  
  [1X6 [33X[0;0YVector Spaces and Algebras[133X[101X
  
  [33X[0;0YThis  chapter  contains  an  introduction into vector spaces and algebras in
  [5XGAP[105X.[133X
  
  
  [1X6.1 [33X[0;0YVector Spaces[133X[101X
  
  [33X[0;0YA  [13Xvector  space[113X  [22XV[122X  over  a  field [22XF[122X is an (abelian) additive group that is
  closed under scalar multiplication by elements in [22XF[122X, such that[133X
  
  [31X1[131X   [33X[0;6Y[22X1v=v[122X,[133X
  
  [31X2[131X   [33X[0;6Y[22Xa(bv)=(ab)v[122X,[133X
  
  [31X3[131X   [33X[0;6Y[22Xa(v+w)=av+aw[122X, and[133X
  
  [31X4[131X   [33X[0;6Y[22X(a+b)v=av+bv[122X,[133X
  
  [33X[0;0Yfor  all  [22Xa,b  ∈  F[122X  and  all  [22Xv,w  ∈ V[122X. In [5XGAP[105X, only those domains that are
  constructed  as  vector spaces are regarded as vector spaces. In particular,
  an  additive  group  that does not know about an acting domain of scalars is
  not regarded as a vector space in [5XGAP[105X.[133X
  
  [33X[0;0YProbably  the  most  common [22XF[122X-vector spaces in [5XGAP[105X are so-called [13Xrow spaces[113X.
  They  consist of row vectors, that is, lists whose elements lie in [22XF[122X. In the
  following  example  we compute the vector space spanned by the row vectors [10X[
  1, 1, 1 ][110X and [10X[ 1, 0, 2 ][110X over the rationals.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:= Rationals;;[127X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( F, [ [ 1, 1, 1 ], [ 1, 0, 2 ] ] );[127X[104X
    [4X[28X<vector space over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27X[ 2, 1, 3 ] in V;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe full row space [22XF^n[122X is created by commands like:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:= GF( 7 );;[127X[104X
    [4X[25Xgap>[125X [27XV:= F^3;   # The full row space over F of dimension 3.[127X[104X
    [4X[28X( GF(7)^3 )[128X[104X
    [4X[25Xgap>[125X [27X[ 1, 2, 3 ] * One( F ) in V;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn  the  same  way we can also create matrix spaces. Here the short notation
  [10X[3Xfield[103X[10X^[[3Xdim1[103X[10X,[3Xdim2[103X[10X][110X can be used:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm1:= [ [ 1, 2 ], [ 3, 4 ] ];; m2:= [ [ 0, 1 ], [ 1, 0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ m1, m2 ] );[127X[104X
    [4X[28X<vector space over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xm1+m2 in V;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XW:= Rationals^[3,2];[127X[104X
    [4X[28X( Rationals^[ 3, 2 ] )[128X[104X
    [4X[25Xgap>[125X [27X[ [ 1, 1 ], [ 2, 2 ], [ 3, 3 ] ] in W;[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YA field is naturally a vector space over itself.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsVectorSpace( Rationals );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  [22XΦ[122X  is  an algebraic extension of [22XF[122X, then [22XΦ[122X is also a vector space over [22XF[122X
  (and  indeed over any subfield of [22XΦ[122X that contains [22XF[122X). This field [22XF[122X is stored
  in the attribute [2XLeftActingDomain[102X ([14XReference: LeftActingDomain[114X). In [5XGAP[105X, the
  default  is  to view fields as vector spaces over their [13Xprime[113X fields. By the
  function  [2XAsVectorSpace[102X  ([14XReference:  AsVectorSpace[114X),  we can view fields as
  vector spaces over fields other than the prime field.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:= GF( 16 );;[127X[104X
    [4X[25Xgap>[125X [27XLeftActingDomain( F );[127X[104X
    [4X[28XGF(2)[128X[104X
    [4X[25Xgap>[125X [27XG:= AsVectorSpace( GF( 4 ), F );[127X[104X
    [4X[28XAsField( GF(2^2), GF(2^4) )[128X[104X
    [4X[25Xgap>[125X [27XF = G;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XLeftActingDomain( G );[127X[104X
    [4X[28XGF(2^2)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YA  vector space has three important attributes: its [13Xfield[113X of definition, its
  [13Xdimension[113X  and a [13Xbasis[113X. We already encountered the function [2XLeftActingDomain[102X
  ([14XReference: LeftActingDomain[114X) in the example above. It extracts the field of
  definition  of a vector space. The function [2XDimension[102X ([14XReference: Dimension[114X)
  provides the dimension of the vector space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:= GF( 9 );;[127X[104X
    [4X[25Xgap>[125X [27Xm:= [ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, Z(3)^0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( F, m );[127X[104X
    [4X[28X<vector space over GF(3^2), with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XDimension( V );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XW:= AsVectorSpace( GF( 3 ), V );[127X[104X
    [4X[28X<vector space over GF(3), with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27XV = W;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XDimension( W );[127X[104X
    [4X[28X4[128X[104X
    [4X[25Xgap>[125X [27XLeftActingDomain( W );[127X[104X
    [4X[28XGF(3)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOne  of  the most important attributes is a [13Xbasis[113X. For a given basis [22XB[122X of [22XV[122X,
  every  vector  [22Xv[122X  in  [22XV[122X can be expressed uniquely as [22Xv = ∑_b ∈ B c_b b[122X, with
  coefficients [22Xc_b ∈ F[122X.[133X
  
