[2]^2*g[1]^2;[127X[104X
    [4X[28X[(-4)*x.1*x.2*x.3^3+(1)*x.1^2*x.2^2*x.3^2+(2)*x.3^3+(2)*x.3^4][128X[104X
  [4X[32X[104X
  
  
  [1X64.11 [33X[0;0YFinitely Presented Lie Algebras[133X[101X
  
  [33X[0;0YFinitely presented Lie algebras can be constructed from free Lie algebras by
  using  the [10X/[110X constructor, i.e., [10XFL/[r1, ..., rk][110X is the quotient of the free
  Lie  algebra [10XFL[110X by the ideal generated by the elements [10Xr1, ..., rk[110X of [10XFL[110X. If
  the  finitely  presented Lie algebra [10XK[110X happens to be finite dimensional then
  an  isomorphic  structure  constants  Lie  algebra  can  be  constructed  by
  [10XNiceAlgebraMonomorphism(K)[110X  (see [2XNiceAlgebraMonomorphism[102X  ([14X62.10-9[114X)),  which
  returns  a  surjective homomorphism. The structure constants Lie algebra can
  then  be  accessed  by  calling  [2XRange[102X  ([14X32.3-7[114X)  for this map. Also limited
  computations  with  elements  of  the  finitely  presented  Lie  algebra are
  possible.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieAlgebra( Rationals, "s", "t" );[127X[104X
    [4X[28X<Lie algebra over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XgL:= GeneratorsOfAlgebra( L );; s:= gL[1];; t:= gL[2];;[127X[104X
    [4X[25Xgap>[125X [27XK:= L/[ s*(s*t), t*(t*(s*t)), s*(t*(s*t))-t*(s*t) ];[127X[104X
    [4X[28X<Lie algebra over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xh:= NiceAlgebraMonomorphism( K );[127X[104X
    [4X[28X[ [(1)*s], [(1)*t] ] -> [ v.1, v.2 ][128X[104X
    [4X[25Xgap>[125X [27XU:= Range( h );[127X[104X
    [4X[28X<Lie algebra of dimension 3 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XIsLieNilpotent( U );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XgK:= GeneratorsOfAlgebra( K );[127X[104X
    [4X[28X[ [(1)*s], [(1)*t] ][128X[104X
    [4X[25Xgap>[125X [27XgK[1]*(gK[2]*gK[1]) = Zero( K );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X64.11-1 FpLieAlgebraByCartanMatrix[101X
  
  [29X[2XFpLieAlgebraByCartanMatrix[102X( [3XC[103X ) [32X function
  
  [33X[0;0YHere  [3XC[103X must be a Cartan matrix. The function returns the finitely-presented
  Lie  algebra  over  the  field  of  rational  numbers defined by this Cartan
  matrix.  By  Serre's  theorem, this Lie algebra is a semisimple Lie algebra,
  and its root system has Cartan matrix [3XC[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XC:= [ [ 2, -1 ], [ -3, 2 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XK:= FpLieAlgebraByCartanMatrix( C );[127X[104X
    [4X[28X<Lie algebra over Rationals, with 6 generators>[128X[104X
    [4X[25Xgap>[125X [27Xh:= NiceAlgebraMonomorphism( K );[127X[104X
    [4X[28X[ [(1)*x1], [(1)*x2], [(1)*x3], [(1)*x4], [(1)*x5], [(1)*x6] ] -> [128X[104X
    [4X[28X[ v.1, v.2, v.3, v.4, v.5, v.6 ][128X[104X
    [4X[25Xgap>[125X [27XSemiSimpleType( Range( h ) );[127X[104X
    [4X[28X"G2"[128X[104X
  [4X[32X[104X
  
  [1X64.11-2 NilpotentQuotientOfFpLieAlgebra[101X
  
  [29X[2XNilpotentQuotientOfFpLieAlgebra[102X( [3XFpL[103X, [3Xmax[103X[, [3Xweights[103X] ) [32X function
  
  [33X[0;0YHere  [3XFpL[103X  is a finitely presented Lie algebra. Let [22XK[122X be the quotient of [3XFpL[103X
  by the [3Xmax[103X+1-th term of its lower central series. This function calculates a
  surjective homomorphism from [3XFpL[103X onto [22XK[122X. When called with the third argument
  [3Xweights[103X,  the  [22Xk[122X-th  generator  of [3XFpL[103X gets assigned the [22Xk[122X-th element of the
  list  [3Xweights[103X.  In  that  case  a quotient is calculated of [3XFpL[103X by the ideal
  generated by all elements of weight [3Xmax[103X+1. If the list [3Xweights[103X only consists
  of  [22X1[122X's then the two calls are equivalent. The default value of [3Xweights[103X is a
  list (of length equal to the number of generators of [3XFpL[103X) consisting of [22X1[122X's.[133X
  
