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<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <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_HTMLor </script> <title>GAP (XModAlg) - Chapter 4: Crossed modules</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" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap4" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <p id="mathjaxlink" class="pcenter"><a href="chap4.html">[MathJax off]</a></p> <p><a id="X808C6B357F8BADC1" name="X808C6B357F8BADC1"></a></p> <div class="ChapSects"><a href="chap4_mj.html#X808C6B357F8BADC1">4 <span class="Heading">C <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X813D94F97D8E71A <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B31475D7C03007 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7CD49D5A84FB05F <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X83C5D98B828DD1D <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X814C89667B997FE <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X824D4DD88323EF7 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7AB09A158148290 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X78400B837A2C8FB <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7CA213397B334CB </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap4_mj.html </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7D575D07810640E <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X83F45A3E8554ABE <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X87F4D35A826599C <span class="ContSS"><br /><span class="nocss"> </span><a href="chap4_mj.html#X7B7975DA7E87069 </div></div> </div> <h3>4 <span class="Heading">Crossed modules</span></h3> <p>In this chapter we will present the notion of crossed modules of commutative algebras and thei <p><a id="X7BB9D67179296AA0" name="X7BB9D67179296AA0"></a></p> <h4>4.1 <span class="Heading">Definition and Examples</span></h4> <p>A <em>crossed module</em> is a <strong class="button">k</strong>-algebra morphism <span class="Si <p class="center">\[ {\bf XModAlg\ 1} ~:~ \partial(r \cdot s) = r(\partial s), \qquad {\bf XModAlg\ 2} ~:~ (\partial s) \cdot s^{\prime} = ss^{\prime}, \]</p> <p>for all <span class="SimpleMath">\(s,s^{\prime }\in S, \ r\in R\)</span>. The morphism <span cla <p>Note that, although in this definition we have used a left action, in the category of commutati <p>When only the first axiom is satisfied, it is a <em>precrossed module</em> which is constructed.</ <p>The details of these implementations can be found in <a href="chapBib_mj.html#biBaodabas1"> <p><a id="X813D94F97D8E71A8" name="X813D94F97D8E71A8"></a></p> <h5>4.1-1 XModAlgebra</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>These two global function call one of the following six operations, depending on the argument <p><a id="X7B31475D7C030075" name="X7B31475D7C030075"></a></p> <h5>4.1-2 XModAlgebraByIdeal</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>Let <span class="SimpleMath">\(A\)</span> be an algebra and <span class="SimpleMath">\(I\)</sp <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">F5 := GF(5);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">id5 := One( F5 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">two := Z(5);; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">z5 := Zero( F5 );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">n0 := [ [id5,z5,z5], [z5,id5,z5], [z <span class="GAPprompt">gap></span> <span class="GAPinput">n1 := [ [z5,id5,two^3], [z5,z5,id5] <span class="GAPprompt">gap></span> <span class="GAPinput">n2 := [ [z5,z5,id5], [z5,z5,z5], [z5 <span class="GAPprompt">gap></span> <span class="GAPinput">An := Algebra( F5, [ n0, n1 ] );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( An, "An" ); </span> <span class="GAPprompt">gap></span> <span class="GAPinput">Bn := Subalgebra( An, [ n1 ] );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Bn, "Bn" ); </span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsIdeal( An, Bn ); </span> true <span class="GAPprompt">gap></span> <span class="GAPinput">actn := AlgebraActionByMultiplier <span class="GAPprompt">gap></span> <span class="GAPinput">Xn := XModAlgebraByIdeal( An, Bn ); </ [ Bn -> An ] <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Xn, "Xn" ); </span> </pre></div> <p><a id="X7CD49D5A84FB05FC" name="X7CD49D5A84FB05FC"></a></p> <h5>4.1-3 AugmentationXMod</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Au <p>As a special case of the previous operation, the attribute <code class="code">AugmentationXM <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Ak4 := GroupRing( GF(5), DihedralGr <algebra-with-one over GF(5), with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">Size( Ak4 );</span> 625 <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Ak4, "GF5[k4]" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IAk4 := AugmentationIdeal( Ak4 );</s <two-sided ideal in GF5[k4], (2 generators)> <span class="GAPprompt">gap></span> <span class="GAPinput">Size( IAk4 );</span> 125 <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( IAk4, "I(GF5[k4])" );</spa <span class="GAPprompt">gap></span> <span class="GAPinput">XIAk4 := XModAlgebraByIdeal( Ak4, I [ I(GF5[k4]) -> GF5[k4] ] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( XIAk4 );</span> Crossed module [I(GF5[k4])->GF5[k4]] :- : Source algebra I(GF5[k4]) has generators: [ (Z(5)^2)*<identity> of ...