products/Sources/formale Sprachen/GAP/tst/testinstall/   (GAP Algebra Version 4.15.1©)  Datei vom 18.9.2025 mit Größe 4 kB image not shown  

Impressum chap6_mj.html   Sprache: HTML

 

 products/Sources/formale Sprachen/GAP/doc/tut/chap6_mj.html




<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//
         "http://www.w3.org/java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 61

<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<script type="text/javascript"
  src="https://cdn.jsdelivr.net/npm/mathjax@2/MathJax.js?config=TeX-AMS-MML_HTMLorMML">
</script>
<title>GAP (tut) - Chapter 6: Vector Spaces and Algebras</title>
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<meta name="generator" content="GAPDoc2HTML" />
<link rel="stylesheet" type="text/css" href="manual.css" />
<.js type="ext/avascript>/script>
<script type="text/javascript">overwriteStyle();< class">></span><pan> < "java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 82
</head>
<body class="chap6"  onload="jscontent()">


<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap5_mj.html">[Previous Chapter]</a><spanclass="GAPprompt>gap><span><span class="">: VectorSpace(,  )<span>

<p id="mathjaxlink" class="pcenter"><a href="chap6.html">[MathJax off]</a></p>
<p><a id="X853484B982F1DF96" name="X853484B982F1DF96"></a></p>
<iv class="ChapSects"><a href="chap6_mj.html#X853484B982F1DF96">6 <span class="Heading">Vector Spaces and Algebras</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap6_mj.html#X7DAD6700787EC845">6.1 <span class="Heading">Vector Spaces</span></a>
</>
</div>
<div class="ContSect"<span class="GAPprompt">gap&;/span> < class"APinput">V = W;</span>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a span =GAPpromptgap&;/><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 84
<java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
</div>
</div>

<h3>6 <span class="Heading">Vector Spaces and Algebras</span></h3>

<p>This chapter contains an introduction into vector spaces and algebras in <strong class="pkg">GAP</strong>.</p>

<p><a 

<h4>6.1 <span class="

<p>an ">&=GAPinput>m1=[ [ 1,2  ]  , ]];m2:= [1] [ , 0 ] ;;</span>

<ol>
<li><p><span class="SimpleMath">\(1v=v\)</span>,</p>

</li>
<li><p><span class="SimpleMath">\(a(bv)=(ab)v\)</span>,</p>

</li>
<li><p><span class="SimpleMath">\(a(v+w)=av+aw\)</span>, and</p>

</li>
<li><p><span class="SimpleMath">\((a+b)v=av+bv\)</span>,</p>

</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>
<span class="GAPprompt">gap></span> <span class="GAPinput">F:= GF( 7 );;</span>
< classGAPprompt">gt</span span class="APinput>= ^3;  # The full row spaceoverF ofdimension 3.span
( GF(7)^3 )
<span class="GAPprompt">gap></span> <pan class"GAPinput"  2,  OneF </pan
rue
</pre></div>

<p>Indiv class"xample><>


<div class="example"><pre>
1:= [ [ 1, 2 ], [3 4 ] ]; 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>


<div =>>
<span span classGAPpromptgapg;<VmodWVW<span
true
</pre></div>

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

<p>Given a <p>Given a vector"gapgt;/span> <span class="GAPinput">m1:= [ [ 1, 1 ], [ 0, 0 ] ];; m2:= [ [ 0, 0 ], [ 0, 1 ] span class"GAPprompt>gap&t;/pan span=GAPinput>= AlgebraRationals [ m1, m2 ] );</span


<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>,
[[1 0 , [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>We can construct 


<div class="example"><pre>
<spanclass=GAPprompt"gt;</span> <span class="GAPinput">V:= Rationals^;<>
( Rationals^4 )
< classGAPprompt>apgt;/span><pan="">:= SubspaceV [[1,2,3,4 ]  0, 9, 8,,7 ]])<span>
<vector space over Rationals, with 2 generators>
<span class="GAPprompt">gap></
( Rationals^2 )
<span class="GAPprompt">gap></span> <span class="GAPinput">h:= NaturalHomomorphismBySubspace( V, W );</span>
< bymatrix, Rationals^4 ) -> Rationals^2&
<span class="     ] ]],[ ][   ]],
 ,  ]
<span class="GAPprompt">gap></span> <span classspan"">><span> span =">SetEntrySCTable(T,1 ,[/21,/,2 ;  ;</>
[1
</pre></div>