  [33X[0;0YIn  [5XGAP[105X,  bases  are  special lists of vectors. They are used mainly for the
  computation of coefficients and linear combinations.[133X
  
  [33X[0;0YGiven  a  vector  space  [22XV[122X,  a basis of [22XV[122X is obtained by simply applying the
  function  [2XBasis[102X ([14XReference: Basis[114X) to [22XV[122X. The vectors that form the basis are
  extracted from the basis by [2XBasisVectors[102X ([14XReference: BasisVectors[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm1:= [ [ 1, 2 ], [ 3, 4 ] ];; m2:= [ [ 1, 1 ], [ 1, 0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ m1, m2 ] );[127X[104X
    [4X[28X<vector space over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V );[127X[104X
    [4X[28XSemiEchelonBasis( <vector space over Rationals, with[128X[104X
    [4X[28X2 generators>, ... )[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors( Basis( V ) );[127X[104X
    [4X[28X[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 0, 1 ], [ 2, 4 ] ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  coefficients  of  a  vector  relative to a given basis are found by the
  function   [2XCoefficients[102X   ([14XReference:   Coefficients[114X).  Furthermore,  linear
  combinations  of  the  basis vectors are constructed using [2XLinearCombination[102X
  ([14XReference: LinearCombination[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ [ 1, 2 ], [ 3, 4 ] ] );[127X[104X
    [4X[28X<vector space over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V );[127X[104X
    [4X[28XSemiEchelonBasis( <vector space over Rationals, with[128X[104X
    [4X[28X2 generators>, ... )[128X[104X
    [4X[25Xgap>[125X [27XBasisVectors( Basis( V ) );[127X[104X
    [4X[28X[ [ 1, 2 ], [ 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XCoefficients( B, [ 1, 0 ] );[127X[104X
    [4X[28X[ 1, -2 ][128X[104X
    [4X[25Xgap>[125X [27XLinearCombination( B, [ 1, -2 ] );[127X[104X
    [4X[28X[ 1, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIn the above examples we have seen that [5XGAP[105X often chooses the basis it wants
  to  work  with. It is also possible to construct bases with prescribed basis
  vectors  by  giving  a  list  of  these  vectors as second argument to [2XBasis[102X
  ([14XReference: Basis[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= VectorSpace( Rationals, [ [ 1, 2 ], [ 3, 4 ] ] );;[127X[104X
    [4X[25Xgap>[125X [27XB:= Basis( V, [ [ 1, 0 ], [ 0, 1 ] ] );[127X[104X
    [4X[28XSemiEchelonBasis( <vector space over Rationals, with 2 generators>,[128X[104X
    [4X[28X[ [ 1, 0 ], [ 0, 1 ] ] )[128X[104X
    [4X[25Xgap>[125X [27XCoefficients( B, [ 1, 2 ] );[127X[104X
    [4X[28X[ 1, 2 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YWe can construct subspaces and quotient spaces of vector spaces. The natural
  projection  map  (constructed  by  [2XNaturalHomomorphismBySubspace[102X ([14XReference:
  NaturalHomomorphismBySubspace[114X)),  connects  a vector space with its quotient
  space.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XV:= Rationals^4;[127X[104X
    [4X[28X( Rationals^4 )[128X[104X
    [4X[25Xgap>[125X [27XW:= Subspace( V, [ [ 1, 2, 3, 4 ], [ 0, 9, 8, 7 ] ] );[127X[104X
    [4X[28X<vector space over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XVmodW:= V/W;[127X[104X
    [4X[28X( Rationals^2 )[128X[104X
    [4X[25Xgap>[125X [27Xh:= NaturalHomomorphismBySubspace( V, W );[127X[104X
    [4X[28X<linear mapping by matrix, ( Rationals^4 ) -> ( Rationals^2 )>[128X[104X
    [4X[25Xgap>[125X [27XImage( h, [ 1, 2, 3, 4 ] );[127X[104X
    [4X[28X[ 0, 0 ][128X[104X
    [4X[25Xgap>[125X [27XPreImagesRepresentative( h, [ 1, 0 ] );[127X[104X
    [4X[28X[ 1, 0, 0, 0 ][128X[104X
  [4X[32X[104X
  
