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<p><a id="X80D10A5D7D73D871" name="X80D10A5D7D73D871"></a></p>
<div class="ChapSects"><a href="chap5_mj.html#X80D10A5D7D73D871">5 <span class="Heading">Finite Order Automorphisms and <span class="SimpleMath">\(\theta\)</span>-Groups</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X8173135A7D187358">5.1 <span class="Heading"> The functions </span></a>
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X856FF71D8565C1E5">5.1-1 FiniteOrderInnerAutomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7E8CCC4885E2A443">5.1-2 FiniteOrderOuterAutomorphisms</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X84F59A2687C62763">5.1-3 Order</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8635ABCD7D5ACED8">5.1-4 KacDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7DCA2568870A2D34">5.1-5 Grading</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D9BDD03811B3C98">5.1-6 NilpotentOrbitsOfThetaRepresentation</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7D4106C37DBD0943">5.1-7 ClosureDiagram</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X8108B6487A3A363B">5.1-8 CarrierAlgebra</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap5_mj.html#X7A8D86667BC7C033">5.1-9 CartanSubspace</a></span>
</div></div>
</div>

<h3>5 <span class="Heading">Finite Order Automorphisms and <span class="SimpleMath">\(\theta\)</span>-Groups</span></h3>

<p>This chapter contains functions for creating and working with finite order automorphisms of simple Lie algebras (or, more precisely, representatives of the conjugacy classes of such automorphisms).</p>

<p>NB: such automorphisms are not created for a given Lie algebra, but the Lie algebra is constructed at the same time as the automorphism. This because the base field may need extending (it needs enough roots of unity).</p>

<p>As noted above the functions give representatives of the conjugacy classes, in the automorphism group of the underlying Lie algebra, of finite order automorphisms. Such conjugacy classes are classified in terms of Kac diagrams. Roughly, this works as follows. A finite order automorphism <span class="SimpleMath">\(f\)</span> corresponds to a diagram automorphism of order <span class="SimpleMath">\(d=1,2,3\)</span>. The inner automorphisms correspond to a diagram automorphism of order 1, the outer automorphisms to a diagram automorphism of order 2 or 3. Let <span class="SimpleMath">\(L_0, L_1\)</span> denote the eigenspaces of the underlying Lie algebra <span class="SimpleMath">\(L\)</span>, with respect to the diagram automorphism, respectively corresponding to the eigenvalues 1 and <span class="SimpleMath">\(w\)</span> (where <span class="SimpleMath">\(w\)</span> is a primitive <span class="SimpleMath">\(d\)</span>-th root of unity). (In case of <span class="SimpleMath">\(d=1\)</span>, we have <span class="SimpleMath">\(L_0=L\)</span>, <span class="SimpleMath">\(L_1=0\)</span>.) Then <span class="SimpleMath">\(L_0\)</span> is semisimple and we choose a set of canonical generators of <span class="SimpleMath">\(L_0\)</span>, denoted <span class="SimpleMath">\(x_i\)</span>, <span class="SimpleMath">\(y_i\)</span>, <span class="SimpleMath">\(h_i\)</span>, for <span class="SimpleMath">\(i=1,\ldots,s\)</span>. Moreover, <span class="SimpleMath">\(L_1\)</span> is an <span class="SimpleMath">\(L_0\)</span>-module. Let <span class="SimpleMath">\(x_0\)</span> be the lowest weight vector in <span class="SimpleMath">\(L_1\)</span>. (If <span class="SimpleMath">\(d=1\)</span> then <span class="SimpleMath">\(x_0\)</span> will be the lowest (negative) root vector.) Let <span class="SimpleMath">\(\alpha_i\)</span> for <span class="SimpleMath">\(i=0,\ldots,s\)</span> be the roots corresponding to <span class="SimpleMath">\(x_i\)</span>, with respect to the subalgebra spanned by the <span class="SimpleMath">\(h_i\)</span>. Let <span class="SimpleMath">\(C\)</span> be the Cartan matrix of these roots. The rows of <span class="SimpleMath">\(C\)</span> are linearly dependent. The Dynkin diagram of <span class="SimpleMath">\(C\)</span> is labeled with integers <span class="SimpleMath">\(a_i\)</span> with greatest common divisor 1, that form the coefficients of a linear dependency of the rows of <span class="SimpleMath">\(C\)</span>. Furthermore, the <span class="SimpleMath">\(x_i\)</span> generate <span class="SimpleMath">\(L\)</span> and the automorphism <span class="SimpleMath">\(f\)</span> is described by <span class="SimpleMath">\(f(x_i) = v^{s_i} x_i \)</span>, where the non-negative integers <span class="SimpleMath">\(s_i\)</span> have greatest common divisor 1, and are such that <span class="SimpleMath">\(m=d\sum a_i s_i\)</span> is the order of <span class="SimpleMath">\(f\)</span>, and where <span class="SimpleMath">\(v\)</span> is a primitive <span class="SimpleMath">\(m\)</span>-th order root of unity. Now the Kac diagram of the automorphism <span class="SimpleMath">\(f\)</span> is the Dynkin diagram of <span class="SimpleMath">\(C\)</span>, labelled with the labels <span class="SimpleMath">\(s_i\)</span>.</p>