<p><a id="X7DDBF6F47A2E021C" name="X7DDBF6F47A2E021C"></re><div>

<h4>6.2 <span class="Heading">Algebras</span></h4

<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

<> <em>full/m matrix associative  canbecreated  follows:/


<div &t; algebraof dimension 2 2 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

<p>Empty structure constants tablesltalgebraof dimension 3  Rationals&;


<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 ] ] ],
     1 2 ], [ 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 a homomorphism, 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>
[ [ 0 span 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


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsDivisionRing( q );</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">Inverse( e+i+j );</span>
(1/3)*e+(-  [1/*5^2+/*(5^,  /*5+2*5^4 -2 ,
<

<> <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


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">c5:= StarCyc( b5 );</span>
E(5)^2+E(5)^3
<class="APprompt"gap></span> <span class="GAPinput">g1:= 1/2*( b5*e + i - c5*j );</span>
(12*E12E5^1/2)*+(-1/2*E(5)^2-1/2*E(5)^3)*j
<span class="GAPprompt">gap></span> <span class="GAPinput">Order( g1 );</span>
5
<span class="GAPprompt">gap></span> <span class="GAPinput">g2:= 1/2*( -c5*e + i + b5*k );</span>
(12*)2-12*)^3**i(/2*5+(5^)k
<span class="GAPprompt<html>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">g:=Group( g1, g2 );;</span>
#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 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( g );</span>
120
<span class="GAPprompt">gap></span> <span class="GAPinput">IsPerfect( g );</span>
true
</pre></div>

<p>Since there is only one perfect group of order 120, up to isomorphism, we see that the group <code class="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>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">cos:= RightCosets( g, Subgroup( g, [ g1 ] ) );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Length( cos );</span>
24
<span class="GAPprompt">gap></span> <span class="GAPinput">hom:= ActionHomomorphism( g, cos, OnRight );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">im:= Image( hom );</span>
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) ])
<span class="GAPprompt">gap></span> <span class="GAPinput">Size( im );</span>
120
</pre></div>

<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>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">bas:= CanonicalBasis( q );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisVectors( bas );</span>
[ e, i, j, k ]
<span class="GAPprompt">gap></span> <span class="GAPinput">op:= OperationAlgebraHomomorphism( q, bas, OnRight );</span>
<op. hom. AlgebraWithOne( NF(5,[ 1, 4 ]),
[ e, i, j, k ] ) -> matrices of dim. 4>
<span class="GAPprompt">gap></span> <span class="GAPinput">ImagesRepresentative( op, e );</span>
[ [ 1, 0, 0, 0 ], [ 0, 1, 0, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImagesRepresentative( op, i );</span>
[ [ 0, 1, 0, 0 ], [ -1, 0, 0, 0 ], [ 0, 0, 0, -1 ], [ 0, 0, 1, 0 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput">ImagesRepresentative( op, g1 );</span>
[ [ 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 ] ]
</pre></div>

<p><a id="X843A8D6F7A7592B5" name="X843A8D6F7A7592B5"></a></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>


<div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a>   <a href="chap0_mj.html#contents">[Contents]</a>    <a href="chap5_mj.html">[Previous Chapter]</a>    <a href="chap7_mj.html">[Next Chapter]</a>   </div>


<div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top</a>  <a href="chap1_mj.html">1</a>  <a href="chap2_mj.html">2</a>  <a href="chap3_mj.html">3</a>  <a href="chap4_mj.html">4</a>  <a href="chap5_mj.html">5</a>  <a href="chap6_mj.html">6</a>  <a href="chap7_mj.html">7</a>  <a href="chap8_mj.html">8</a>  <a href="chapBib_mj.html">Bib</a>  <a href="chapInd_mj.html">Ind</a>  </div>

<hr />
<p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GAPDoc">GAPDoc2HTML</a></p>
<body>
</html>

Messung V0.5 in Prozent
C=86 H=92 G=88

¤ Dauer der Verarbeitung: 0.40 Sekunden  ¤

*© Formatika GbR, Deutschland






Wurzel

Suchen

PVS Prover

Isabelle Prover

NIST Cobol Testsuite

Cephes Mathematical Library

Vienna Development Method

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.

Bemerkung:

Die farbliche Syntaxdarstellung und die Messung sind noch experimentell.