  [33X[0;0YIf  the  relators  of  [3XFpL[103X  are  homogeneous,  then the resulting algebra is
  naturally graded.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= FreeLieAlgebra( Rationals, "x", "y" );;[127X[104X
    [4X[25Xgap>[125X [27Xg:= GeneratorsOfAlgebra(L);; x:= g[1]; y:= g[2];[127X[104X
    [4X[28X(1)*x[128X[104X
    [4X[28X(1)*y[128X[104X
    [4X[25Xgap>[125X [27Xrr:=[ ((y*x)*x)*x-6*(y*x)*y, [127X[104X
    [4X[25X>[125X [27X         3*((((y*x)*x)*x)*x)*x-20*(((y*x)*x)*x)*y ];[127X[104X
    [4X[28X[ (-1)*(x*(x*(x*y)))+(6)*((x*y)*y), [128X[104X
    [4X[28X  (-3)*(x*(x*(x*(x*(x*y)))))+(20)*(x*(x*((x*y)*y)))+([128X[104X
    [4X[28X    -20)*((x*(x*y))*(x*y)) ][128X[104X
    [4X[25Xgap>[125X [27XK:= L/rr;[127X[104X
    [4X[28X<Lie algebra over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xh:=NilpotentQuotientOfFpLieAlgebra(K, 50, [1,2] );[127X[104X
    [4X[28X[ [(1)*x], [(1)*y] ] -> [ v.1, v.2 ][128X[104X
    [4X[25Xgap>[125X [27XL:= Range( h );[127X[104X
    [4X[28X<Lie algebra of dimension 50 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XGrading( L );[127X[104X
    [4X[28Xrec( hom_components := function( d ) ... end, max_degree := 50, [128X[104X
    [4X[28X  min_degree := 1, source := Integers )[128X[104X
  [4X[32X[104X
  
  
  [1X64.12 [33X[0;0YModules over Lie Algebras and Their Cohomology[133X[101X
  
  [33X[0;0YRepresentations  of  Lie  algebras  are  dealt  with  in  the  same  way  as
  representations  of ordinary algebras (see [14X62.11[114X). In this section we mainly
  deal  with  modules over general Lie algebras and their cohomology. The next
  section  is devoted to modules over semisimple Lie algebras. An [22Xs[122X-cochain of
  a module [22XV[122X over a Lie algebra [22XL[122X is an [22Xs[122X-linear map[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yc: L × ⋯ × L → V ,[133X[124X[133X
  
  
  [33X[0;0Ywith  [22Xs[122X  factors  [22XL[122X,  that  is  skew-symmetric  (meaning  that if any of the
  arguments are interchanged, [22Xc[122X changes to [22X-c[122X).[133X
  
  [33X[0;0YLet [22X(x_1, ..., x_n)[122X be a basis of [22XL[122X. Then any [22Xs[122X-cochain is determined by the
  values  [22Xc(  x_{i_1},  ..., x_{i_s} )[122X, where [22X1 ≤ i_1 < i_2 < ⋯ < i_s ≤ dim L[122X.
  Now  this  value  again  is  a linear combination of basis elements of [22XV[122X: [22Xc(
  x_{i_1},  ..., x_{i_s} ) = ∑ λ^k_{i_1,..., i_s} v_k[122X. Denote the dimension of
  [22XV[122X  by  [22Xr[122X.  Then  we represent an [22Xs[122X-cocycle by a list of [22Xr[122X lists. The [22Xj[122X-th of
  those lists consists of entries of the form[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y[ [ i_1, i_2, ..., i_s ], λ^j_{i_1, ..., i_s} ][133X[124X[133X
  
  
  [33X[0;0Ywhere  the  coefficient  on  the second position is non-zero. (We only store
  those entries for which this coefficient is non-zero.) It follows that every
  [22Xs[122X-tuple [22X(i_1, ..., i_s)[122X gives rise to [22Xr[122X basis elements.[133X
  
  [33X[0;0YSo the zero cochain is represented by a list of the form [10X[ [ ], [ ], \ldots,
  [  ]  ][110X.  Furthermore,  if  [22XV[122X  is,  e.g.,  [22X4[122X-dimensional, then the [22X2[122X-cochain
  represented by[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28X[ [ [ [1,2], 2] ], [ ], [ [ [1,2], 1/2 ] ], [ ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0Ymaps  the  pair  [22X(x_1,  x_2)[122X to [22X2v_1 + 1/2 v_3[122X (where [22Xv_1[122X is the first basis
  element of [22XV[122X, and [22Xv_3[122X the third), and all other pairs to zero.[133X
  