+(Z(5)^0)*f1, (Z(5)^2)*<identity> of ...+(Z(5)^ 0)*f2 ] : Range algebra GF5[k4] has generators: [ (Z(5)^0)*<identity> of ..., (Z(5)^0)*f1, (Z(5)^0)*f2 ] : Boundary homomorphism maps source generators to: [ (Z(5)^2)*<identity> of ...+(Z(5)^0)*f1, (Z(5)^2)*<identity> of ...+(Z(5)^ 0)*f2 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Size2d( XIAk4 );</span> [ 125, 625 ] </pre></div> <p><a id="X83C5D98B828DD1DD" name="X83C5D98B828DD1DD"></a></p> <h5>4.1-4 Source</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ So <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bo <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>These four attributes are used in the construction of a crossed module <span class="SimpleMat <ul> <li><p><code class="code">Source(X)</code> and <code class="code">Range(X)</code> are the <em>source</ </li> <li><p><code class="code">Boundary(X)</code> is the boundary map of the crossed module <span class=" </li> <li><p><code class="code">XModAlgebraAction(X)</code> is the action used in the crossed module. Th </li> </ul> <p>The following standard <strong class="pkg">GAP</strong> operations have special <strong class <ul> <li><p><code class="code">Display(X)</code> is used to list the components of <span class="SimpleMa </li> <li><p><code class="code">Size2d(X)</code> for a crossed module <span class="SimpleMath">\(\mathc </li> <li><p><code class="code">Dimension(X)</code> for a crossed module <span class="SimpleMath">\(\ma </li> <li><p><code class="code">Name(X)</code> is used for giving a name to the crossed module <span class=" </li> </ul> <p>In the following example, we construct a crossed module by using the algebra <span class="Simp <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">RepresentationsOfObject( XIAk4 );< [ "IsComponentObjectRep", "IsAttributeStoringRep", "IsPreXModAlgebraObj" ] <span class="GAPprompt">gap></span> <span class="GAPinput">KnownPropertiesOfObject( XIAk4 );< [ "CanEasilyCompareElements", "CanEasilySortElements", "IsDuplicateFree", "IsLeftActedOnByDivisionRing", "IsAdditivelyCommutative", "IsLDistributive", "IsRDistributive", "IsPreXModDomain", "Is2dAlgebraObject", "IsPreXModAlgebra", "IsXModAlgebra" ] <span class="GAPprompt">gap></span> <span class="GAPinput">KnownAttributesOfObject( XIAk4 );< [ "Name", "LeftActingDomain", "Range", "Source", "Boundary", "Size2d", "XModAlgebraAction" ] </pre></div> <p><a id="X814C89667B997FE8" name="X814C89667B997FE8"></a></p> <h5>4.1-5 XModAlgebraByMultiplierAlgebra</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>When <span class="SimpleMath">\(A\)</span> is an algebra with multiplier algebra <span class=" <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">XAn := XModAlgebraByMultiplierAlg [ An -> <algebra of dimension 3 over GF(5)> ] <span class="GAPprompt">gap></span> <span class="GAPinput">XModAlgebraAction( XAn );</span> IdentityMapping( <algebra of dimension 3 over GF(5)> ) </pre></div> <p><a id="X824D4DD88323EF70" name="X824D4DD88323EF70"></a></p> <h5>4.1-6 XModAlgebraBySurjection</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>Let <span class="SimpleMath">\(\partial : S\rightarrow R\)</span> be a surjective algebra hom <p>Continuing with the example in that section,</p> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">X2 := XModAlgebraBySurjection( nat <span class="GAPprompt">gap></span> <span class="GAPinput">Display( X2 ); </span> Crossed module [A2->Q2] :- : Source algebra A2 has generators: [ [ [ 0, 1, 2, 3 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ] ] ] : Range algebra Q2 has generators: [ v.1, v.2 ] : Boundary homomorphism maps source generators to: [ v.1 ] </pre></div> <p><a id="X7AB09A158148290F" name="X7AB09A158148290F"></a></p> <h5>4.1-7 XModAlgebraByBoundaryAndAction</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <p>When a suitable pair of algebra homomorphisms are available, these operations may be used. Th <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">bdy3 := AlgebraHomomorphismByImag [ (1)*(1,2,3) ] -> [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">X3 := XModAlgebraByBoundaryAndAct [ GR(c3) -> A3 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( X3 );</span> Crossed module [GR(c3) -> A3] :- : Source algebra GR(c3) has generators: [ (1)*(), (1)*(1,2,3) ] : Range algebra A3 has generators: [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ] : Boundary homomorphism