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%W algvspc.tex GAP documentation Thomas Breuer
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%H @(#)$Id: algvspc.tex,v 4.21 2001/10/11 14:44:07 gap Exp $
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%Y Copyright (C) 1997, Lehrstuhl D fuer Mathematik, RWTH Aachen, Germany
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\Chapter{Vector Spaces and Algebras}
This chapter contains an introduction into vector spaces and
algebras in {\GAP}.
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\Section{Vector Spaces}
A *vector space* over the field $F$ is an additive group that is closed
under scalar multiplication with elements in $F$.
In {\GAP}, 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 {\GAP}.
Probably the most common $F$-vector spaces in {\GAP} are so-called
*row spaces*.
They consist of row vectors, that is, lists whose elements lie in $F$.
In the following example we compute the vector space spanned by the
row vectors `[ 1, 1, 1 ]' and `[ 1, 0, 2 ]' over the rationals.
\beginexample
gap> F:= Rationals;;
gap> V:= VectorSpace( F, [ [ 1, 1, 1 ], [ 1, 0, 2 ] ] );
gap> [ 2, 1, 3 ] in V;
true
\endexample
The full row space $F^n$ is created by commands like:
\beginexample
gap> F:= GF( 7 );;
gap> V:= F^3; # The full row space over F of dimension 3.
( GF(7)^3 )
gap> [ 1, 2, 3 ] * One( F ) in V;
true
\endexample
In the same way we can also create matrix spaces. Here the short notation
`^[,]' can be used:
\beginexample
gap> m1:= [ [ 1, 2 ], [ 3, 4 ] ];; m2:= [ [ 0, 1 ], [ 1, 0 ] ];;
gap> V:= VectorSpace( Rationals, [ m1, m2 ] );
gap> m1+m2 in V;
true
gap> W:= Rationals^[3,2];
( Rationals^[ 3, 2 ] )
gap> [ [ 1, 1 ], [ 2, 2 ], [ 3, 3 ] ] in W;
true
\endexample
A field is naturally a vector space over itself.
\beginexample
gap> IsVectorSpace( Rationals );
true
\endexample
If $\Phi$ is an algebraic extension of $F$, then $\Phi$ is
also a vector space over $F$ (and indeed over any subfield of $\Phi$
that contains $F$). This field $F$ is stored in the attribute
`LeftActingDomain'.
In {\GAP}, the default is to view fields as vector spaces
over their *prime* fields.
By the function `AsVectorSpace', we can view fields
as vector spaces over fields other than the prime field.
\beginexample
gap> F:= GF( 16 );;
gap> LeftActingDomain( F );
GF(2)
gap> G:= AsVectorSpace( GF( 4 ), F );
AsField( GF(2^2), GF(2^4) )
gap> F = G;
true
gap> LeftActingDomain( G );
GF(2^2)
\endexample
A vector space has three important attributes: its *field* of definition,
its *dimension* and a *basis*. We already encountered the function
`LeftActingDomain' in the example above. It extracts the field of definition
of a vector space.
The function `Dimension' provides the dimension of the space.
Here is one more example.
\beginexample
gap> F:= GF( 9 );;
gap> m:= [ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, Z(3)^0 ] ];;
gap> V:= VectorSpace( F, m );
gap> Dimension( V );
2
gap> W:= AsVectorSpace( GF( 3 ), V );
gap> V = W;
true
gap> Dimension( W );
4
gap> LeftActingDomain( W );
GF(3)
\endexample
One of the most important attributes is a *basis*. For a given basis $B$ of
$V$, every vector $v$ in $V$ can be expressed uniquely as
$v = \sum_{b \in B} c_b b$, with coefficients $c_b \in F$.
In {\GAP}, bases are special lists of vectors.
They are used mainly for the computation of coefficients and linear
combinations.
Given a vector space $V$, a basis of $V$ is obtained by
simply applying the function `Basis' to $V$. The vectors that form
the basis are extracted from the basis by `BasisVectors'.
\beginexample
gap> m1:= [ [ 1, 2 ], [ 3, 4 ] ];; m2:= [ [ 1, 1 ], [ 1, 0 ] ];;
gap> V:= VectorSpace( Rationals, [ m1, m2 ] );
gap> B:= Basis( V );
SemiEchelonBasis( , ... )
gap> BasisVectors( Basis( V ) );
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 0, 1 ], [ 2, 4 ] ] ]
\endexample
The coefficients of
a vector relative to a given basis are found by the function
`Coefficients'. Furthermore, linear combinations of the basis vectors
are constructed using `LinearCombination'.