  [33X[0;0YBy  definition, [22X0[122X-cochains are constant maps [22Xc( x ) = v_c ∈ V[122X for all [22Xx ∈ L[122X.
  So  [22X0[122X-cochains have a different representation: they are just represented by
  the list [10X[ v_c ][110X.[133X
  
  [33X[0;0YCochains  are  constructed  using  the function [2XCochain[102X ([14X64.12-2[114X), if [3Xc[103X is a
  cochain, then its corresponding list is returned by [10XExtRepOfObj( [3Xc[103X[10X )[110X.[133X
  
  [1X64.12-1 IsCochain[101X
  
  [29X[2XIsCochain[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsCochainCollection[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YCategories of cochains and of collections of cochains.[133X
  
  [1X64.12-2 Cochain[101X
  
  [29X[2XCochain[102X( [3XV[103X, [3Xs[103X, [3Xobj[103X ) [32X operation
  
  [33X[0;0YConstructs  a  [3Xs[103X-cochain  given  by the data in [3Xobj[103X, with respect to the Lie
  algebra module [3XV[103X. If [3Xs[103X is non-zero, then [3Xobj[103X must be a list.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "A", 1, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XV:= AdjointModule( L );[127X[104X
    [4X[28X<3-dimensional left-module over <Lie algebra of dimension [128X[104X
    [4X[28X3 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27Xc1:= Cochain( V, 2, [127X[104X
    [4X[25X>[125X [27X              [ [ [ [ 1, 3 ], -1 ] ], [ ], [ [ [ 2, 3 ], 1/2 ] ] ]);[127X[104X
    [4X[28X<2-cochain>[128X[104X
    [4X[25Xgap>[125X [27XExtRepOfObj( c1 );[127X[104X
    [4X[28X[ [ [ [ 1, 3 ], -1 ] ], [  ], [ [ [ 2, 3 ], 1/2 ] ] ][128X[104X
    [4X[25Xgap>[125X [27Xc2:= Cochain( V, 0, Basis( V )[1] );[127X[104X
    [4X[28X<0-cochain>[128X[104X
    [4X[25Xgap>[125X [27XExtRepOfObj( c2 );[127X[104X
    [4X[28Xv.1[128X[104X
    [4X[25Xgap>[125X [27XIsCochain( c2 );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X64.12-3 CochainSpace[101X
  
  [29X[2XCochainSpace[102X( [3XV[103X, [3Xs[103X ) [32X operation
  
  [33X[0;0YReturns the space of all [3Xs[103X-cochains with respect to [3XV[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "A", 1, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XV:= AdjointModule( L );;[127X[104X
    [4X[25Xgap>[125X [27XC:=CochainSpace( V, 2 );[127X[104X
    [4X[28X<vector space of dimension 9 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors( Basis( C ) );[127X[104X
    [4X[28X[ <2-cochain>, <2-cochain>, <2-cochain>, <2-cochain>, <2-cochain>, [128X[104X
    [4X[28X  <2-cochain>, <2-cochain>, <2-cochain>, <2-cochain> ][128X[104X
    [4X[25Xgap>[125X [27XExtRepOfObj( last[1] );[127X[104X
    [4X[28X[ [ [ [ 1, 2 ], 1 ] ], [  ], [  ] ][128X[104X
  [4X[32X[104X
  
  [1X64.12-4 ValueCochain[101X
  
  [29X[2XValueCochain[102X( [3Xc[103X, [3Xy1[103X, [3Xy2[103X, [3X...[103X, [3Xys[103X ) [32X function
  
  [33X[0;0YHere [3Xc[103X is an [10Xs[110X-cochain. This function returns the value of [3Xc[103X when applied to
  the  [10Xs[110X  elements  [3Xy1[103X to [3Xys[103X (that lie in the Lie algebra acting on the module
  corresponding  to  [3Xc[103X).  It  is  also possible to call this function with two
  arguments: first [3Xc[103X and then the list containing [10X[3Xy1[103X[10X,...,[3Xys[103X[10X[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "A", 1, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XV:= AdjointModule( L );;[127X[104X
    [4X[25Xgap>[125X [27XC:= CochainSpace( V, 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xc:= Basis( C )[1];[127X[104X
    [4X[28X<2-cochain>[128X[104X
    [4X[25Xgap>[125X [27X ValueCochain( c, Basis(L)[2], Basis(L)[1] );[127X[104X
    [4X[28X(-1)*v.1[128X[104X
  [4X[32X[104X
  