<p><a id v.14v.5+38 -)*.41-1)42,v+v.34+v.35 java.lang.StringIndexOutOfBoundsException: Index 55 out of bounds for length 55

<h4>5.1 <span class="Heading"> The functions </span></h4>

<p><a id="X856FF71D8565C1E5" name="X856FF71D8565C1E5"></a></p>

<h5>5.1-1 FiniteOrderInnerAutomorphisms</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FiniteOrderInnerAutomorphisms</code>( <var class="Arg">type</var>, <var class="Arg">rank</var>, <var class="Arg">m</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <span class="SimpleMath">\(L\)</span> be the simple Lie algebra of type <var class="Arg">type</var> and rank <var class="Arg">rank</var>. The function returns representatives of the conjugacy classes of inner automorphisms of<span class="APprompt">apgt</span> <span class="GAPinput">NilpotentOrbitsOfThetaRepresentation( L, [0,1,0,0,0,0] );</span>


<div class="example"><pre>
<span class=" [ v.65+v.66+v.672*.+3)v74()v.5+()v.+(4)*v.77()v., 
[ [ v.72, v.1, v.2, v.3, v.4, v.5, v.6 ] -> [ (E(3))*v.72, (E(3)^2)*v.1, v.2,       v.+v.+.31] 
      .3,v., v., v. ],[ v., v.1,v., v.,v., v.5, v. ]-&t 
    [ v.72, (E(3))*v.1, (E(3))*v.2, v.3, v.4, v.5, v.6 ], 
  [      19+30]
      v.[v.66v70,2)v.+2*.74()v.75()v.76+()v.+2*.78
    [ v.72, v.1, v.      .30+v.4 ][v.71, v.3+v.+2)*v.5+(3).76+2)77v. v.  ]
  [ v.72, </re>/>
      v.3, v.4
spanclass="APprompt>gap></span> <span class="GAPinput">Source(f[1]);</span>
<Lie algebra of dimension 78 over CF(3)>
</pre></div>

<p><a id="X7E8CCC4885E2A443" name="X7E8CCC4885E2A443"></a></

<h5>5.1-2 div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#8227; ClosureDiagram</code>( <var classp>Here var "">/>is anautomorphismof  simpleLiealgebra<pan ="SimpleMath">\(L\)</span> of order <span class="SimpleMath">\(m\)</span>, and <var class="Arg">s</var> a list of <span class="SimpleMath">