  
  [1X6.2 [33X[0;0YAlgebras[133X[101X
  
  [33X[0;0YIf  a bilinear multiplication is defined for the elements of a vector space,
  and  if  the  vector  space  is  closed under this multiplication then it is
  called an [13Xalgebra[113X. For example, every field is an algebra:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xf:= GF(8); IsAlgebra( f );[127X[104X
    [4X[28XGF(2^3)[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YOne  of  the  most  important classes of algebras are sub-algebras of matrix
  algebras.  On the set of all [22Xn × n[122X matrices over a field [22XF[122X it is possible to
  define  a  multiplication  in  many ways. The most frequent are the ordinary
  matrix multiplication and the Lie multiplication.[133X
  
  [33X[0;0YEach  matrix  constructed  as  [22X[  [3Xrow1[103X, [3Xrow2[103X, ... ][122X is regarded by [5XGAP[105X as an
  [13Xordinary[113X  matrix,  its  multiplication  is  the  ordinary associative matrix
  multiplication.  The  sum  and  product  of  two ordinary matrices are again
  ordinary matrices.[133X
  
  [33X[0;0YThe [13Xfull[113X matrix associative algebra can be created as follows:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:= GF( 9 );;[127X[104X
    [4X[25Xgap>[125X [27XA:= F^[3,3];[127X[104X
    [4X[28X( GF(3^2)^[ 3, 3 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAn  algebra  can  be  constructed from generators using the function [2XAlgebra[102X
  ([14XReference:  Algebra[114X). It takes as arguments the field of coefficients and a
  list  of generators. Of course the coefficient field and the generators must
  fit  together;  if  we want to construct an algebra of ordinary matrices, we
  may take the field generated by the entries of the generating matrices, or a
  subfield or extension field.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm1:= [ [ 1, 1 ], [ 0, 0 ] ];; m2:= [ [ 0, 0 ], [ 0, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XA:= Algebra( Rationals, [ m1, m2 ] );[127X[104X
    [4X[28X<algebra over Rationals, with 2 generators>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YAn  interesting  class  of  algebras  for  which many special algorithms are
  implemented is the class of [13XLie algebras[113X. They arise for example as algebras
  of matrices whose product is defined by the Lie bracket [22X[ A, B ] = A * B - B
  * A[122X, where [22X*[122X denotes the ordinary matrix product.[133X
  