  [1X64.12-5 LieCoboundaryOperator[101X
  
  [29X[2XLieCoboundaryOperator[102X( [3Xc[103X ) [32X function
  
  [33X[0;0YThis  is  a  function that takes an [10Xs[110X-cochain [3Xc[103X, and returns an [10Xs+1[110X-cochain.
  The coboundary operator is applied.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "A", 1, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XV:= AdjointModule( L );;[127X[104X
    [4X[25Xgap>[125X [27XC:= CochainSpace( V, 2 );;[127X[104X
    [4X[25Xgap>[125X [27Xc:= Basis( C )[1];;[127X[104X
    [4X[25Xgap>[125X [27Xc1:= LieCoboundaryOperator( c );[127X[104X
    [4X[28X<3-cochain>[128X[104X
    [4X[25Xgap>[125X [27Xc2:= LieCoboundaryOperator( c1 );[127X[104X
    [4X[28X<4-cochain>[128X[104X
  [4X[32X[104X
  
  [1X64.12-6 Cocycles[101X
  
  [29X[2XCocycles[102X( [3XV[103X, [3Xs[103X ) [32X operation
  
  [33X[0;0Yis  the  space  of  all [3Xs[103X-cocycles with respect to the Lie algebra module [3XV[103X.
  That  is  the kernel of the coboundary operator when restricted to the space
  of [3Xs[103X-cochains.[133X
  
  [1X64.12-7 Coboundaries[101X
  
  [29X[2XCoboundaries[102X( [3XV[103X, [3Xs[103X ) [32X operation
  
  [33X[0;0Yis the space of all [3Xs[103X-coboundaries with respect to the Lie algebra module [3XV[103X.
  That  is  the image of the coboundary operator, when applied to the space of
  [3Xs[103X-1-cochains. By definition the space of all 0-coboundaries is zero.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 3, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 2, [ 1, 3 ] );[127X[104X
    [4X[25Xgap>[125X [27XL:= LieAlgebraByStructureConstants( Rationals, T );;[127X[104X
    [4X[25Xgap>[125X [27XV:= FaithfulModule( L );[127X[104X
    [4X[28X<left-module over <Lie algebra of dimension 3 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27XCocycles( V, 2 );[127X[104X
    [4X[28X<vector space of dimension 7 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XCoboundaries( V, 2 );[127X[104X
    [4X[28X<vector space over Rationals, with 9 generators>[128X[104X
    [4X[25Xgap>[125X [27XDimension( last );[127X[104X
    [4X[28X5[128X[104X
  [4X[32X[104X
  
  
  [1X64.13 [33X[0;0YModules over Semisimple Lie Algebras[133X[101X
  
  [33X[0;0YThis   section   contains   functions   for   calculating   information   on
  representations of semisimple Lie algebras. First we have some functions for
  calculating  some  combinatorial data (set of dominant weights, the dominant
  character,  the  decomposition  of  a  tensor  product,  the  dimension of a
  highest-weight  module). Then there is a function for creating an admissible
  lattice  in  the  universal  enveloping algebra of a semisimple Lie algebra.
  Finally  we  have a function for constructing a highest-weight module over a
  semisimple Lie algebra.[133X
  
  [1X64.13-1 DominantWeights[101X
  
  [29X[2XDominantWeights[102X( [3XR[103X, [3Xmaxw[103X ) [32X operation
  
  [33X[0;0YReturns  a  list  consisting  of  two lists. The first of these contains the
  dominant  weights  (written  on  the  basis  of  fundamental weights) of the
  irreducible  highest-weight  module,  with highest weight [3Xmaxw[103X, over the Lie
  algebra  with  the root system [3XR[103X. The [22Xi[122X-th element of the second list is the
  level  of  the [22Xi[122X-th dominant weight. (Where the level is defined as follows.
  For  a  weight  [22Xμ[122X we write [22Xμ = λ - ∑_i k_i α_i[122X, where the [22Xα_i[122X are the simple
  roots, and [22Xλ[122X the highest weight. Then the level of [22Xμ[122X is [22X∑_i k_i[122X.)[133X
  
  [1X64.13-2 DominantCharacter[101X
  
  [29X[2XDominantCharacter[102X( [3XL[103X, [3Xmaxw[103X ) [32X operation
  [29X[2XDominantCharacter[102X( [3XR[103X, [3Xmaxw[103X ) [32X operation
  
  [33X[0;0YFor  a  highest weight [3Xmaxw[103X and a semisimple Lie algebra [3XL[103X, this returns the
  dominant  weights  of  the highest-weight module over [3XL[103X, with highest weight
  [3Xmaxw[103X.  The  output  is  a  list  of  two  lists, the first list contains the
  dominant weights; the second list contains their multiplicities.[133X
  
  [33X[0;0YThe  first  argument  can  also be a root system, in which case the dominant
  character of the highest-weight module over the corresponding semisimple Lie
  algebra is returned.[133X
  
  [1X64.13-3 DecomposeTensorProduct[101X
  
  [29X[2XDecomposeTensorProduct[102X( [3XL[103X, [3Xw1[103X, [3Xw2[103X ) [32X operation
  
