</li>
</ol>
<p>for all <span class="SimpleMath">\(a,b \in F\)</span> and all <span class="SimpleMath">\(v,w \in V\)</span>. In <strong class="pkg">GAP</strong>, only those domains that are constructed as vector spaces are regarded as vector <span =>&;</> < class">:VectorSpace , m1, ;/>
<p>Probably the most common <span class="SimpleMath">\(F\)</span>/>>
<div class="example"><pre>
<span class="GAPprompt">gap></span span=">&;<span> <="java.lang.StringIndexOutOfBoundsException: Range [64, 63) out of bounds for length 120
<vector space over Rationals, with 2 generators>
<="java.lang.StringIndexOutOfBoundsException: Range [84, 22) out of bounds for length 84
true
</pre></div>
<p>The full row space <span class="SimpleMath">\(F^n\)</span<pan =G"gapgt;<>span"APinput>Coefficients, 1 ];/java.lang.StringIndexOutOfBoundsException: Index 97 out of bounds for length 97
<div class="example"><pre> 1:= [ [ 1, 2 ], [34 ] ]; m2= [0, ] ,0 ]];;/pan
<span class="GAPprompt">gap></span> <span class="GAPinput">V( &;vectorspace , 2 &t,
<vector =GAPprompt>g<>span="GAPinput"( B [,2 <span
<span class="java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds for length 12
java.lang.StringIndexOutOfBoundsException: Range [4, 5) out of bounds for length 4
<span class
( <div class="java.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="(java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 15
true
</pre></div>
<p>A field is naturally a vector space over itself.</p>
pIf span=impleMath\(Phi)<> span="impleMath>(\<span>, < java.lang.StringIndexOutOfBoundsException: Range [132, 131) out of bounds for length 844
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:= GF( 16 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">LeftActingDomain( F&;java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 71
GF)
<span class="GAPprompt">gap></span> <span class="GAPinput">G:= AsVectorSpace( GF( 4 ), F );</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">F = java.lang.StringIndexOutOfBoundsException: Range [0, 67) out of bounds for length 14
true
<span class="GAPprompt">gap></span> <span class="GAPinput">LeftActingDomain( G );</span>
GF(2^2)
</pre></div>
<p>A vector space has three important attributes: its
<div class="example"><pre>
<GF(^)
<span java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 4
<span class="GAPprompt<>ne the important of are - algebras.On set all<span=S>( n)<span over < =">(\/> it is to a in waysThe are multiplication .<p>
<vector space over GF(3^2 "APprompt&t;/ <pan class=GAPinput")<pan 2
<span class="GAPprompt">gap></span> <span class="GAPinput">W:= AsVectorSpace( GF( 3 ), V );</span>
<vector space over GF(3), with 4 generators>
<span class="GAPprompt">gap&
true
<span class="GAPprompt" 4
<spandiv class"example"><pre
F3
</pre></div>
<p>One of the most important attributes is a <em>basis</em>. (3^)[3,3] )
<p>In <strong class="pkg">GAP</strong>, bases are special lists of vectors. They
<div class
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Range [126, 27) out of bounds for length 615 ""gap<>=""=(Rationalsm1 )<span
<vector space over Rationals, with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V );</span>
SemiEchelonBasis( <vector space over Rationals, with 2 generators>, ... )
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisVectors( Basis( V ) );</span>
[ [ [ 1, 2 ], [ 3, 4 ] ], [ [ 0, 1 ], [ 2, 4 ] ] ]
</pre></div>
<p>The coefficients of a vector relative to a given basis are found by the function <code class="func">Coefficients</code> (<span class="java.lang.StringIndexOutOfBoundsException: Index 141 out of bounds for length 35
<div class="example"><pre>
<span class="GAPprompt">&<span <pan=GAPinput>sOrdinaryMatrixm1) IsOrdinaryMatrix(m ;<span>
< space over Rationals,with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">B:= Basis( V );</span>
SemiEchelonBasis( <vector space over Rationals, with 2 generators>, ... )
<span class=GAPprompt"gapgt;</span <pan class"GAPinput">asisVectors(Basis( V ) );</span>
[ [ 1, 2 ], [ 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Coefficients( B, [ 1, 0 ] );</
,- ]