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; FiniteOrderOuterAutomorphisms</code>( <var class="Arg">type</var>, <var class="Arg">rank</var>, <var class="Arg">m</var>, <var class="Arg">d</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Let <span class="SimpleMath">\(L\)</span> be the simple Lie algebra of type <var class="Arg">type</var> and rank <var class="Arg">rank</var>. The function returns representatives of the conjugacy classes of outer automorphisms of <span class="SimpleMath">\(L\)</span> of order <var class="Arg">m</var>, corresponding to a diagram automorphism of order <var class="Arg">d</var>.</p>

<p><a id="X84F59A2687C62763" name="X84F59A2687C62763"></a></p>

<h5>5.1-3 Order</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Order</code>( <var class="Arg">f</var> )</td><td class="tdright">(
<>< class="rg">f<var>is afinite orderautomorphism Thisreturnsits order<>

<p><a id="X8635ABCD7D5ACED8" name="X8635ABCD7D5ACED8"></a></p>

<h5>5.1-4 KacDiagram</h5>

<div class="func"><table class="func" width="100%"><tr><td class=
<p>="gt;span> span ="h=f8;<


<  Selected .
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= FiniteOrderOuterAutomorphisms( "A", 5, 4, 2 )java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 82
<span class="GAPprompt">gap></span> <span class="GAPinput">r:= KacDiagram(27
rec( 
  CM := [ [ 2, 0, -1, 0 ], [ 0, 2, -1, 0 ], [ -1, -1, 2, -1 ], 
      [ 0, 0, -2, 2 ] ], labels := [ 1, 1, 2, 1#IAll(-  proved!
<span class  ,1][ ,1 ,[, 2][4  ,[ 5  ,[ ,5 ] ,2]
[ 0, 0, 0, 0 ]
</>

<p><a id="X7DCA2568870A2D34" name="X7DCA2568870A2D34"></a></p>

<h5>5.1-5 Grading</h5>

< =func><able class""width"100"<trclass"tdleft">codeclass""&8227;Gradingcode var="/java.lang.StringIndexOutOfBoundsException: Range [141, 140) out of bounds for length 215
<>ere < class=Arg></var isa  order automorphism  order< =S>(m\)<span>. This returnsa  length <class"">\(m\)/>.  <pan ="impleMath">\i)/span- element containsa  ofeigenspaceof <class""f/var  eigenvalue <spanclass="SimpleMath"\v^\</>  <span =SimpleMath>()/pan>is ajava.lang.StringIndexOutOfBoundsException: Range [413, 412) out of bounds for length 510

<p><a id="X7D9BDD03811B3C98" name="X7D9BDD03811B3C98"></a></p>

<h5>5.1-6 NilpotentOrbitsOfThetaRepresentation</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; NilpotentOrbitsOfThetaRepresentation</code>( <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 270
p java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 1391

<p>It is possible to add an extra optional argument: <var class="Arg">method:= "Carrier"</var>, or <var class="Arg">method:= "WeylOrbit"</var>. Then a method based on finding carrier algebras (respectively, computing orbits under the Weyl group) is chosen. If no optional argument is chosen, then the system will make its own choice. (In the case of outer automorphisms, currently the only available method is the one based on orbits of the Weyl group.) The method based on carrier algebras tends to work better for the higher order automorphisms.</p>

<p>This function prints somespanclass""gapgt;><class"APinput">:NilpotentOrbits);/