  [33X[0;0YHere  [3XL[103X is a semisimple Lie algebra and [3Xw1[103X, [3Xw2[103X are dominant weights. Let [22XV_i[122X
  be  the irreducible highest-weight module over [3XL[103X with highest weight [22Xw_i[122X for
  [22Xi  = 1, 2[122X. Let [22XW = V_1 ⊗ V_2[122X. Then in general [22XW[122X is a reducible [3XL[103X-module. Now
  this function returns a list of two lists. The first of these is the list of
  highest weights of the irreducible modules occurring in the decomposition of
  [22XW[122X  as  a  direct  sum  of  irreducible modules. The second list contains the
  multiplicities  of  these  weights  (i.e.,  the  number  of  copies  of  the
  irreducible  module  with the corresponding highest weight that occur in [22XW[122X).
  The algorithm uses Klimyk's formula (see [Kli68] or [Kli66] for the original
  Russian version).[133X
  
  [1X64.13-4 DimensionOfHighestWeightModule[101X
  
  [29X[2XDimensionOfHighestWeightModule[102X( [3XL[103X, [3Xw[103X ) [32X operation
  
  [33X[0;0YHere  [3XL[103X  is a semisimple Lie algebra, and [3Xw[103X a dominant weight. This function
  returns  the  dimension  of  the  highest-weight  module over [3XL[103X with highest
  weight [3Xw[103X. The algorithm uses Weyl's dimension formula.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "F", 4, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XR:= RootSystem( L );;[127X[104X
    [4X[25Xgap>[125X [27XDominantWeights( R, [ 1, 1, 0, 0 ] );[127X[104X
    [4X[28X[ [ [ 1, 1, 0, 0 ], [ 2, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [128X[104X
    [4X[28X      [ 1, 0, 0, 0 ], [ 0, 0, 0, 0 ] ], [ 0, 3, 4, 8, 11, 19 ] ][128X[104X
    [4X[25Xgap>[125X [27XDominantCharacter( L, [ 1, 1, 0, 0 ] );[127X[104X
    [4X[28X[ [ [ 1, 1, 0, 0 ], [ 2, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 1, 0, 0 ], [128X[104X
    [4X[28X      [ 1, 0, 0, 0 ], [ 0, 0, 0, 0 ] ], [ 1, 1, 4, 6, 14, 21 ] ][128X[104X
    [4X[25Xgap>[125X [27XDecomposeTensorProduct( L, [ 1, 0, 0, 0 ], [ 0, 0, 1, 0 ] );[127X[104X
    [4X[28X[ [ [ 1, 0, 1, 0 ], [ 1, 0, 0, 0 ], [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [128X[104X
    [4X[28X      [ 2, 0, 0, 0 ], [ 0, 0, 1, 0 ], [ 1, 1, 0, 0 ] ], [128X[104X
    [4X[28X  [ 1, 1, 1, 1, 1, 1, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XDimensionOfHighestWeightModule( L, [ 1, 2, 3, 4 ] );[127X[104X
    [4X[28X79316832731136[128X[104X
  [4X[32X[104X
  
  
  [1X64.14 [33X[0;0YAdmissible Lattices in UEA[133X[101X
  
  [33X[0;0YLet  [22XL[122X be a semisimple Lie algebra over a field of characteristic [22X0[122X, and let
  [22XR[122X  be  its root system. For a positive root [22Xα[122X we let [22Xx_α[122X and [22Xy_α[122X be positive
  and  negative  root vectors, respectively, both from a fixed Chevalley basis
  of  [22XL[122X.  Furthermore,  [22Xh_1,  ...,  h_l[122X  are the Cartan elements from the same
  Chevalley basis. Also we set[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yx_α^(n) = {x_α^n over n!}, y_α^(n) = {y_α^n over n!} .[133X[124X[133X
  
  
  [33X[0;0YFurthermore,  let  [22Xα_1,  ...,  α_s[122X  denote  the  positive  roots  of  [22XR[122X. For
  multi-indices  [22XN  =  (n_1, ..., n_s)[122X, [22XM = (m_1, ..., m_s)[122X and [22XK = (k_1, ...,
  k_s)[122X (where [22Xn_i, m_i, k_i ≥ 0[122X) set[133X
  
        [22Xx^N[122X   =   [22Xx_{α_1}^(n_1) ⋯ x_{α_s}^(n_s)[122X,     
        [22Xy^M[122X   =   [22Xy_{α_1}^(m_1) ⋯ y_{α_s}^(m_s)[122X,     
        [22Xh^K[122X   =   [22X{h_1 choose k_1} ⋯ {h_l choose k_l}[122X
  
  [33X[0;0YThen  by a theorem of Kostant, the [22Xx_α^(n)[122X and [22Xy_α^(n)[122X generate a subring of
  the  universal  enveloping  algebra [22XU(L)[122X spanned (as a free [22XZ[122X-module) by the
  elements[133X
  