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[1,0]
</pre></div>
<p>In the above examples we have seen that <strong class="pkg">GAP</strong> often chooses the basis it wants to work java.lang.StringIndexOutOfBoundsException: Index 118 out of bounds for length 98
<div class="example"><pre>
< class="GAPprompt">gap></span> <span class="GAPinput">V:= VectorSpace( Rationals, [ [ 1, 2 ], [ 3, 4 ] ] );;</span>
< "APprompt>></> "GAPinputBB(V 0,[0 ]);</pan
SemiEchelonBasis( <vector space over Rationals, with 2 generators>,
[[10 , [0, 1] ] )
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 0
java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 8
</pre></div>
<p>If a bilinear span class="GAPprompt">gap>span class="GAPinput">A:= AlgebraByStructureConstants( Rationals, T );</span>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class= java.lang.StringIndexOutOfBoundsException: Range [110, 109) out of bounds for length 401
GF(2^3)
true
</pre></div>
<p>One of the most important classes of algebras are sub-algebras of matrix algebras. On the set of all <span class="SimpleMath">\(n \times n\)</span> matrices over a field <span class="SimpleMath">\(F\)</span> it is possible to define a multiplication in many ways<
<p>Each matrix constructed as <span class="SimpleMath">\([ \textit{row1}, \textit{row2}, \ldots ]\)</span> is regarded by <strong class="pkg">GAP</strong> as an <em>ordinary</em> java.lang.StringIndexOutOfBoundsException: Index 181 out of bounds for length 105
<div &t; algebraof dimension 22 over >
<span class
<pan =GAPprompt>gt;/pan>< class=""A=F[3,3]</pan>
( GF(java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</pre></div>
<p>An algebra can be constructed pan> <span class="GAPinput">z:= Zero( F );; o:= One( F );;</span>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:= [ [ 1, 1 ], [ ;/span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals, [ m1, m2 ] );</span=">apjava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 112
<algebra over Rationals, with 2 generators>
/>/iv
<p>spanclass=">&;/java.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 109
<p>Since the java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 5
<divpre<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 12
<java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 109
LieObject( [ [ 1, 1 ]
<span class=div ="<prejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 26
java.lang.StringIndexOutOfBoundsException: Range [62, 63) out of bounds for length 35
<span class="GAPprompt">span class="GAPprompt">gapspan=">A:=Algebra( Rationals, m];/>
spa classGAPprompt">><span <span =>ubA:=( A, m-m^2 ] );</span>
false
<span class="GAPprompt">gap></span> <span class="GAPinputjava.lang.StringIndexOutOfBoundsException: Index 60 out of bounds for length 48
true
</pre></div
<p>Given a field <code class="code">F</lt;two-sided ideal in <algebra of dimension 3 over Rationals>,
<div class="example"><pre>
< GAPpromptgapjava.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 98
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:= [ [ 0, 0 ], [ 1, 0 ] ];;</span>
<span class="GAPprompt">
&;Liealgebra java.lang.StringIndexOutOfBoundsException: Range [36, 30) out of bounds for length 53
<span class="class="GAPprompt">gap></span> <span class="GAPinput">m1:=[[1span classgap>/span><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 100
java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 5
</pre></ss=>gapgt;/pan< class"">imension(A)<>
<p>Aspan=">gap&<span> <span class=GAPinput"R=RadicalOfAlgebra(java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 95
<p>In ="java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 108
<p>Finally two 6 over RationalsRationals;-gt; ;java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 76
<div class="example"><pre>
<span class="GAPprompt1+-2).2+3).3 ]
[ [ [ [ ], [ ] ], [ [ ], s ">gap&;/spanspanclass">PreImagesRepresentative( h, id[1] );</span>
[ [ [ ], [ ] ], [ [ ], [ ] ] 0,0,0 ] , java.lang.StringIndexOutOfBoundsException: Index 105 out of bounds for length 105
> <spanjava.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 115