\beginexample
gap> V:= VectorSpace( Rationals, [ [ 1, 2 ], [ 3, 4 ] ] );
gap> B:= Basis( V );
SemiEchelonBasis( , ... )
gap> BasisVectors( Basis( V ) );
[ [ 1, 2 ], [ 0, 1 ] ]
gap> Coefficients( B, [ 1, 0 ] );
[ 1, -2 ]
gap> LinearCombination( B, [ 1, -2 ] );
[ 1, 0 ]
\endexample
In the above examples we have seen that {\GAP} 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 `Basis'.
\beginexample
gap> V:= VectorSpace( Rationals, [ [ 1, 2 ], [ 3, 4 ] ] );;
gap> B:= Basis( V, [ [ 1, 0 ], [ 0, 1 ] ] );
SemiEchelonBasis( ,
[ [ 1, 0 ], [ 0, 1 ] ] )
gap> Coefficients( B, [ 1, 2 ] );
[ 1, 2 ]
\endexample
We can construct subspaces and quotient spaces of vector spaces. The
natural projection map (constructed by `NaturalHomomorphismBySubspace'),
connects a vector space with its quotient space.
\beginexample
gap> V:= Rationals^4;
( Rationals^4 )
gap> W:= Subspace( V, [ [ 1, 2, 3, 4 ], [ 0, 9, 8, 7 ] ] );
gap> VmodW:= V/W;
( Rationals^2 )
gap> h:= NaturalHomomorphismBySubspace( V, W );
( Rationals^2 )>
gap> Image( h, [ 1, 2, 3, 4 ] );
[ 0, 0 ]
gap> PreImagesRepresentative( h, [ 1, 0 ] );
[ 1, 0, 0, 0 ]
\endexample
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\Section{Algebras}
If a 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 *algebra*. For example, every field is an algebra:
\beginexample
gap> f:= GF(8); IsAlgebra( f );
GF(2^3)
true
\endexample
One of the most important classes of algebras are sub-algebras of matrix
algebras. On the set of all $n\times n$ matrices over a field $F$
it is possible to define a multiplication in many ways.
The most frequent are the ordinary matrix multiplication and the Lie
multiplication.
Each matrix constructed as `[ , , ... ]' is regarded by {\GAP}
as an *ordinary* matrix, its multiplication is the ordinary associative
matrix multiplication.
The sum and product of two ordinary matrices are again ordinary matrices.
The *full* matrix associative algebra can be created as follows:
\beginexample
gap> F:= GF( 9 );;
gap> A:= F^[3,3];
( GF(3^2)^[ 3, 3 ] )
\endexample
An algebra can be constructed from generators using the function `Algebra'.
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.
\beginexample
gap> m1:= [ [ 1, 1 ], [ 0, 0 ] ];; m2:= [ [ 0, 0 ], [ 0, 1 ] ];;
gap> A:= Algebra( Rationals, [ m1, m2 ] );
\endexample
An interesting class of algebras for which many special algorithms
are implemented is the class of *Lie algebras*.
They arise for example as algebras of matrices whose product is defined
by the Lie bracket $[ A, B ] = A \* B - B \* A$,
where $\*$ denotes the ordinary matrix product.
Since the multiplication of objects in {\GAP} is always assumed to be
the operation `\\\*' (resp. the infix operator `\*'),
and since there is already the ``ordinary'' 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 `LieObject' from ordinary matrices,
the sum and product of Lie matrices are again Lie matrices.
\beginexample
gap> m:= LieObject( [ [ 1, 1 ], [ 1, 1 ] ] );
LieObject( [ [ 1, 1 ], [ 1, 1 ] ] )
gap> m*m;
LieObject( [ [ 0, 0 ], [ 0, 0 ] ] )
gap> IsOrdinaryMatrix( m1 ); IsOrdinaryMatrix( m );
true
false
gap> IsLieMatrix( m1 ); IsLieMatrix( m );
false
true
\endexample
Given a field `F' and a list `mats' of Lie objects over `F', we can construct
the Lie algebra generated by `mats' using the function `Algebra'.