  
        [33X[1;6Y[24X[33X[0;0Yy^M h^K x^N[133X[124X[133X
  
  
  [33X[0;0Y(see, e.g., [Hum72] or [Hum78, Section 26]) So by the Poincare-Birkhoff-Witt
  theorem  this  subring  is  a  lattice in [22XU(L)[122X. Furthermore, this lattice is
  invariant  under  the  [22Xx_α^(n)[122X  and  [22Xy_α^(n)[122X.  Therefore,  it  is  called an
  admissible lattice in [22XU(L)[122X.[133X
  
  [33X[0;0YThe  next  functions  enable  us  to  construct  the  generators  of such an
  admissible lattice.[133X
  
  [1X64.14-1 IsUEALatticeElement[101X
  
  [29X[2XIsUEALatticeElement[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsUEALatticeElementCollection[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsUEALatticeElementFamily[102X( [3Xfam[103X ) [32X Category
  
  [33X[0;0Yis  the  category  of  elements  of  an  admissible lattice in the universal
  enveloping algebra of a semisimple Lie algebra [10XL[110X.[133X
  
  [1X64.14-2 LatticeGeneratorsInUEA[101X
  
  [29X[2XLatticeGeneratorsInUEA[102X( [3XL[103X ) [32X attribute
  
  [33X[0;0YHere  [3XL[103X  must be a semisimple Lie algebra of characteristic [22X0[122X. This function
  returns  a  list  of  generators  of  an admissible lattice in the universal
  enveloping  algebra  of  [3XL[103X,  relative  to  the  Chevalley basis contained in
  [10XChevalleyBasis(  [3XL[103X[10X  )[110X  (see [2XChevalleyBasis[102X  ([14X64.6-2[114X)).  First are listed the
  negative  root  vectors  (denoted  by [22Xy_1, ..., y_s[122X), then the positive root
  vectors  (denoted  by  [22Xx_1,  ..., x_s[122X). At the end of the list there are the
  Cartan elements. They are printed as [10X( hi/1 )[110X, which means[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y{h_i choose 1}.[133X[124X[133X
  
  
  [33X[0;0YIn general the printed form [10X( hi/ k )[110X means[133X
  
  
        [33X[1;6Y[24X[33X[0;0Y{h_i choose k}.[133X[124X[133X
  
  
  [33X[0;0YAlso [22Xy_i^(m)[122X is printed as [10Xyi^(m)[110X, which means that entering [10Xyi^m[110X at the [5XGAP[105X
  prompt results in the output [10Xm!*yi^(m)[110X.[133X
  
  [33X[0;0YProducts  of  lattice  generators  are  collected using the following order:
  first come the [22Xy_i^(m_i)[122X (in the same order as the positive roots), then the
  [22Xh_i  choose  k_i[122X,  and then the [22Xx_i^(n_i)[122X (in the same order as the positive
  roots).[133X
  
  [1X64.14-3 ObjByExtRep[101X
  
  [29X[2XObjByExtRep[102X( [3XF[103X, [3Xdescr[103X ) [32X method
  
  [33X[0;0YAn UEALattice element is represented by a list of the form [10X[ m1, c1, m2, c2,
  ...  ][110X,  where  the  [10Xc1[110X,  [10Xc2[110X  etc. are coefficients, and the [10Xm1[110X, [10Xm2[110X etc. are
  monomials.  A  monomial  is  a  list of the form [10X[ ind1, e1, ind2, e2, ... ][110X
  where  [10Xind1[110X,  [10Xind2[110X  are indices, and [10Xe1[110X, [10Xe2[110X etc. are exponents. Let [22XN[122X be the
  number  of  positive  roots of the underlying Lie algebra [10XL[110X. The indices lie
  between 1 and [22Xdim(L)[122X. If an index lies between 1 and [10XN[110X, then it represents a
  negative  root  vector  (corresponding  to the root [10XNegativeRoots( R )[ind][110X,
  where  [10XR[110X is the root system of [10XL[110X; see [2XNegativeRoots[102X ([14X64.6-8[114X)). This leads to
  a  factor  [10Xyind1^(e1)[110X  in  the  printed  form  of the monomial (which equals
  [10Xz^e1/e1![110X, where [10Xz[110X is a basis element of [10XL[110X). If an index lies between [22XN+1[122X and
  [22X2N[122X,  then it represents a positive root vector. Finally, if ind lies between
  [22X2N+1[122X  and  [22X2N+rank[122X,  then it represents an element of the Cartan subalgebra.
  This  is  printed  as  [22X( h_1/ e_1 )[122X, meaning [22Xh_1 choose e_1[122X, where [22Xh_1, ...,
  h_rank[122X are the canonical Cartan generators.[133X
  