[ [ [ [ ], [ ] ], [ [ 1, 2 ], [ 1/2, 1/3 ] ] ], 12 ], [ 1/2,1/3]],[[],[] ,1,
<pre</
<p>If we have<p>java.lang.StringIndexOutOfBoundsException: Range [13, 12) out of bounds for length 336
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= AlgebraByStructureConstants( Rationals, T );</span>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</
<p>If we know that a structure constants table defines a Lie algebra, then we can construct the corresponding Lie algebra by <code class="func">LieAlgebraByStructureConstants</code> (<span class="RefLink">Reference: LieAlgebraByStructureConstants</span>); the algebra returned by this function knows that it is a Lie algebra, so <strong class="pkg">GAP</strong> need not check the Jacobi identity.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">T:= EmptySCTable( 2, 0, "antisymmetric" &troot 2gt; ""SetEntrySCTableT,1, 2, 2/,]<java.lang.StringIndexOutOfBoundsException: Range [105, 104) out of bounds for length 105
=">g;<span <class=": ( Rationals,T );<>
<Lie algebra of dimension 2 over Rationals>
</pre></div>
<p>In
<div class="example"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<spanclass"java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 80
<span classE5+5^4
<classGjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 112
<span class="GAPprompt">;algebra-with-one of dimensionone 4overNF5,[ 1 ]>
<="gapgt< < ="java.lang.StringIndexOutOfBoundsException: Range [79, 77) out of bounds for length 106
<span class="GAPprompt">gap></span> <span class[e i
<span class="<pan class="GAPprompt">gap></span> <span class="GAPinput">e:= gens[1];; i:= gens[2];; j:= gens[3];; k:= gens[4];;</span>
&;
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( A );</span> 3
<span class="GAPprompt">gap></span> <span class="GAPinput">LeftActingDomain( A );</span>
GF(2)
<span class="GAPprompt">gap></span> <span class
CanonicalBasis( <algebra of dimension-1*
<<div>
<p>java.lang.StringIndexOutOfBoundsException: Index 9 out of bounds for length 1
<div class="examplempt">java.lang.StringIndexOutOfBoundsException: Range [78, 27) out of bounds for length 89
<span class="GAPprompt">gap></span> <span class="GAPinput">m:= [ [ 1, 2, 3 ], [ 0, 1, 6 ], [ 0, 0, 1 ] ];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals1/)+-1/)(1/)
<span class="GAPprompt">gap></span> <span class="GAPinput">subA:= Subalgebra( A, [ m-m^2 ] );</span>
<algebra over Rationals, with 1 generator>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( subA );</span> 2
<span class="GAPprompt">gap
ltidealla of3Rationalsjava.lang.StringIndexOutOfBoundsException: Index 69 out of bounds for length 69
(1 generator)>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( idA );</span> 2 span class="GAPprompt">gap></span> <span class="GAPinput">B:= A/idA;</span>
<algebra of dimension 1 over Rationals>
</pre></div>
<p>The call <code class="code">B:= A/idA</code> creates a new algebra that does not <q>know</q> about its connection with <code class="code">A</code>. If we want to connect an algebra with its factor via ahomomorphism, then we first have to create the homomorphism (<code class="func">NaturalHomomorphismByIdeal</code> (<span class="RefLink">Reference: NaturalHomomorphismByIdeal</span>)). After this we create the factor algebra from the homomorphism by the function <code class="func">ImagesSource</code> (<span class="RefLink">Reference: ImagesSource</span>). In the next example we divide an algebra <code class="code">A</code> by its radical and lift the central idempotents of the factor to the original algebra <code class="code">A</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">m1:=[[1,0,0],[0,2,0],[0,0,3]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">m2:=[[0,1,0],[0,0,2],[0,0,0]];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">A:= Algebra( Rationals, [ m1, m2 ] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Dimension( A );</span> 6
<span class="GAPprompt">gap></span> <span class="GAPinput">R:= RadicalOfAlgebra( A );</span>
<algebra of dimension 3 over Rationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:= NaturalHomomorphismByIdeal( A, R );</span>