  [33X[0;0YSince  the  multiplication  of  objects  in  [5XGAP[105X is always assumed to be the
  operation  [10X*[110X  (resp.  the  infix operator [10X*[110X), and since there is already the
  [21Xordinary[121X  matrix  product defined for ordinary matrices, as mentioned above,
  we  must use a different construction for matrices that occur as elements of
  Lie  algebras. Such Lie matrices can be constructed by [2XLieObject[102X ([14XReference:
  LieObject[114X)  from  ordinary matrices, the sum and product of Lie matrices are
  again Lie matrices.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= LieObject( [ [ 1, 1 ], [ 1, 1 ] ] );[127X[104X
    [4X[28XLieObject( [ [ 1, 1 ], [ 1, 1 ] ] )[128X[104X
    [4X[25Xgap>[125X [27Xm*m;[127X[104X
    [4X[28XLieObject( [ [ 0, 0 ], [ 0, 0 ] ] )[128X[104X
    [4X[25Xgap>[125X [27XIsOrdinaryMatrix( m1 ); IsOrdinaryMatrix( m );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XIsLieMatrix( m1 ); IsLieMatrix( m );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YGiven  a field [10XF[110X and a list [10Xmats[110X of Lie objects over [10XF[110X, we can construct the
  Lie  algebra  generated  by  [10Xmats[110X  using  the  function  [2XAlgebra[102X ([14XReference:
  Algebra[114X).  Alternatively, if we do not want to be bothered with the function
  [2XLieObject[102X  ([14XReference:  LieObject[114X),  we  can  use  the  function  [2XLieAlgebra[102X
  ([14XReference:  LieAlgebra for an associative algebra[114X) that takes a field and a
  list  of  ordinary matrices, and constructs the Lie algebra generated by the
  corresponding  Lie matrices. Note that this means that the ordinary matrices
  used  in  the  call  of [2XLieAlgebra[102X ([14XReference: LieAlgebra for an associative
  algebra[114X) are not contained in the returned Lie algebra.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm1:= [ [ 0, 1 ], [ 0, 0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27Xm2:= [ [ 0, 0 ], [ 1, 0 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XL:= LieAlgebra( Rationals, [ m1, m2 ] );[127X[104X
    [4X[28X<Lie algebra over Rationals, with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27Xm1 in L;[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [33X[0;0YA second way of creating an algebra is by specifying a multiplication table.
  Let  [22XA[122X be a finite dimensional algebra with basis [22X(x_1, x_2, ..., x_n)[122X, then
  for  [22X1  ≤  i,  j  ≤  n[122X  the product [22Xx_i x_j[122X is a linear combination of basis
  elements,  i.e.,  there  are  [22Xc_ij^k[122X in the ground field such that [22Xx_i x_j =
  ∑_k=1^n  c_ij^k  x_k.[122X  It is not difficult to show that the constants [22Xc_ij^k[122X
  determine  the  multiplication  completely. Therefore, the [22Xc_ij^k[122X are called
  [13Xstructure  constants[113X.  In  [5XGAP[105X we can create a finite dimensional algebra by
  specifying an array of structure constants.[133X
  
  [33X[0;0YIn  [5XGAP[105X  such a table of structure constants is represented using lists. The
  obvious way to do this would be to construct a [21Xthree-dimensional[121X list [10XT[110X such
  that  [10XT[i][j][k][110X  equals  [22Xc_ij^k[122X.  But  it  often happens that many of these
  constants vanish. Therefore a more complicated structure is used in order to
  be  able  to  omit  the  zeros.  A  multiplication table of an [22Xn[122X-dimensional
  algebra  is  an [22Xn × n[122X array [10XT[110X such that [10XT[i][j][110X describes the product of the
  [10Xi[110X-th  and  the  [10Xj[110X-th basis element. This product is encoded in the following
  way.  The  entry  [10XT[i][j][110X is a list of two elements. The first of these is a
  list  of indices [22Xk[122X such that [22Xc_ij^k[122X is nonzero. The second list contains the
  corresponding constants [22Xc_ij^k[122X. Suppose, for example, that [10XS[110X is the table of
  an algebra with basis [22X(x_1, x_2, ..., x_8)[122X and that [10XS[3][7][110X equals [10X[ [ 2, 4,
  6  ],  [ 1/2, 2, 2/3 ] ][110X. Then in the algebra we have the relation [22Xx_3 x_7 =
  (1/2)  x_2  + 2 x_4 + (2/3) x_6.[122X Furthermore, if [10XS[6][1] = [ [ ], [ ] ][110X then
  the product of the sixth and first basis elements is zero.[133X
  
