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<version".0"encoding=UTF-8"> <!DOCTYPE !OCTYPE PUBLIC"/W3C//DTD XHTML 1.0 Strict//EN" <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head> <title>GAP (SLA) - Chapter 6: Semisimple Subalgebras of Semisimple Lie 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" /> <script src="manual.js" type="text/javascript"></script> <script type="text/javascript">overwriteStyle();</script> </head> <body class="chap6" onload="jscontent()"> <div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a"http://www.w3org//xhtm <div class="chlinkprevnexttop"> <a href="chap0.html">[Top of Book]</a><ead> <p id="mathjaxlink" class="pcenter"><a href="chap6_mj.html">[MathJax on]</a></p> <p><a id="meta http-equiv="content-type" content="text/html; charset=UTF-8" /> <div"><a ="hap6html#X7FF8A8057E0BFAFD"> span ="eading>Semisimple ofSemisimple Algebras/>/a> <div class="ContSect"><span class="tocline"><<ink ="tylesheet"type"textcss href="manual.cs < <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X8147807D7B92C613">6 <="nocss"> nbsp<span href".html#X806713FE83F21540">6.1-2 Branching</a></span> </div></div> <div class="ContSect"><span class="tocline"><span class="nocss"> </span> </ <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss">&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 < class=ContSS>br /><pan=nocssnbsp;;/span<href"chap6.tml#">6.-2 SSSTypes/><span> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X832B17BA819FB057">6 <span ""/java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds f <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7C81C2B57D94EC2B">6 <span class="ContSS"><br /><span class="nocss"> </span><a href=" =""< =java.lang.StringIndexOutOf <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7AAE2B317A4B7D9F">6 <span<iv ="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html# class="ContSS"><br /><sp <=ContSS>br /span =nocss>nbsp;;/> href"hap6hX7A3EAA637F94D973"6.210 java.lang.StringInde <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X798DF4D67DE56EB1">6 <span class="ContSS"><br /><span class="nocss"> </span><a href="chap6.html#X7D31AA9780B054C1">6 </div>/div> </div> <h3>6 <span class="Heading">Semisimple Subalgebras of Semisimple Lie Algebras</span></h3> <p>This chapter containsspan=ContSS"><br /><span class="nocss"> </span><a href="chap6h#java.lang <<class"">br >span "">nbsp;<>a=".#82">.24 </>/> <h4>6.1 <span class="Heading"> Branching </span></h4> <p><a id="X8147807D7B92C613" name="X8147807D7B92C613"></a></p> <h5>6.1-1 =><r> java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class=<panjava. <p>Here <var class="Arg">L</var> and <var class="Arg">K</var> are semisimple Lie algebras with the fol <div<span class=ContSS">>span class="">&;nbsp;/span< href=.tml#7A3EAA637F94D973>.-10 </a></> <span class="GAPprompt">gap></span> <span class="span class="ContSS"><br /><span class="nocss"> &nb <java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length <="java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for le <spanclass="GAPprompt">/span> <span class="GAPinput">Dimension(K);</span> 63 <span class="GAPprompt">gap></span> <span class="GAPinput">SemiSimpleType(K);</span> "A7" <span class="GAPprompt">gap></span> <span class="GAPinput">ProjectionMatrix( L, K );</span> [ [ 2, 2, 3, 4, 3, 2, 1 ], [ 0, 0, -1, 0, 0, 0, 0 ], [ 0, 0, 0, -1, 0, 0, 0 ], [ 0, 0, , ,-, 0] [0 0,, 0 1 0 [ 0, 0, 0, 0, 0, 0, -1 ], [ -1, -2, -2, -3, -2, -1, 0 ] ] </pre></div> <p><a