  [33X[0;0YThe  zero  element is represented by the empty list, the identity element by
  the list [10X[ [], 1 ][110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra( "G", 2, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27Xg:=LatticeGeneratorsInUEA( L );[127X[104X
    [4X[28X[ y1, y2, y3, y4, y5, y6, x1, x2, x3, x4, x5, x6, ( h13/1 ), [128X[104X
    [4X[28X  ( h14/1 ) ][128X[104X
    [4X[25Xgap>[125X [27XIsUEALatticeElement( g[1] );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xg[1]^3;[127X[104X
    [4X[28X6*y1^(3)[128X[104X
    [4X[25Xgap>[125X [27Xq:= g[7]*g[1]^2;[127X[104X
    [4X[28X-2*y1+2*y1*( h13/1 )+2*y1^(2)*x1[128X[104X
    [4X[25Xgap>[125X [27XExtRepOfObj( q );[127X[104X
    [4X[28X[ [ 1, 1 ], -2, [ 1, 1, 13, 1 ], 2, [ 1, 2, 7, 1 ], 2 ][128X[104X
  [4X[32X[104X
  
  [1X64.14-4 IsWeightRepElement[101X
  
  [29X[2XIsWeightRepElement[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsWeightRepElementCollection[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsWeightRepElementFamily[102X( [3Xfam[103X ) [32X Category
  
  [33X[0;0YIs   a   category  of  vectors,  that  is  used  to  construct  elements  of
  highest-weight modules (by [2XHighestWeightModule[102X ([14X64.14-5[114X)).[133X
  
  [33X[0;0Y[10XWeightRepElement[110Xs  are  represented  by a list of the form [10X[ v1, c1, v2, c2,
  ....][110X,  where  the  [10Xvi[110X  are  basis  vectors,  and  the  [10Xci[110X are coefficients.
  Furthermore a basis vector [10Xv[110X is a weight vector. It is represented by a list
  of  the  form  [10X[  k, mon, wt ][110X, where [10Xk[110X is an integer (the basis vectors are
  numbered  from  [22X1[122X to [22Xdim V[122X, where [22XV[122X is the highest weight module), [10Xmon[110X is an
  [10XUEALatticeElement[110X  (which means that the result of applying [10Xmon[110X to a highest
  weight  vector is [10Xv[110X; see [2XIsUEALatticeElement[102X ([14X64.14-1[114X)) and [10Xwt[110X is the weight
  of  [10Xv[110X.  A  [10XWeightRepElement[110X  is  printed as [10Xmon*v0[110X, where [10Xv0[110X denotes a fixed
  highest weight vector.[133X
  
  [33X[0;0YIf  [10Xv[110X is a [10XWeightRepElement[110X, then [10XExtRepOfObj( v )[110X returns the corresponding
  list,  and  if  [10Xlist[110X  is  such a list and [3Xfam[103X a [10XWeightRepElementFamily[110X, then
  [10XObjByExtRep( [3Xlist[103X[10X, [3Xfam[103X[10X )[110X returns the corresponding [10XWeightRepElement[110X.[133X
  
  [1X64.14-5 HighestWeightModule[101X
  
  [29X[2XHighestWeightModule[102X( [3XL[103X, [3Xwt[103X ) [32X function
  
  [33X[0;0Yreturns  the  highest weight module with highest weight [3Xwt[103X of the semisimple
  Lie algebra [3XL[103X of characteristic [22X0[122X.[133X
  
  [33X[0;0YNote   that   the   elements   of   such   a  module  lie  in  the  category
  [2XIsLeftAlgebraModuleElement[102X  ([14X62.11-9[114X)  (and in particular they do not lie in
  the  category  [2XIsWeightRepElement[102X ([14X64.14-4[114X)). However, if [10Xv[110X is an element of
  such a module, then [10XExtRepOfObj( v )[110X is a [10XWeightRepElement[110X.[133X
  
  [33X[0;0YNote  that  for  the following examples of this chapter we increase the line
  length  limit  from  its  default  value 80 to 81 in order to make some long
  output expressions fit into the lines.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XK1:= SimpleLieAlgebra( "G", 2, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XK2:= SimpleLieAlgebra( "B", 2, Rationals );;[127X[104X
    [4X[25Xgap>[125X [27XL:= DirectSumOfAlgebras( K1, K2 );[127X[104X
    [4X[28X<Lie algebra of dimension 24 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( L, [ 0, 1, 1, 1 ] );[127X[104X
    [4X[28X<224-dimensional left-module over <Lie algebra of dimension [128X[104X
    [4X[28X24 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27Xvv:= GeneratorsOfLeftModule( V );;[127X[104X
    [4X[25Xgap>[125X [27Xvv[100];[127X[104X
    [4X[28Xy5*y7*y10*v0[128X[104X
    [4X[25Xgap>[125X [27Xe:= ExtRepOfObj( vv[100] );[127X[104X
    [4X[28Xy5*y7*y10*v0[128X[104X
    [4X[25Xgap>[125X [27XExtRepOfObj( e );[127X[104X
    [4X[28X[ [ 100, y5*y7*y10, [ -3, 2, -1, 1 ] ], 1 ][128X[104X
    [4X[25Xgap>[125X [27XBasis(L)[17]^vv[100];[127X[104X
    [4X[28X-1*y5*y7*y8*v0-1*y5*y9*v0[128X[104X
  [4X[32X[104X
  