Alternatively, if we do not want to be bothered with the function
`LieObject', we can use the function `LieAlgebra' 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
`LieAlgebra' are not contained in the returned Lie algebra.
\beginexample
gap> m1:= [ [ 0, 1 ], [ 0, 0 ] ];;
gap> m2:= [ [ 0, 0 ], [ 1, 0 ] ];;
gap> L:= LieAlgebra( Rationals, [ m1, m2 ] );
gap> m1 in L;
false
\endexample
A second way of creating an algebra is by specifying a multiplication
table. Let $A$ be a finite dimensional algebra with basis
$\{x_1,\ldots,x_n\}$, then for $1\leq i,j\leq n$ the product $x_ix_j$ is
a linear combination of basis elements, i.e., there are $c_{ij}^k$ in the
ground field such that
$$
x_i x_j = \sum_{k=1}^n c_{ij}^k x_k\.
$$
It is not difficult to show that the constants $c_{ij}^k$
determine the multiplication completely. Therefore, the $c_{ij}^k$ are
called *structure constants*. In {\GAP} we can create a finite dimensional
algebra by specifying an array of structure constants.
In {\GAP} such a table of structure constants is represented using
lists. The obvious way to do this
would be to construct a ``three-dimensional'' list `T' such that
`T[i][j][k]' equals
$c_{ij}^k$. 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 $n$-dimensional algebra is an
$n\times n$ array `T' such that `T[i][j]' describes the product
of the `i'-th and the `j'-th basis element. This product is encoded
in the following way. The entry `T[i][j]' is a list of two elements.
The first of these is a list of
indices $k$ such that $c_{ij}^k$ is nonzero. The second list contains the
corresponding constants $c_{ij}^k$. Suppose, for example, that `S'
is the table
of an algebra with basis $\{x_1,\ldots ,x_8\}$ and that `S[3][7]'
equals `[ [ 2, 4, 6 ], [ 1/2, 2, 2/3 ] ]'. Then in the algebra we
have the relation
$$
x_3 x_7 = (1/2) x_2 + 2 x_4 + (2/3) x_6\.
$$
Furthermore, if `S[6][1] = [ [ ], [ ] ]' then the product of the
sixth and first basis elements is zero.
Finally 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., $c_{ij}^k=c_{ji}^k$. 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.
Empty structure constants tables are created by the function
`EmptySCTable', which takes a dimension $d$, a zero element $z$,
and optionally one of the strings `"symmetric"', `"antisymmetric"',
and returns an empty structure constants table $T$ corresponding to
a $d$-dimensional algebra over a field with zero element $z$.
Structure constants can be entered into the table $T$ using the function
`SetEntrySCTable'.
It takes four arguments, namely $T$, two indices $i$ and $j$,
and a list of the form `[$c_{ij}^{k_1}$,$k_1$,$c_{ij}^{k_2}$,$k_2$,...]'.
In this call to `SetEntrySCTable',
the product of the $i$-th and the $j$-th basis vector
in any algebra described by $T$ is set to $\sum_l c_{ij}^{k_l} x_{k_l}$.
(Note that in the empty table, this product was zero.)
If $T$ knows that it is (anti)symmetric, then at the same time also
the product of the $j$-th and the $i$-th basis vector is set appropriately.
In the following example we temporarily increase the line length limit from
its default value 80 to 82 in order to make the long output expression fit
into one line.
\beginexample
gap> SizeScreen([ 82, ]);;
gap> T:= EmptySCTable( 2, 0, "symmetric" );
[ [ [ [ ], [ ] ], [ [ ], [ ] ] ], [ [ [ ], [ ] ], [ [ ], [ ] ] ], 1, 0 ]
gap> SetEntrySCTable( T, 1, 2, [1/2,1,1/3,2] ); T;
[ [ [ [ ], [ ] ], [ [ 1, 2 ], [ 1/2, 1/3 ] ] ],
[ [ [ 1, 2 ], [ 1/2, 1/3 ] ], [ [ ], [ ] ] ], 1, 0 ]
gap> SizeScreen([ 80, ]);;
\endexample
If we have defined a structure constants table, then we can construct
the corresponding algebra by `AlgebraByStructureConstants'.