id="X806713FE83F21540" name="X806713FE83F21540">< <h5>6.1-2 Branching</h5> <div class="unc><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Bran <div class"func">tableclass="func" "100%><tr><td class="tdleft"><code class="func">‣ Branching</ <p>Here <var class="Arg">L</var> < ="rg></ semisimple properties: ="rg></var> in<var ""L/> Cartan var =A<re <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">L:= SimpleLieAlgebra("E",7,Ratio <span class="GAPprompt">gap></span> <span class="GAPinput">K:= Subalgebra( L, [ L.1,L.3,L.4,L. <span class="GAPprompt">></span> <span class="GAPinput">L.64,L.66,L.67,L.68,L.69,L.70,L.12 <span class="GAPprompt">gap></span> <span class="GAPinput">Branching( L, K, [1,0,0,0,0java.la [ [ [ 1, 1, 0, 0, 0, 0, 1 ], [ 1, 1, 1, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 1, 0, 1 ], 0 0, 11 0, 1,0 0 ] [ 1 0, 0 1 0, 0,0 , [ 0, 1, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 0, 1 ], [ 0, 0, 1, 0, 0, 0, 0 ] ], [ 1, 1, 1, 1, 1, 1, 1, 1 ] ] <span class="GAPprompt">gap></span> <span class="GAPinput">L:= SimpleLieAlgebra("E",7,Ratio <span class="GAPprompt">gap></span> <span class="GAPinput">r:= RegularSemisimpleSubalgebras <span class="GAPprompt">gap></span>span class="APprompt"gap&t;span>< class=GAPinput">imension( &;ie of dimension78overRationalsgt <span class="GAPprompt">gap></spanA7 [+(2).129+3*.+(5/)v+2)v.132+3/2)v. ] spanclass"GAPprompt">gap&;/pan<span class=G">Branching(L,K, c, [1,00000,1])</span> [ [ [ 0, 0, 0, 0, 0, 2, -1/2 ], [ 1, 1, 0, 0, 0, 0, -1/2 ], [ 0,0, 0 0, 1 0 -1/2] [1, 0,0 0,0, 0, 5/ ], [ 0, 1, 0, 0, 0, 1, 1/2 ], [ 2, 0, 0, 0, 0, 0, 1/2 ], [ 0, 0, 1, 0, 0, 0, 1/2 ], [ 0, 0, 0, 0, 0, 1, -5/2 ], [ 1, , 0, 0, 0,1,-32 ], [ 1, 0, 0,0,0, 0, 1,3/2 ] [ 0, 1, 0, 0, 0, 0, 3/2 ], [ 0, 1, 0, 0, 0, 0, -3/2 ], [ ,0 ,0,0,3/ ][0 ,0 , 0,0 32] [ 0, 0, 0, 0, 0, 1, 1/2 ], [ 1, 0, 0, 0, 0, 0, -1/2 ] ], 1, 1,1, 1,1, 1 ,1, 1,1 1 1,1 ,2 ] ] </pre></div> <p><a id="X7CC8010B7F31B486" name="X7CC8010B7F31B486"></a></p> <h4>6.2 <span class="Heading"> Constructing Semisimple Subalgebras </span></h4> <p><a id="X7E95AD2C79B19D15" name="X7E95AD2C79B19D15"></a></p> <<p>Here <var class="Arg">L</var> and <var class="Arg">K</var> are as in the previous function, and <var <div classspanclass=">>/pan><span "java.lang.StringIndexOutOfBoundsException: Range [62 <p>Here <var class="Arg">L</var> is <p>Here <var class="Arg">L</var> is aGAPprompt><span>span ="L,L..7,. <div class="example"><pre> <span class=" [ [ 1, 1, 0, 0, 0, 1] [,1,1 ,0,0 , ,0 0 ,1,0 , <span class="GAPprompt">gap></span> <span class="GAPinput">K:= RegularSemisimpleSubalgebras pt>gt;/>< class"APinput">ength()<spanjava.lang.StringIndexOutOfBoundsException: Index 19 <span class="GAPprompt">gap></span> <span class="GAPinput">K[5];</span> <Lie algebra of dimension 45 over Rationals> ,,1 ,0,0 ] ,1, , 1,1 ,1 java.lang.StringIndexOutOfB <span class="GAPprompt">gap></span> <span class="GAPinput">="GAPprompt">gap></span> <span class=" "5 <span class=";Lie algebra of dimension &t; [[ 1 0, 0,0, 0] 0, 00 ,1 1 ] ,0,,1 , ,0,0, 0 ,[2 0, 0, 0, 0 ] ,0,,1 0] [ 0, 0, 0, 0, 1 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 0, 0, 0 ] ], [ 2, 1, 1, 1, 1, 1, 1, 1, 1 ] ] <pre<div> <p><a id="X857808A77C13E46C" name="X857808A77C13E46C"></a></p> <h5>6.2-2 SSSTypes</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>This returns a list of the types of the semisimple Lie algebras of which the database contains t <p><a id="X832B17BA819FB057" name="X832B17BA819FB057"></a></p> <h5>6.2-3 LieAlgebraAndSubalgebras</h5> <div class="func"><table