<linear mapping by matrix, <algebra of dimension 6 over Rationals> -> <algebra of dimension 3 over Rationals>>
<span class="GAPprompt">gap></span> <span class="GAPinput">AmodR:= ImagesSource( h );</span>
<algebra of dimension 3 over Rationals>
<span class"GAPprompt>gap>/span><span class=GAPinput">id:= CentralIdempotentsOfAlgebra AmodR ;<>
[ v.3, v.2+(-3)*v.3, v.1+(-2)*v.2+(3)*v.3 ]
<span class="GAPprompt">gap></span> <gt;</span> <span>(g1 );span>
[ [ 0span class=GAPprompt"gap&t;<span> < class="GAPinput">g2:= 1/2*( -c5*e + i + b5*k );</span>
<span class="GAPprompt">gap></span> <span class=(1/*5)2)*+1/)i(/*()+1*()4)*java.lang.StringIndexOutOfBoundsException: Range [58, 59) out of bounds for length 58
[ [ 0, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 0 ] ]
<></span> span =GAPinput"( h,[3 ;<span>
[ [ 1, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ]
</pre></div>
<p>Structure constants tables for the simple Lie algebras are present in <strong class="pkg">GAP</strong>. They can be constructed using the function <code class="func">SimpleLieAlgebra</code> (<span class="RefLink">Reference: SimpleLieAlgebra</span>). The Lie algebras constructed by this function (1*52-*()3)+1/*+/()E4k]
<div class="example"><pre>
<span class="GAPprompt"
<Lie algebra of dimension 14 over Rationals>span java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 116
<span class="GAPprompt">gap class=GAPprompt"&;/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 83
<root system of rank 2&java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<span class="GAPprompt">gap></span> <span class= =GAPprompt>&t<java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 87
[ 2 -1 ,[ -3,2] -1, ][1 ] [3 - ] , ]
<span class="GAPprompt">gap></span> <span class="GAPinput">CartanMatrix( R );</span>
[ [ 2, -1 ], [ -3, 2 ] ]
</pre></div>
<p>Anotherspan =""&;<>< classclass=>();/span>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">b5:= EB
E(+E(5)4
<span class="GAPprompt">gap></span> <span class="GAPinput">q:= QuaternionAlgebra( FieldByGenerators( [ b5 ] ) );</span>
<algebra-with-one of dimension 4 over NFspan class="gap>/pan>< =GAPinput">asCanonicalBasis( )</>
< class="GAPprompt">gap></span> <span class="GAPinput">gens:= GeneratorsOfAlgebra( q );</span>
[ e, i, j, k ]
<">gt</pan> <span class=GAPinput">:= gens];i:[];;j: gens3; k:=gens[4];;/pan>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAssociative( q );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsCommutative( q )[, i, j, ]) -gt;matrices ofdim.4>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">i*j; j*i;</span>
k
(-1)*k
<span class="GAPprompt">gap></span> <span class="GAPinput">One( q ) =>/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 99
e
</pre></div>
<p>If the coefficient field is a real subfield of the complex numbers then the quaternion java.lang.StringIndexOutOfBoundsException: Range [24, 11) out of bounds for length 100
<> <strong class="pkg">GAP</strong> knows about this fact. As in any ring, we can look at groups of units. (The function <code class="func">StarCyc</code> (<span class="RefLink
<p>Since there is only one perfect group of order 120, up to isomorphism, we see that the group <codeclass="code">g</code> is isomorphic to <span class="SimpleMath">\(SL_2(5)\)</span>. As usual, a permutation representation of the group can be constructed using a suitable action of the group.</p>
<p>To get a matrix representation of <code class="code">g</code> or of the whole algebra <code class="code">q</code>, 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.</p>
<h4>6.3 <span class="Heading">Further Information about Vector Spaces and Algebras</span></h4>
<p>More information about vector spaces can be found in Chapter <span class="RefLink">Reference: Vector Spaces</span>. Chapter <span class="RefLink">Reference: Algebras</span> deals with the functionality for general algebras. Furthermore, concerning special functions for Lie algebras, there is Chapter <span class="RefLink">Reference: Lie Algebras</span>.</p>
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