maps source generators to: [ [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ], [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ] </pre></div> <p><a id="X78400B837A2C8FB9" name="X78400B837A2C8FB9"></a></p> <h5>4.1-8 XModAlgebraByModule</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <p>Let <span class="SimpleMath">\(M\)</span> be an <span class="SimpleMath">\(A\)</span>-module. <p>Conversely, given a crossed module <span class="SimpleMath">\(\mathcal{X} = (\partial :M\r <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Y3 := XModAlgebraByModule( A3, M3 );< [ A(M3) -> A3 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Image( XModAlgebraAction( Y3 ), m3 ) true <span class="GAPprompt">gap></span> <span class="GAPinput">Display( Y3 );</span> Crossed module [A(M3) -> A3] :- : Source algebra A(M3) has generators: [ [[ 1, 0, 0 ]], [[ 0, 1, 0 ]], [[ 0, 0, 1 ]] ] : Range algebra A3 has generators: [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ] ] : Boundary homomorphism maps source generators to: [ [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ], [ [ 0, 0, 0 ], [ 0, 0, 0 ], [ 0, 0, 0 ] ] ] </pre></div> <p><a id="X7CA213397B334CBC" name="X7CA213397B334CBC"></a></p> <h5>4.1-9 SubXModAlgebra</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Su <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>A crossed module <span class="SimpleMath">\(\mathcal{X}^{\prime } = (\partial ^{\prime }:S <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">e4 := Elements( IAk4 )[4];</span> (Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 <span class="GAPprompt">gap></span> <span class="GAPinput">Je4 := Ideal( IAk4, [e4] );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Je4, "<e4>" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">Ke4 := Subalgebra( Ak4, [e4] );;</spa <span class="GAPprompt">gap></span> <span class="GAPinput">[ Size( Je4 ), Size( Ke4 ) ];</span> [ 5, 5 ] <span class="GAPprompt">gap></span> <span class="GAPinput">Se4 := SubXModAlgebra( XIAk4, Je4, K [ <e4> -> <algebra of dimension 1 over GF(5)> ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubXModAlgebra( XIAk4, Se4 ); </sp true <span class="GAPprompt">gap></span> <span class="GAPinput">Display( Se4 );</span> Crossed module [<e4> -> ..] :- : Source algebra <e4> has generators: [ (Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ] : Range algebra has generators: [ (Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ] : Boundary homomorphism maps source generators to: [ (Z(5)^0)*<identity> of ...+(Z(5)^0)*f1+(Z(5)^2)*f2+(Z(5)^2)*f1*f2 ] </pre></div> <p><a id="X866610BC7C41E1EC" name="X866610BC7C41E1EC"></a></p> <h4>4.2 <span class="Heading">(Pre-)Crossed Module Morphisms</span></h4> <p>Let <span class="SimpleMath">\(\mathcal{X} = (\partial:S\rightarrow R)\)</span> and <span cl <p class="center">\[ \varphi \circ \partial = \partial ^{\prime } \circ \theta, \qquad \theta (r\cdot s)=\varphi(r) \cdot \theta (s), \]</p> <p>for all <span class="SimpleMath">\(r\in R\)</span>, <span class="SimpleMath">\(s\in S,\)</spa <p>The conditions can be thought as the commutativity of the following diagrams:</p> <p class="center">\[ \xymatrix@R=40pt@C=40pt{ S \ar[d]_{\partial} \ar[r]^{\theta} & S^{\prime } \ar[d]^{\partial^{\prime }} \\ R \ar[r]_{\varphi} & R^{\prime } } \ \ \ \ \xymatrix@R=40pt@C=40pt{ R \times S \ar[d] \ar[r]^{ \varphi \times \theta } & R^{\prime } \times S^{\prime } \ar[d] \\ S \ar[r]_{ \theta } & S^{\prime }. } \]</p> <p>In <strong class="pkg">GAP</strong> we define the morphisms between algebraic structures such <p><a id="X7D575D07810640EB" name="X7D575D07810640EB"></a></p> <h5>4.2-1 XModAlgebraMorphism</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Id <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Pr <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ XM <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ So <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Na <p>These operations construct crossed module homomorphisms, which may have the attributes lis <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">c4 := CyclicGroup( 4 );;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">Ac4 := GroupRing( GF(2), c4 );</span> <algebra-with-one over GF(2), with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Ac4, "GF2[c4]" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IAc4 := AugmentationIdeal( Ac4 );</s <two-sided ideal in GF2[c4], (dimension 3)> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( IAc4, "I(GF2[c4])" );</spa <span class="GAPprompt">gap></span> <span class="GAPinput">XIAc4 := XModAlgebra( Ac4, IAc4 );</s [ I(GF2[c4]) -> GF2[c4] ] <span class="GAPprompt">gap></span> <span class="GAPinput">Bk4 := GroupRing( GF(2), SmallGroup <algebra-with-one over GF(2), with 2 generators> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( Bk4, "GF2[k4]" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IBk4 := AugmentationIdeal( Bk4 );</s <two-sided ideal in GF2[k4], (dimension 3)> <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( IBk4, "I(GF2[k4])" );</spa <span class="GAPprompt">gap></span> <span class="GAPinput">XIBk4 := XModAlgebra( Bk4, IBk4 );</s [ I(GF2[k4]) -> GF2[k4] ] <span class="GAPprompt">gap></span> <span class="GAPinput">IAc4 = IBk4;</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">homIAIB := AllAlgebraHomomorphism <span class="GAPprompt">gap></span> <span class="GAPinput">theta := homIAIB[3];; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">homAB := AllAlgebraHomomorphisms( <span class="GAPprompt">gap></span> <span class="GAPinput">phi := homAB[7];; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">mor := XModAlgebraMorphism( XIAc4, [[I(GF2[c4])->GF2[c4]] => [I(GF2[k4])->GF2[k4]]] <span class="GAPprompt">gap></span> <span class="GAPinput">Display( mor );</span> Morphism of crossed modules :- : Source = [I(GF2[c4])->GF2[c4]] : Range = [I(GF2[k4])->GF2[k4]] : Source Homomorphism maps source generators to: [ <zero> of ..., (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^ 0)*f1*f2, (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^ 0)*f1*f2 ] : Range Homomorphism maps range generators to: [ (Z(2)^0)*<identity> of ..., (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2, (Z(2)^0)*<identity> of ... ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsTotal( mor );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">IsSingleValued( mor );</span> true </pre></div> <p><a id="X83F45A3E8554ABE9" name="X83F45A3E8554ABE9"></a></p> <h5>4.2-2 Kernel</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ke <p>Let <span class="SimpleMath">\((\theta,\varphi) : \mathcal{X} = (\partial : S \rightarrow R <p class="center">\[ \ker(\theta,\varphi) = (\partial| : \ker\theta \rightarrow \ker\varphi ) \]</p> <p>is called the <em>kernel</em> of <span class="SimpleMath">\((\theta,\varphi)\)</span>. Also, <sp <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">Xmor := Kernel( mor );</span> [ <algebra of dimension 2 over GF(2)> -> <algebra of dimension 2 over GF(2)> ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsXModAlgebra( Xmor );</span> true <span class="GAPprompt">gap></span> <span class="GAPinput">Size2d( Xmor );</span> [ 4, 4 ] <span class="GAPprompt">gap></span> <span class="GAPinput">IsSubXModAlgebra( XIAc4, Xmor );</s true </pre></div> <p><a id="X87F4D35A826599C6" name="X87F4D35A826599C6"></a></p> <h5>4.2-3 Image</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Im <p>Let <span class="SimpleMath">\((\theta,\varphi) : \mathcal{X} = (\partial : S \rightarrow R <p class="center">\[ \Im(\theta,\varphi) = (\partial^{\prime}| : \Im\theta \rightarrow \Im\varphi) \]</p> <p>is called the image of <span class="SimpleMath">\((\theta,\varphi)\)</span>. Further, <span c <p>In this package, the image of a crossed module homomorphism can be obtained by the command <code <p><a id="X7B7975DA7E870690" name="X7B7975DA7E870690"></a></p> <h5>4.2-4 SourceHom</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ So <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Ra <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Is <p>Let <span class="SimpleMath">\((\theta,\varphi)\)</span> be a homomorphism of crossed modul <p>The attributes <code class="code">SourceHom</code> and <code class="code">RangeHom</code> stor <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">ic4 := One( Ac4 );; </span> <span class="GAPprompt">gap></span> <span class="GAPinput">e1 := ic4*c4.1 + ic4*c4.2;</span> (Z(2)^0)*f1+(Z(2)^0)*f2 <span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( theta, e1 ); </span> (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 <span class="GAPprompt">gap></span> <span class="GAPinput">e2 := ic4*c4.1;</span> (Z(2)^0)*f1 <span class="GAPprompt">gap></span> <span class="GAPinput">ImageElm( phi, e2 );</span> (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 <span class="GAPprompt">gap></span> <span class="GAPinput">IsInjective( mor );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">IsSurjective( mor );</span> false <span class="GAPprompt">gap></span> <span class="GAPinput">immor := ImagesSource2Dimensional <span class="GAPprompt">gap></span> <span class="GAPinput">Display( immor );</span> Crossed module [..->..] :- : Source algebra has generators: [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ] : Range algebra has generators: [ (Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2, (Z(2)^0)*<identity> of ... ] : Boundary homomorphism maps source generators to: [ (Z(2)^0)*<identity> of ...+(Z(2)^0)*f1+(Z(2)^0)*f2+(Z(2)^0)*f1*f2 ] </pre></div> <div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0_mj.html">Top< <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </body> </html> [ zur Elbe Produktseite wechseln0.17Quellennavigators Analyse erneut starten ] |
2026-03-28