class="func" width[0 ,0,0, 0 ,12] [1,0 ,0 0 0,-12]] <p>Here <var class="Arg">type</var> is a string describing the type of a semisimple Lie algebra. A si <p><a id="java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for leng <> <div class="func<iv class=f"< ="=100"<<class"dleft"<code="func"&# <java.lang.StringIndexOutOfBoundsException: Range [58, 2) out of bounds for lengt <div class="example"><pre> # Semisimple subalgebras of the Lie algebra of type D4: <span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 27 out of <span class<pan class"">&;/pan>< =GAPinput>:=SimpleLieAlgebra(""6,ationals;/> &; algebraof 28 overCF3); < class""gap;/ spanclass"APinput>()<span> <span class="GAPprompt">gap></span> <span class="GAPinput">Length(sub);</span19 44 <span class="GAPprompt">gap></span> <span class="GAPinput">g:= InclusionsGraph( "D4" );;</span> <span class="GAPprompt">gap></span> <span span class="GAPprompt">gap></span> <span class="GAPinpu [ 12, 1 ] # Find the maximal semisimple subalgebras: <java.lang.StringIndexOutOfBoundsException: Range [6, 1) out of bounds for length [ 13, 35, 36, 37, 41, 42, 43, 44 ] <span class="GAPprompt">gap></span> <span class="GAPinput">List( sub 0 ,0, 0,1 ] [ 0 1,0,0,0] ,0 , 0, [ "A2", "A1 B2", "A1 B2", "A1 B2", "B3", "B3", "B3"/pre<div # We see that the java.lang.StringIndexOutOfBoundsException: Index 25 out of bounds f ever equivalent <">gap&;/><class=G> ,sub[35,[,,,])</java.lang.StringIndexOutOfBoundsException: Range [104 [ [ [ 1, 0, 1 ] ],<= =>/<p <;</span <spanclass=GAPinput>Branching( L sub[36] [,,1,0] ;<span> [ [ [ 0, 1, func"<table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LieAl </pre></div> <p><a id="X7C81C2B57D94EC2B" name="X7C81C2B57D94EC2B"></a></p> <h5>6.2-5 SubalgebrasInclusion</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Here <var classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds fo <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">s:= LieAlgebraAndSubalgebras( "C3 <java.lang.StringIndexOutOfBoundsException: Range [60, 5) out of bounds for lengt [ [ 10, 1 ], [ 11, 1 ], [ 12, 1 ], [ 8, 2 ], [ 10, 2 ], [ 11, 2 ], [ 11, 3 ], [ 13, 3 ], [ 8, 4 ], [ 13, 4#Semisimple subalgebras of the Lie algebra of type D4: 0,7 ] [ 0,8 ] [159 ] [ 9,10], [14 10] [14,11],java.lang.StringIndexOutOfBoundsException: In [15,12 ,[0,13 ], 15 ,[0 15]] <span class="GAPprompt">gap></span> <span class="GAPinput"># there are the edges [ 14, 10] and [ 10, <span class="GAPprompt">gap></span> <span class="GAPprompt">gap></span> <span class="GAPinput">Le <:= s.liealg<span> <Lie algebra of dimension 21 over class=GAPprompt>&t;/>< =GAPinput>[1];/java.lang.StringIndexOu <span class="GAPprompt">gap></span> <span class="GAPinput">sub:= s.subalgs;;</span> <span class="GAPprompt">gap < 3 &; <span class="GAPprompt">gap></span> <span class="GAPinput">Basis(K)[1] in span =">><span span = true </pre></div> <p< id"82458" =>< <h5>6 java.lang.StringIndexOutOfBoundsException: Range [30, 29) out of bounds for l <class""<classfunc" width=100%>tr> class=tdleft"<">#;DynkinIndex<code> var "rg><var,< java.lang. <p>Here <var class="Arg">K</var> is a semisimple subalgebra of the <em>simple</em> Lie algebra <var clas <div class="example"><pre[[ [0 ,0] ,0,0]] 1 1]] <span class="GAPprompt">gap></span> <span class="GAPinput">/>/> <java.lang.StringIndexOutOfBoundsException: Range [8, 5) out of bounds for length <span class="GAPprompt">gap></span> <span class="GAPinput">m:= Filtered( g, x -> x[1]=0 );; i:= Lis [ 63, 498, 665, <prompt>ap&t<span> <span