  
  [1X64.15 [33X[0;0YTensor Products and Exterior and Symmetric Powers[133X[101X
  
  [1X64.15-1 TensorProductOfAlgebraModules[101X
  
  [29X[2XTensorProductOfAlgebraModules[102X( [3Xlist[103X ) [32X operation
  [29X[2XTensorProductOfAlgebraModules[102X( [3XV[103X, [3XW[103X ) [32X operation
  
  [33X[0;0YHere  the  elements  of  [3Xlist[103X must be algebra modules. The tensor product is
  returned  as  an  algebra module. The two-argument version works in the same
  way and returns the tensor product of its arguments.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("G",2,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( L, [ 1, 0 ] );;[127X[104X
    [4X[25Xgap>[125X [27XW:= TensorProductOfAlgebraModules( [ V, V, V ] );[127X[104X
    [4X[28X<343-dimensional left-module over <Lie algebra of dimension [128X[104X
    [4X[28X14 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27Xw:= Basis(W)[1];[127X[104X
    [4X[28X1*(1*v0<x>1*v0<x>1*v0)[128X[104X
    [4X[25Xgap>[125X [27XBasis(L)[1]^w;[127X[104X
    [4X[28X<0-tensor>[128X[104X
    [4X[25Xgap>[125X [27XBasis(L)[7]^w;[127X[104X
    [4X[28X1*(1*v0<x>1*v0<x>y1*v0)+1*(1*v0<x>y1*v0<x>1*v0)+1*(y[128X[104X
    [4X[28X1*v0<x>1*v0<x>1*v0)[128X[104X
  [4X[32X[104X
  
  [1X64.15-2 ExteriorPowerOfAlgebraModule[101X
  
  [29X[2XExteriorPowerOfAlgebraModule[102X( [3XV[103X, [3Xk[103X ) [32X operation
  
  [33X[0;0YHere  [3XV[103X must be an algebra module, defined over a Lie algebra. This function
  returns the [3Xk[103X-th exterior power of [3XV[103X as an algebra module.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("G",2,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( L, [ 1, 0 ] );;[127X[104X
    [4X[25Xgap>[125X [27XW:= ExteriorPowerOfAlgebraModule( V, 3 );[127X[104X
    [4X[28X<35-dimensional left-module over <Lie algebra of dimension [128X[104X
    [4X[28X14 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27Xw:= Basis(W)[1];[127X[104X
    [4X[28X1*(1*v0/\y1*v0/\y3*v0)[128X[104X
    [4X[25Xgap>[125X [27XBasis(L)[10]^w;[127X[104X
    [4X[28X1*(1*v0/\y1*v0/\y6*v0)+1*(1*v0/\y3*v0/\y5*v0)+1*(y1*v0/\y3*v0/\y4*v0)[128X[104X
  [4X[32X[104X
  
  [1X64.15-3 SymmetricPowerOfAlgebraModule[101X
  
  [29X[2XSymmetricPowerOfAlgebraModule[102X( [3XV[103X, [3Xk[103X ) [32X operation
  
  [33X[0;0YHere  [3XV[103X  must be an algebra module. This function returns the [3Xk[103X-th symmetric
  power of [3XV[103X (as an algebra module).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XL:= SimpleLieAlgebra("G",2,Rationals);;[127X[104X
    [4X[25Xgap>[125X [27XV:= HighestWeightModule( L, [ 1, 0 ] );;[127X[104X
    [4X[25Xgap>[125X [27XW:= SymmetricPowerOfAlgebraModule( V, 3 );[127X[104X
    [4X[28X<84-dimensional left-module over <Lie algebra of dimension [128X[104X
    [4X[28X14 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27Xw:= Basis(W)[1];[127X[104X
    [4X[28X1*(1*v0.1*v0.1*v0)[128X[104X
    [4X[25Xgap>[125X [27XBasis(L)[2]^w;[127X[104X
    [4X[28X<0-symmetric element>[128X[104X
    [4X[25Xgap>[125X [27XBasis(L)[7]^w;[127X[104X
    [4X[28X3*(1*v0.1*v0.y1*v0)[128X[104X
  [4X[32X[104X
  