<> theargumentversion the firstargument< class=Arg><var hastobeasemisimpleLie,andthesecond varclass="rg"dvar>a  ofnonnegative.Then< class="rg"></ar>is <span class=SimpleMath">\()/- by  the root space corresponding to the <span class="SimpleMath">\(i\)</span>-th simple root the degree <var class="Arg">d[i]</var>. Apart from this the function works the same in this case as in the one-argument version.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span#  All (non-inclusionsproved!
<panclass="GAPprompt">&t/>< class="GAPinput"ResetGlobalMersenneTwister1)<spanjava.lang.StringIndexOutOfBoundsException: Index 102 out of bounds for length 102
<span class="GAPprompt">gap&t<span> span class=GAPinput"f: ( "D,5,3 );   </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:= NilpotentOrbitsOfThetaRepresentation( f[2] : method:= "Carrier" );;</span>
#I  Selected carrier algebra method.
#I  Constructed 123 root bases of possible flat subalgebras, now checking them...
#I  Obtained 30 Cartan elements, weeding out equivalent copies...
<pan class"">gapgt<span <class"GAPinput">Length(s);</span>
10
<span class="GAPprompt">gap></span> <span class="GAPinput">s[4];</span>
[ v.14+p the version the second argument varclass=Arg"d<> a  list of non-negative Then <ar class="Arg"L</var>is <span ">(\/pan>graded bygiving  to spanjava.lang.StringIndexOutOfBoundsException: Range [243, 242) out of bounds for length 379
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= SimpleLieAlgebra("E",6,Rationals);;</span>
<span class="GAPprompt">gap></span> <div class==example>pre
# method
#I  Constructed a Weyl transversal =""gapgt;/>< class""h= f[4];/>
#I  Obtained 5 Cartan elements, weeding =G">ap&;/> span class=GAPinput"sl2=NilpotentOrbitsOfThetaRepresentation(  ; </pan>
[ [ v.65+v.66+v.67, (2)*v.73+(3)*v.74+(4)*v.75+(6)*v.76+(4)*v.77+(2)*v.78, 
      v.29+v.30+v.31 ], 
  +(4)*v.4+4*.+6)v.+(4)*v.77()v78 
      v.19+v.30 ],
 [.66v.70 2)*73()v.4+3*v.75+()v.76+3*77()v.78,
      v.30+v.34 ], [  52 overCF5)&;
</></div

<p><a rec( g0 := [ v.49+(2)*v.50+(2)*v.51+(3)*v.52, v.50+(1/2)*v.51+v.52 ], 

<gp :=   .9 v48][ . ] [v.9   )

<classf>table=f""100%">><=t"java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 274
<class"unc>table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ClosureDiagram</code>( <var class="Arg">L</var>, <var class="Arg">d</var>, <var class="Arg">s</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="<Lie algebra over CF5) with 10 generators>