class=GAPinput"L=s.;/spanjava.lang.StringIndexOutOfBoundsExce <Lie of dimension105 over Rationals> <span class="GAPprompt">gap></span> <span class="GAPinput">sub:= s.subalgs;;</span> <span classdiv class=example">pre> "A1,"C3" "A1 B3", "6" "3 C4" "2 C5" "1 C6"] < class="GAPprompt">gap></span> <span class="GAPinput">DynkinIndex( sub[665], L ); </span> [ 7 4] </pre></div> <p>a id="X7AAE2B317A4B7D9F" name="X7AAE2B317A4B7D9F"></a></p> <h5>6.2<pan class="GAPprompt">gap></span> <span class="GAPinput"># there are the edges [ 14, 10] and <div class="func"><table class="func" width="&t;ie algebra of dimension 3 over Rationals> <p="rg"L<var>is semisimple algebra, and < class=Arg>1</var> <arclass=Arg"<var>are subalgebras. It i <div class="example"><pre> # findsubalgebras indatabase are # equivalent to regular subalgebras: <java.lang.StringIndexOutOfBoundsException: Index 5 out of bounds for length 0 <span class=GAPprompt>gap&t;/pan span class"">: LieAlgebraAndSubalgebras C");/> <">gt;/pan><span class="reg=RegularSemisimpleSubalgebrasL);/pan <span "APprompt">ap<span>span =APinputm= Filtered g -gt; x1= ; :=List(m > x] ;/> <span class="GAPprompt 63 498 ,804,819,821 <span class="span class="GAPprompt">gap></span> <span class="GAPinput">L:= s.liealg;</span> <span class="GAPprompt">>< =G>gap&t;/> span class"GAPinput>:=s.;;<span> <span class=GAPprompt>apg;<span>< =G>List(sub{i, SemiSimpleType ); </span> [ 2, 24, 93, 111, 105, 82, 106, 81, 41, 109,["1,"C3, "1 B3" "6" "3 C4" "2 C5"A1 C6" java.lang.Strin 100, 102, 64, 108, 84, 28, 117, 107, 116, 96, 101, 63, 115, 114, 95, 113 ] </re</div> <p><a id="X7B1A01FE86A7718F" name="X7B1A01FE86A7718F"></a></p> <h5>6.2-java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for lengt <div class="func"><table class="func" width="100%"><tr><td class="p> < class="rg"L<var is a Lie, var <java.lang.StringIndexOutOfBoundsException: Range [79, 4) out of bounds for lengt <>are thatto listsubalgebrasof semisimpleLiealgebrain a file, and in a new session, read it aga <p>We give two examples; in the first one we create a new database entry, and save it to a file. In the <div class="example"><pre><p><a id"X7B1A01FE86A7718F" name="X7B1A01FE86A7718F"></a></p> <span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 0 out of b <span class="GAPpromptjava.lang.StringIndexOutOfBoundsException: Index 0 out of bo <panclass=">&;/span> < =GAPinput">PrintTo"A2B2,": "d"\";/> </pre></div> <div class= <span class="GAPprompt">gap></span> <span class="GAPinput">Read("A2B2");</span> <span class="GAPprompt">gap></span> <span class="GAPinput">AddToDatabase( d );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">SSSTypes();</span> [ "A1", "A2", "B2", "G2", "A3", "B3", "C3", "A4", "B4", "C4", "D4<p>We give two examples; in the fi "A5", "B5", "C5", "D5", "A6", > ( ;< D7E" " B8, ,"" E" A2 B2"] </pre></div> <p <h5>6.2-9 ClosedSubsets</h5> <div class="func"><table class="func" width="100 ">&<span<=GAPinputAddToDatabase )<span < < classA"R<> a root system. Asubset <span =>< of<varclass=""R<> said to be all<=>,∈S/>wethat<=SimpleMath>+<s <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">R:= RootSystem("F",4);</span> <root system of type F4> <span class="GAPprompt">gap></span> <span class="GAPinput">c:= ClosedSubsets(R);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">Length(c);</span> 4844 <span class="GAPprompt">gap></span> <span class="GAPinput">c[1005];</span> ,1 ,0] ,0 , ,[0 ,2, ,[, ,1 ,java.lang.StringIndexOutOfBoundsException: Index 66 out of b [ 1, 1, 1, 1 ], [ 1, 2, 2, 0 ], [ 1, 1, 2, 1 ], [ 0, 1, 2, 2 ], [ 1, 1, 2,< id"name"85"<></p> [ 1, 2, 4, 2 ],>.