\beginexample
gap> A:= AlgebraByStructureConstants( Rationals, T );
\endexample
If we know that a structure constants table defines a Lie algebra,
then we can construct the corresponding Lie algebra by
`LieAlgebraByStructureConstants';
the algebra returned by this function knows that it is a Lie algebra,
so {\GAP} need not check the Jacobi identity.
\beginexample
gap> T:= EmptySCTable( 2, 0, "antisymmetric" );;
gap> SetEntrySCTable( T, 1, 2, [2/3,1] );
gap> L:= LieAlgebraByStructureConstants( Rationals, T );
\endexample
In {\GAP} an algebra is naturally a vector space. Hence all the functionality
for vector spaces is also available for algebras.
\beginexample
gap> F:= GF(2);;
gap> z:= Zero( F );; o:= One( F );;
gap> T:= EmptySCTable( 3, z, "antisymmetric" );;
gap> SetEntrySCTable( T, 1, 2, [ o, 1, o, 3 ] );
gap> SetEntrySCTable( T, 1, 3, [ o, 1 ] );
gap> SetEntrySCTable( T, 2, 3, [ o, 3 ] );
gap> A:= AlgebraByStructureConstants( F, T );
gap> Dimension( A );
3
gap> LeftActingDomain( A );
GF(2)
gap> Basis( A );
CanonicalBasis( )
\endexample
Subalgebras 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.
In the following example we temporarily increase the line length limit from
its default value 80 to 81 in order to make the long output expression fit
into one line.
\beginexample
gap> m:= [ [ 1, 2, 3 ], [ 0, 1, 6 ], [ 0, 0, 1 ] ];;
gap> A:= Algebra( Rationals, [ m ] );;
gap> subA:= Subalgebra( A, [ m-m^2 ] );
gap> Dimension( subA );
2
gap> SizeScreen([ 81, ]);;
gap> idA:= Ideal( A, [ m-m^3 ] );
, (1 generators)>
gap> SizeScreen([ 80, ]);;
gap> Dimension( idA );
2
gap> B:= A/idA;
\endexample
The call `B:= A/idA' creates a new algebra that does not ``know'' about
its connection with `A'. If we want to connect an algebra with its factor
via a homomorphism, then we first have to create the homomorphism
(`NaturalHomomorphismByIdeal'). After this we create the factor algebra
from the homomorphism by the function `ImagesSource'. In the next example
we divide an algebra `A' by its radical and lift the central idempotents
of the factor to the original algebra `A'.
\beginexample
gap> m1:=[[1,0,0],[0,2,0],[0,0,3]];;
gap> m2:=[[0,1,0],[0,0,2],[0,0,0]];;
gap> A:= Algebra( Rationals, [ m1, m2 ] );;
gap> Dimension( A );
6
\endexample
\beginexample
gap> R:= RadicalOfAlgebra( A );
gap> h:= NaturalHomomorphismByIdeal( A, R );
-> >
\endexample
\beginexample
gap> AmodR:= ImagesSource( h );
gap> id:= CentralIdempotentsOfAlgebra( AmodR );
[ v.3, v.2+(-3)*v.3, v.1+(-2)*v.2+(3)*v.3 ]
gap> PreImagesRepresentative( h, id[1] );
[ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 1 ] ]
gap> PreImagesRepresentative( h, id[2] );
[ [ 0, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 0 ] ]
gap> PreImagesRepresentative( h, id[3] );
[ [ 1, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ]
\endexample
Structure constants tables for the simple Lie algebras are present in {\GAP}.
They can be constructed using the function `SimpleLieAlgebra'. The Lie
algebras constructed by this function come with a root system attached.
\beginexample
gap> L:= SimpleLieAlgebra( "G", 2, Rationals );
gap> R:= RootSystem( L );
gap> PositiveRoots( R );
[ [ 2, -1 ], [ -3, 2 ], [ -1, 1 ], [ 1, 0 ], [ 3, -1 ], [ 0, 1 ] ]
gap> CartanMatrix( R );
[ [ 2, -1 ], [ -3, 2 ] ]
\endexample
Another example of algebras is provided by *quaternion algebras*.
We define a quaternion algebra over an extension field of the
rationals, namely the field generated by $\sqrt{5}$.
(The number `EB(5)' is equal to $1/2 (-1+\sqrt{5})$.