<p>This span classGAPprompt>>/span> <spanclass"">emiSimpleTypeK );/>

<p>During/re>/div>

<p>In the second version, the p><a id="X7A8D86667BC7C033" name="></a></>

<<h5>5.- <h5>


<div class="example"><pre>
<class"APprompt>&t<span> <span ="APinput">:=FiniteOrderInnerAutomorphisms("" 8 8 ); /span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:= f[8]<p> <varclass""f/>isan  of simpleLie span class=">(\<span>of  <span class="SimpleMath>(\<span.Then< class"rg>f<var>defines a grading  <pan =SimpleMath">\L)/>   homogeneous   this bedenoted<spanclass=SimpleMath>(\)/> for span class=SimpleMath">i0..m-\)</span>. Let <span class="SimpleMath">\(G_0\)</span> be the group corresponding to <span class="SimpleMath">\(L_0\)</span> (i.e., the connected subgroup of the adjoint group of <span class="SimpleMath">\(L\)</span> with Lie algebra <span class="SimpleMath">\(L_0\)</span>). This function computes a maximal subspace of <span class="SimpleMath">\(L_1\)</span> consisting ofjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">sl2:= NilpotentOrbitsOfThetaRepresentation(h);;  </span>
#I  Selected carrier algebra method.
#I  Constructed 2782 root bases of possible flat subalgebras, now checking them...
#I  Obtained 58 Cartan elements, weeding out equivalent copies...
< class">gap><span>< java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 81
27
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 12
<span class="GAPprompt">gap></span> <span class="GAPinput">r:= ClosureDiagram class="chlinkprevnextbot>n;a ="hap0_mjhtml>TopBook<a   ahref"hap0_mj.html#ontents"[ontents</>nbsp  nbsp;a ="chap4_mj.tml"Previous]/>n;  &bsp;ahref=.">Next Chapter]/>;  </div>
#I  All (non-) div class="chlinkbot"><span"hlink1"Chapter /<href"hap0_mj.tml"Top/>< href=c."1/> <href"hap2_mjhtml">/><href"hap3_mj.tml></> ahref="chap4_mj.tml"4<a> a =".>5<a><a ="chap6_mjhtml6/>< c.html"Bib</>< href="chapInd_mjhtml"/>  /div>
<span class="GAPprompt">gap></span> <span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
 [2  ]  3  ,[4, 2] [4,3]  5,1 ], [6,5 ],[7, 2 ] 
  [ 7, 5 ], [ 8, 4 ], [ 9, 4 ], [ 9, 7 ], [ 10, 6 ], [ 10, 7 ], [ 11, 3 ], 
  [ 11, 6 ], [ 12, 7 ], [ 13, 9 ], [ 13, 10 ], [ 13, 11 ], [ 14, 9 ], 
  [ 14, 12 ], [ 15, 8 ], [ 15, 9 ], [ 16, 6 ], [ 17, 10 ], [ 17, 12 ], 
  [ 17, 16 ], [ 18, 13 ], [ 18, 16 ], [ 19, 13 ], [ 19, 15 ], [ 20, 11 ], 
  [ 20, 16 ], [ 21, 14 ], [ 21, 17 ], [ 21, 18 ], [ 22, 14 ], [ 22, 15 ], 
  [ 23, 18 ], [ 23, 20 ], [ 24, 18 ], [ 24, 19 ], [ 25, 21 ], [ 25, 22 ], 
  [ 25, 24 ], [ 26, 23 ], [ 26, 24 ], [ 27, 21 ], [ 27, 23 ] ]
<span class="GAPprompt">gap></span> <span class="GAPinput"># Now we do the adjoint representation of the Lie algebra of type F4:</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= SimpleLieAlgebra("F",4,Rationals);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">o:= NilpotentOrbits(L);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sl2:= List( o, SL2Triple );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">r:= ClosureDiagram( L, [0,0,0,0], sl2 );;      </span>
#I  All (non-) inclusions proved!
<span class="GAPprompt">gap></span> <span class="GAPinput">r.diag;</span>
[ [ 2, 1 ], [ 3, 2 ], [ 4, 3 ], [ 5, 3 ], [ 6, 4 ], [ 6, 5 ], [ 7, 6 ], 
  [ 8, 7 ], [ 9, 7 ], [ 10, 8 ], [ 10, 9 ], [ 11, 8 ], [ 12, 10 ], 
  [ 13, 11 ], [ 13, 12 ], [ 14, 13 ], [ 15, 14 ] ]
</pre></div>

<p><a id="X8108B6487A3A363B" name="X8108B6487A3A363B"></a></p>

<h5>5.1-8 CarrierAlgebra</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CarrierAlgebra</code>( <var class="Arg">L</var>, <var class="Arg">f</var>, <var class="Arg">e</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CarrierAlgebra</code>( <var class="Arg">L</var>, <var class="Arg">d</var>, <var class="Arg">e</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">f</var> is an automorphism of a simple Lie algebra <span class="SimpleMath">\(L\)</span> of order <span class="SimpleMath">\(m\)</span>, and <var class="Arg">e</var> a nilpotent element of <span class="SimpleMath">\(L_1\)</span>. This function returns the carrier algebra of <var class="Arg">e</var>. This is a <span class="SimpleMath">\(\mathbb{Z}\)</span>-graded semisimple subalgebra <span class="SimpleMath">\(K\)</span> of <span class="SimpleMath">\(L\)</span>, such that <var class="Arg">e</var> lies in <span class="SimpleMath">\(K_1\)</span>. For the precise definition we refer to <a href="chapBib_mj.html#biBvinberg2">[Vin79]</a>, <a href="chapBib_mj.html#biBvinberg3">[Vin75]</a>. The output is given in the form of a record, with three components: <var class="Arg">g0</var>, a basis of <span class="SimpleMath">\(K_0\)</span>, <var class="Arg">gp</var> a list containing bases of <span class="SimpleMath">\(K_1\)</span>, <span class="SimpleMath">\(K_2\)</span> and so on, and <var class="Arg">gn</var> a list containing bases of <span class="SimpleMath">\(K_{-1}\)</span>, <span class="SimpleMath">\(K_{-2}\)</span> and so on.</p>