  [33X[0;0YFinally  two  numbers are added to the table. The first number can be 1, -1,
  or  0.  If  it is 1, then the table is known to be symmetric, i.e., [22Xc_ij^k =
  c_ji^k[122X.  If  this  number is -1, then the table is known to be antisymmetric
  (this happens for instance when the algebra is a Lie algebra). The remaining
  case,  0,  occurs in all other cases. The second number that is added is the
  zero element of the field over which the algebra is defined.[133X
  
  [33X[0;0YEmpty  structure  constants  tables are created by the function [2XEmptySCTable[102X
  ([14XReference:  EmptySCTable[114X), which takes a dimension [22Xd[122X, a zero element [22Xz[122X, and
  optionally  one  of the strings [10X"symmetric"[110X, [10X"antisymmetric"[110X, and returns an
  empty  structure  constants table [22XT[122X corresponding to a [22Xd[122X-dimensional algebra
  over  a  field  with zero element [22Xz[122X. Structure constants can be entered into
  the table [22XT[122X using the function [2XSetEntrySCTable[102X ([14XReference: SetEntrySCTable[114X).
  It  takes  four  arguments, namely [22XT[122X, two indices [22Xi[122X and [22Xj[122X, and a list of the
  form   [22X[   c_ij^{k_1},  k_1,  c_ij^{k_2},  k_2,  ...  ][122X.  In  this  call  to
  [10XSetEntrySCTable[110X,  the  product  of the [22Xi[122X-th and the [22Xj[122X-th basis vector in any
  algebra  described  by [22XT[122X is set to [22X∑_l c_ij^{k_l} x_{k_l}[122X. (Note that in the
  empty  table, this product was zero.) If [22XT[122X knows that it is (anti)symmetric,
  then at the same time also the product of the [22Xj[122X-th and the [22Xi[122X-th basis vector
  is set appropriately.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 2, 0, "symmetric" );[127X[104X
    [4X[28X[ [ [ [  ], [  ] ], [ [  ], [  ] ] ],[128X[104X
    [4X[28X  [ [ [  ], [  ] ], [ [  ], [  ] ] ], 1, 0 ][128X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 2, [1/2,1,1/3,2] );  T;[127X[104X
    [4X[28X[ [ [ [  ], [  ] ], [ [ 1, 2 ], [ 1/2, 1/3 ] ] ],[128X[104X
    [4X[28X  [ [ [ 1, 2 ], [ 1/2, 1/3 ] ], [ [  ], [  ] ] ], 1, 0 ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  we  have  defined a structure constants table, then we can construct the
  corresponding    algebra    by    [2XAlgebraByStructureConstants[102X    ([14XReference:
  AlgebraByStructureConstants[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XA:= AlgebraByStructureConstants( Rationals, T );[127X[104X
    [4X[28X<algebra of dimension 2 over Rationals>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  we  know that a structure constants table defines a Lie algebra, then we
  can      construct      the      corresponding      Lie      algebra      by
  [2XLieAlgebraByStructureConstants[102X  ([14XReference: LieAlgebraByStructureConstants[114X);
  the algebra returned by this function knows that it is a Lie algebra, so [5XGAP[105X
  need not check the Jacobi identity.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 2, 0, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 2, [2/3,1] );[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
  