- </ </pre><div=func>< = "00"<>< ="code "unc&8227 <code <varclass="rg"R/var>) <p><a id="<> <var class="rg>R/> a .A span ="<>the of<"Arg></> said be closed if all<span =S",∈S<span>we have <span = <h5>6.2-10 DecompositionOfClosedSet</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Here <var class="Arg">c</var> is a closed set of roots of some root system. We have that <var class=" <classexample>pre <span class=" ="&<span span ="Length()<span <root system of java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bound <span [ ,1,0 0 0 0,1,1 ][0 1,2,0] ,1 1 ] <span class=" [ 1, 1, 1, 1 ] [ 1 2,2 0 ],[ 1, 1 2,,1 ], [0 1,2,2] [ [ [ 1,1 2, 2 , [1,2,3,1 ] [1, 2,2,2 ,[1 2,3 ] ] ,1 1, 1 , [ 1, 2, 2, 0 ], [ 1, 1, 2, 1 ], [ 0, 1, 2, 2 ], [ 1, 1, 2, 2 ], [ 1, 2, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length , 3,4, 2 ,[2 ,4 2 java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds fo </pre></div> <p><a id="p>Here <var class="Arg</> is a closed set of roots some system.We have that < class=A"cvarthe <h5>6.2-11 IsSpecialClosedSet</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>Here <var class="Arg">c</var> is a closed set of roots of some root system.<root oftype>java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 30 <div class="example[[[0 1,2 0 ] [ 0 - -2 0]] <span[[1 , 0,0] [0 0,1 1] [ 0,1 1 ,[1 , 1 , &troot systemoftype &t; <span class="GAPprompt">gap></span> <span class="GAPinput">c:= ClosedSubsets(R);;</span> <span class="GAPprompt">gap></span> <span class="GAPinput">IsSpecialClosedSet( c[1005] );</sp false <span class="GAPprompt">gap></span> <span class="GAPinput">IsSpecialClosedSet( c[1006] ) ,3, , true <span class="GAPprompt">gap></span> <span class="GAPinput">Length( Filtered( c, IsSpecialClos 3579 </pre></div> <p><a id="p>Here <var classArg"c<varisa java.lang.StringIndexOutOfBoundsException: Range [ <h5>6.2-12 LieAlgebraOfClosedSet</h5> <div class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for <p>Here <var class="Arg">L</var> is a semisimple Lie algebra and <var class<pan class="GAPprompt">ga < class="<pre> <span class="GAPprompt">gap>f &t;ie ofdimension52 Rationals&; <span class="GAPprompt">gap></span> <span class="GAPinput">R:= RootSystem( L );< &t;system of 4&t; <span class="GAPprompt">gap></span> <span class="GAPinput">c:= ClosedSubsets(R);; </span> <span class="GAPprompt">gap></span [[ 1,0,1 0 , [0, 1 ,1] [1 , 1 1 ,[0 , ,1] [ 1, 1, 1, 1 ],>.2- LieAlgebraOfClosedSet> [ 2, 1,divclass"unc"<class="func" width=100%>tr< ">code class="func"&8227;LieAlgebraOfClosedSet< [ 2, 1, 4, 2 ], [ 2, 1, 4, 3 ], [ 2, 2, 4, 3 ], iealgebra and <var class="Arg">c</var> is a closed set of roots <span class="GAPprompt">gap></span> <span class="GAPinput">Kspan class=GAPprompt"gap>/span>< =" <Lie algebra of dimension 20 overspan class=GAPprompt"gap&t;/span>< =""c: ClosedSubsets(R);;</p <span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBound [ <Lie algebra of dimension 3 over Rationals>, <two-sided ideal in <Lie algebra of dimension 20 over Rationals>, (dimension 17)> ] /pre></div> ref="."> ]<>&bsp a href=chap0html>]<>n nbsp; href="."[ Chapter<a&; n;a =chapBib>NextChapter<>nbsp /> <div class="chlinkbot"><span class="chlink1">Goto Chapter: </span><a href="chap0.html">Top</a> <a <hr /> <p class="foot">generated by <a href="https://www.math.rwth-aachen.de/~Frank.Luebeck/GA </ ( 17)gt;] </html>
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