The field is printed as `NF(5,[ 1, 4 ])'.)
\beginexample
gap> b5:= EB(5);
E(5)+E(5)^4
gap> q:= QuaternionAlgebra( FieldByGenerators( [ b5 ] ) );
gap> gens:= GeneratorsOfAlgebra( q );
[ e, i, j, k ]
gap> e:= gens[1];; i:= gens[2];; j:= gens[3];; k:= gens[4];;
gap> IsAssociative( q );
true
gap> IsCommutative( q );
false
gap> i*j; j*i;
k
(-1)*k
gap> One( q );
e
\endexample
If the coefficient field is a real subfield of the complex numbers
then the quaternion algebra is in fact a division ring.
\beginexample
gap> IsDivisionRing( q );
true
gap> Inverse( e+i+j );
(1/3)*e+(-1/3)*i+(-1/3)*j
\endexample
So {\GAP} knows about this fact.
As in any ring, we can look at groups of units.
(The function `StarCyc' used below computes the unique algebraic
conjugate of an element in a quadratic subfield of a cyclotomic field.)
\beginexample
gap> c5:= StarCyc( b5 );
E(5)^2+E(5)^3
gap> g1:= 1/2*( b5*e + i - c5*j );
(1/2*E(5)+1/2*E(5)^4)*e+(1/2)*i+(-1/2*E(5)^2-1/2*E(5)^3)*j
gap> Order( g1 );
5
gap> g2:= 1/2*( -c5*e + i + b5*k );
(-1/2*E(5)^2-1/2*E(5)^3)*e+(1/2)*i+(1/2*E(5)+1/2*E(5)^4)*k
gap> Order( g2 );
10
gap> g:=Group( g1, g2 );;
#I default `IsGeneratorsOfMagmaWithInverses' method returns `true' for
[ (1/2*E(5)+1/2*E(5)^4)*e+(1/2)*i+(-1/2*E(5)^2-1/2*E(5)^3)*j,
(-1/2*E(5)^2-1/2*E(5)^3)*e+(1/2)*i+(1/2*E(5)+1/2*E(5)^4)*k ]
gap> Size( g );
120
gap> IsPerfect( g );
true
\endexample
Since there is only one perfect group of order 120, up to isomorphism,
we see that the group `g' is isomorphic to $SL_2(5)$.
As usual, a permutation representation of the group can be constructed
using a suitable action of the group.
\beginexample
gap> cos:= RightCosets( g, Subgroup( g, [ g1 ] ) );;
gap> Length( cos );
24
gap> hom:= OperationHomomorphism( g, cos, OnRight );;
gap> im:= Image( hom );
Group([ (2,3,5,9,15)(4,7,12,8,14)(10,17,23,20,24)(11,19,22,16,13),
(1,2,4,8,3,6,11,20,17,19)(5,10,18,7,13,22,12,21,24,15)(9,16)(14,23) ])
gap> Size( im );
120
\endexample
To get a matrix representation of `g' or of the whole algebra `q',
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.
\beginexample
gap> bas:= CanonicalBasis( q );;
gap> BasisVectors( bas );
[ e, i, j, k ]
gap> op:= OperationAlgebraHomomorphism( q, bas, OnRight );
matrices of dim. 4>
gap> ImagesRepresentative( op, e );
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ]
gap> ImagesRepresentative( op, i );
[ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ] ]
gap> ImagesRepresentative( op, g1 );
[ [ 1/2*E(5)+1/2*E(5)^4, 1/2, -1/2*E(5)^2-1/2*E(5)^3, 0 ],
[ -1/2, 1/2*E(5)+1/2*E(5)^4, 0, -1/2*E(5)^2-1/2*E(5)^3 ],
[ 1/2*E(5)^2+1/2*E(5)^3, 0, 1/2*E(5)+1/2*E(5)^4, -1/2 ],
[ 0, 1/2*E(5)^2+1/2*E(5)^3, 1/2, 1/2*E(5)+1/2*E(5)^4 ] ]
\endexample
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\Section{Further Information about Vector Spaces and Algebras}
More information about vector spaces can be found in
Chapter~"ref:Vector Spaces".
Chapter~"ref:Algebras" deals with the functionality for general algebras.
Furthermore, concerning special functions for Lie algebras,
there is Chapter~"ref:Lie Algebras".
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