  [33X[0;0YIn  [5XGAP[105X  an algebra is naturally a vector space. Hence all the functionality
  for vector spaces is also available for algebras.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XF:= GF(2);;[127X[104X
    [4X[25Xgap>[125X [27Xz:= Zero( F );;  o:= One( F );;[127X[104X
    [4X[25Xgap>[125X [27XT:= EmptySCTable( 3, z, "antisymmetric" );;[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 2, [ o, 1, o, 3 ] );[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 1, 3, [ o, 1 ] );[127X[104X
    [4X[25Xgap>[125X [27XSetEntrySCTable( T, 2, 3, [ o, 3 ] );[127X[104X
    [4X[25Xgap>[125X [27XA:= AlgebraByStructureConstants( F, T );[127X[104X
    [4X[28X<algebra of dimension 3 over GF(2)>[128X[104X
    [4X[25Xgap>[125X [27XDimension( A );[127X[104X
    [4X[28X3[128X[104X
    [4X[25Xgap>[125X [27XLeftActingDomain( A );[127X[104X
    [4X[28XGF(2)[128X[104X
    [4X[25Xgap>[125X [27XBasis( A );[127X[104X
    [4X[28XCanonicalBasis( <algebra of dimension 3 over GF(2)> )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSubalgebras  and ideals of an algebra can be constructed by specifying a set
  of  generators for the subalgebra or ideal. The quotient space of an algebra
  by an ideal is naturally an algebra itself.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm:= [ [ 1, 2, 3 ], [ 0, 1, 6 ], [ 0, 0, 1 ] ];;[127X[104X
    [4X[25Xgap>[125X [27XA:= Algebra( Rationals, [ m ] );;[127X[104X
    [4X[25Xgap>[125X [27XsubA:= Subalgebra( A, [ m-m^2 ] );[127X[104X
    [4X[28X<algebra over Rationals, with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XDimension( subA );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XidA:= Ideal( A, [ m-m^3 ] );[127X[104X
    [4X[28X<two-sided ideal in <algebra of dimension 3 over Rationals>,[128X[104X
    [4X[28X  (1 generator)>[128X[104X
    [4X[25Xgap>[125X [27XDimension( idA );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XB:= A/idA;[127X[104X
    [4X[28X<algebra of dimension 1 over Rationals>[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  call  [10XB:=  A/idA[110X  creates  a  new  algebra that does not [21Xknow[121X about its
  connection  with  [10XA[110X.  If we want to connect an algebra with its factor via a
  homomorphism,    then   we   first   have   to   create   the   homomorphism
  ([2XNaturalHomomorphismByIdeal[102X  ([14XReference: NaturalHomomorphismByIdeal[114X)). After
  this  we  create  the  factor  algebra from the homomorphism by the function
  [2XImagesSource[102X  ([14XReference:  ImagesSource[114X).  In  the next example we divide an
  algebra  [10XA[110X  by its radical and lift the central idempotents of the factor to
  the original algebra [10XA[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xm1:=[[1,0,0],[0,2,0],[0,0,3]];;[127X[104X
    [4X[25Xgap>[125X [27Xm2:=[[0,1,0],[0,0,2],[0,0,0]];;[127X[104X
    [4X[25Xgap>[125X [27XA:= Algebra( Rationals, [ m1, m2 ] );;[127X[104X
    [4X[25Xgap>[125X [27XDimension( A );[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27XR:= RadicalOfAlgebra( A );[127X[104X
    [4X[28X<algebra of dimension 3 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27Xh:= NaturalHomomorphismByIdeal( A, R );[127X[104X
    [4X[28X<linear mapping by matrix, <algebra of dimension[128X[104X
    [4X[28X6 over Rationals> -> <algebra of dimension 3 over Rationals>>[128X[104X
    [4X[25Xgap>[125X [27XAmodR:= ImagesSource( h );[127X[104X
    [4X[28X<algebra of dimension 3 over Rationals>[128X[104X
    [4X[25Xgap>[125X [27Xid:= CentralIdempotentsOfAlgebra( AmodR );[127X[104X
    [4X[28X[ v.3, v.2+(-3)*v.3, v.1+(-2)*v.2+(3)*v.3 ][128X[104X
    [4X[25Xgap>[125X [27XPreImagesRepresentative( h, id[1] );[127X[104X
    [4X[28X[ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XPreImagesRepresentative( h, id[2] );[127X[104X
    [4X[28X[ [ 0, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XPreImagesRepresentative( h, id[3] );[127X[104X
    [4X[28X[ [ 1, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YStructure  constants  tables for the simple Lie algebras are present in [5XGAP[105X.
  They  can  be  constructed  using  the function [2XSimpleLieAlgebra[102X ([14XReference:
  SimpleLieAlgebra[114X). The Lie algebras constructed by this function come with a
  root system attached.[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 [27XPositiveRoots( R );[127X[104X
    [4X[28X[ [ 2, -1 ], [ -3, 2 ], [ -1, 1 ], [ 1, 0 ], [ 3, -1 ], [ 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XCartanMatrix( R );[127X[104X
    [4X[28X[ [ 2, -1 ], [ -3, 2 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YAnother  example of algebras is provided by [13Xquaternion algebras[113X. We define a
  quaternion  algebra  over  an  extension  field of the rationals, namely the
  field  generated by [22Xsqrt{5}[122X. (The number [10XEB(5)[110X is equal to [22X1/2 (-1+sqrt{5})[122X.
  The field is printed as [10XNF(5,[ 1, 4 ])[110X.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xb5:= EB(5);[127X[104X
    [4X[28XE(5)+E(5)^4[128X[104X
    [4X[25Xgap>[125X [27Xq:= QuaternionAlgebra( FieldByGenerators( [ b5 ] ) );[127X[104X
    [4X[28X<algebra-with-one of dimension 4 over NF(5,[ 1, 4 ])>[128X[104X
    [4X[25Xgap>[125X [27Xgens:= GeneratorsOfAlgebra( q );[127X[104X
    [4X[28X[ e, i, j, k ][128X[104X
    [4X[25Xgap>[125X [27Xe:= gens[1];; i:= gens[2];; j:= gens[3];; k:= gens[4];;[127X[104X
    [4X[25Xgap>[125X [27XIsAssociative( q );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsCommutative( q );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xi*j; j*i;[127X[104X
    [4X[28Xk[128X[104X
    [4X[28X(-1)*k[128X[104X
    [4X[25Xgap>[125X [27XOne( q );[127X[104X
    [4X[28Xe[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  the coefficient field is a real subfield of the complex numbers then the
  quaternion algebra is in fact a division ring.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsDivisionRing( q );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XInverse( e+i+j );[127X[104X
    [4X[28X(1/3)*e+(-1/3)*i+(-1/3)*j[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSo  [5XGAP[105X  knows  about  this  fact.  As in any ring, we can look at groups of
  units.  (The  function  [2XStarCyc[102X ([14XReference: StarCyc[114X) used below computes the
  unique  algebraic  conjugate  of  an  element  in  a quadratic subfield of a
  cyclotomic field.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xc5:= StarCyc( b5 );[127X[104X
    [4X[28XE(5)^2+E(5)^3[128X[104X
    [4X[25Xgap>[125X [27Xg1:= 1/2*( b5*e + i - c5*j );[127X[104X
    [4X[28X(1/2*E(5)+1/2*E(5)^4)*e+(1/2)*i+(-1/2*E(5)^2-1/2*E(5)^3)*j[128X[104X
    [4X[25Xgap>[125X [27XOrder( g1 );[127X[104X
    [4X[28X5[128X[104X
    [4X[25Xgap>[125X [27Xg2:= 1/2*( -c5*e + i + b5*k );[127X[104X
    [4X[28X(-1/2*E(5)^2-1/2*E(5)^3)*e+(1/2)*i+(1/2*E(5)+1/2*E(5)^4)*k[128X[104X
    [4X[25Xgap>[125X [27XOrder( g2 );[127X[104X
    [4X[28X10[128X[104X
    [4X[25Xgap>[125X [27Xg:=Group( g1, g2 );;[127X[104X
    [4X[28X#I  default `IsGeneratorsOfMagmaWithInverses' method returns `true' for[128X[104X
    [4X[28X  [ (1/2*E(5)+1/2*E(5)^4)*e+(1/2)*i+(-1/2*E(5)^2-1/2*E(5)^3)*j,[128X[104X
    [4X[28X  (-1/2*E(5)^2-1/2*E(5)^3)*e+(1/2)*i+(1/2*E(5)+1/2*E(5)^4)*k ][128X[104X
    [4X[25Xgap>[125X [27XSize( g );[127X[104X
    [4X[28X120[128X[104X
    [4X[25Xgap>[125X [27XIsPerfect( g );[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0YSince  there  is  only one perfect group of order 120, up to isomorphism, we
  see  that  the  group  [10Xg[110X  is  isomorphic to [22XSL_2(5)[122X. As usual, a permutation
  representation  of  the  group can be constructed using a suitable action of
  the group.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xcos:= RightCosets( g, Subgroup( g, [ g1 ] ) );;[127X[104X
    [4X[25Xgap>[125X [27XLength( cos );[127X[104X
    [4X[28X24[128X[104X
    [4X[25Xgap>[125X [27Xhom:= ActionHomomorphism( g, cos, OnRight );;[127X[104X
    [4X[25Xgap>[125X [27Xim:= Image( hom );[127X[104X
    [4X[28XGroup([ (2,3,5,9,15)(4,7,12,8,14)(10,17,23,20,24)(11,19,22,16,13),[128X[104X
    [4X[28X  (1,2,4,8,3,6,11,20,17,19)(5,10,18,7,13,22,12,21,24,15)(9,16)(14,23) ])[128X[104X
    [4X[25Xgap>[125X [27XSize( im );[127X[104X
    [4X[28X120[128X[104X
  [4X[32X[104X
  