<p>In the second version, the second argument <var class="Arg">d</var> is a list of non-negative integers. Then <var class="Arg">L</var> is <span class="SimpleMath">\(Z\)</span>-graded by giving the root space corresponding to the <span class="SimpleMath">\(i\)</span>-th simple root the degree <var class="Arg">d[i]</var>. Apart from this the function works in the same way.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= FiniteOrderInnerAutomorphisms( "F", 4, 5 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">h:= f[4];;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">sl2:= NilpotentOrbitsOfThetaRepresentation( h );;  </span>
#I  Selected Weyl orbit method.
#I  Constructed a Weyl transversal of 144 elements.
#I  Constructed 621 Cartan elements to be checked.
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= Source(h);   </span>
<Lie algebra of dimension 52 over CF(5)>
<span class="GAPprompt">gap></span> <span class="GAPinput">r:=CarrierAlgebra( L, h, sl2[1][3] );   </span>
rec( g0 := [ v.49+(2)*v.50+(2)*v.51+(3)*v.52, v.50+(1/2)*v.51+v.52 ], 
  gn := [ [ v.24, v.33 ], [ v.21 ], [ v.15 ] ], 
  gp := [ [ v.9, v.48 ], [ v.45 ], [ v.39 ] ] )
<span class="GAPprompt">gap></span> <span class="GAPinput">K:= Subalgebra( L, Concatenation( r.g0, Flat(r.gp), Flat(r.gn) ) );</span>
<Lie algebra over CF(5), with 10 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">SemiSimpleType( K );</span>
"B2"
</pre></div>

<p><a id="X7A8D86667BC7C033" name="X7A8D86667BC7C033"></a></p>

<h5>5.1-9 CartanSubspace</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; CartanSubspace</code>( <var class="Arg">f</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">f</var> is an automorphism of a simple Lie algebra <span class="SimpleMath">\(L\)</span> of order <span class="SimpleMath">\(m\)</span>. Then <var class="Arg">f</var> defines a grading on <span class="SimpleMath">\(L\)</span>. Let the homogeneous components of this grading be denoted <span class="SimpleMath">\(L_i\)</span> for <span class="SimpleMath">\(i=0,...,m-1\)</span>. Let <span class="SimpleMath">\(G_0\)</span> be the group corresponding to <span class="SimpleMath">\(L_0\)</span> (i.e., the connected subgroup of the adjoint group of <span class="SimpleMath">\(L\)</span> with Lie algebra <span class="SimpleMath">\(L_0\)</span>). This function computes a maximal subspace of <span class="SimpleMath">\(L_1\)</span> consisting of commuting semisimple elements. (Such a subspace is called a <em>Cartan subspace</em>.)</p>

<p>Every semisimple orbit of <span class="SimpleMath">\(G_0\)</span> in <span class="SimpleMath">\(L_1\)</span> contains an element of a fixed Cartan subspace.</p>


<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">f:= FiniteOrderInnerAutomorphisms( "A", 3, 3 );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">c:= CartanSubspace( f[3] ); </span>
<vector space of dimension 1 over CF(3)>
<span class="GAPprompt">gap></span> <span class="GAPinput">BasisVectors( Basis( c ) );</span>
[ v.1+v.5+v.12 ]
</pre></div>


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