  [33X[0;0YTo  get  a  matrix  representation  of  [10Xg[110X or of the whole algebra [10Xq[110X, we must
  specify  a  basis of the vector space on which the algebra acts, and compute
  the linear action of elements w.r.t. this basis.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xbas:= CanonicalBasis( q );;[127X[104X
    [4X[25Xgap>[125X [27XBasisVectors( bas );[127X[104X
    [4X[28X[ e, i, j, k ][128X[104X
    [4X[25Xgap>[125X [27Xop:= OperationAlgebraHomomorphism( q, bas, OnRight );[127X[104X
    [4X[28X<op. hom. AlgebraWithOne( NF(5,[ 1, 4 ]),[128X[104X
    [4X[28X[ e, i, j, k ] ) -> matrices of dim. 4>[128X[104X
    [4X[25Xgap>[125X [27XImagesRepresentative( op, e );[127X[104X
    [4X[28X[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ][128X[104X
    [4X[25Xgap>[125X [27XImagesRepresentative( op, i );[127X[104X
    [4X[28X[ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ] ][128X[104X
    [4X[25Xgap>[125X [27XImagesRepresentative( op, g1 );[127X[104X
    [4X[28X[ [ 1/2*E(5)+1/2*E(5)^4, 1/2, -1/2*E(5)^2-1/2*E(5)^3, 0 ],[128X[104X
    [4X[28X  [ -1/2, 1/2*E(5)+1/2*E(5)^4, 0, -1/2*E(5)^2-1/2*E(5)^3 ],[128X[104X
    [4X[28X  [ 1/2*E(5)^2+1/2*E(5)^3, 0, 1/2*E(5)+1/2*E(5)^4, -1/2 ],[128X[104X
    [4X[28X  [ 0, 1/2*E(5)^2+1/2*E(5)^3, 1/2, 1/2*E(5)+1/2*E(5)^4 ] ][128X[104X
  [4X[32X[104X
  
  
  [1X6.3 [33X[0;0YFurther Information about Vector Spaces and Algebras[133X[101X
  
  [33X[0;0YMore  information  about  vector  spaces can be found in Chapter [14X'Reference:
  Vector  Spaces'[114X.  Chapter [14X'Reference: Algebras'[114X deals with the functionality
  for  general  algebras.  Furthermore,  concerning  special functions for Lie
  algebras, there is Chapter [14X'Reference: Lie Algebras'[114X.[133X
  
