<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealFormsInformation</code>( <var class="Arg">type</var>, <var class="Arg">rank</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function displays information regarding the simple real Lie algebras that can be constructed from the complex Lie algebra of type <var class="Arg">type</var> (which is a string) and rank <var class="Arg">rank</var> (a positive integer). Each Lie algebra is given an index which is an integer, and for each index some information is given on the Lie algebra, such as a commonly used name. In all cases the index 0 refers to the realification of the complex Lie algebra.</p>
There are 4 simple real forms with complexification A4
1 is of type su(5), compact form
2 - 3 are of type su(p,5-p) with 1 <= p <= 2
4 is of type sl(5,R)
Index '0' returns the realification of A4
There are 5 simple real forms with complexification E6
1 is the compact form
2 is EI = E6(6), with k_0 of type sp(4) (C4)
3 is EII = E6(2), with k_0 of type su(6)+su(2) (A5+A1)
4 is EIII = E6(-14), with k_0 of type so(10)+R (D5+R)
5 is EIV = E6(-26), with k_0 of type f_4 (F4)
Index '0' returns the realification of E6
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NumberRealForms</code>( <var class="Arg">type</var>, <var class="Arg">rank</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns the number of (isomorphism types of) all real forms of the simple complex Lie algebras of type <var class="Arg">type</var> and rank <var class="Arg">rank</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealFormById</code>( <var class="Arg">type</var>, <var class="Arg">rank</var>, <var class="Arg">id</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealFormById</code>( <var class="Arg">type</var>, <var class="Arg">rank</var>, <var class="Arg">id</var>, <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealFormById</code>( [<var class="Arg">type</var>, <var class="Arg">rank</var>, <var class="Arg">id</var>] )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealFormById</code>( [<var class="Arg">type</var>, <var class="Arg">rank</var>, <var class="Arg">id</var>, ]<var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealFormById</code>( <var class="Arg">list</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealFormById</code>( <var class="Arg">list</var>, <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <span class="SimpleMath">L</span> be the complex Lie algebra of type <var class="Arg">type</var> and rank <var class="Arg">rank</var>. This function constructs the real form of <span class="SimpleMath">L</span> with index <var class="Arg">id</var> (see <code class="func">RealFormsInformation</code> (<a href="chap3.html#X7BB53454857133FF"><span class="RefLink">3.1-1</span></a>)). By default this Lie algebra is constructed over the field <var class="Arg">SqrtField</var>. However, by adding as an optional fourth argument the field <var class="Arg">F</var>, it is possible to construct the Lie algebra output by this function over <var class="Arg">F</var>. It is required that the complex unit <var class="Arg">E(4)</var> is contained in <var class="Arg">F</var>. The function also accepts <var class="Arg">type</var>, <var class="Arg">rank</var>, and <var class="Arg">id</var> in a list as an argument. Moreover, if <var class="Arg">list</var> is a list of such triples, then the function constructs the direct sum of the simple real forms specified by the individual triples. If the index <var class="Arg">ind</var> is 0, then the realification of <span class="SimpleMath">L</span> is constructed, which, strictly speaking is not a real form of <span class="SimpleMath">L</span>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">RealFormById( "A", 4, 2 );</span>
<Lie algebra of dimension 24 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput">RealFormById( "A", 4, 2, CF(4) );</span>
<Lie algebra of dimension 24 over GaussianRationals>
<span class="GAPprompt">gap></span> <span class="GAPinput">RealFormById( [ ["A", 4, 2], ["D", 5, 2] ], SqrtField);</span>
<Lie algebra of dimension 69 over SqrtField>
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IdRealForm</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Let <span class="SimpleMath">L</span> be a semisimple real Lie algebra, where each simple summand is a real form of a simple complex Lie algebra. This function returns (a list of) the id(s) of the simple real form(s).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ NameRealForm</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a reductive real Lie algebra whose centre is stable under the Cartan involution of <var class="Arg">L</var>. This function returns a string giving the names of the real forms of the simple components of the derived subalgebra of <var class="Arg">L</var>, as well as the number of compact and non-compact dimensions of the centre of <var class="Arg">L</var>. For the simple real Lie algebras we use the naming conventions as in <a href="chapBib.html#biBknapp">[Kna02]</a>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AllRealForms</code>( <var class="Arg">type</var>, <var class="Arg">rank</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>This function returns all real forms of the simple complex Lie algebras of type <var class="Arg">type</var> and rank <var class="Arg">rank</var> up to isomorphism. In the same way as with <code class="func">RealFormById</code> (<a href="chap3.html#X8443E03C868CA7D3"><span class="RefLink">3.1-3</span></a>) it is possible to add the base field as an optional third argument.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealFormParameters</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>For a real Lie algebra <var class="Arg">L</var> constructed by the function <code class="func">RealFormById</code> (<a href="chap3.html#X8443E03C868CA7D3"><span class="RefLink">3.1-3</span></a>), this function returns a list of the parameters defining <var class="Arg">L</var> as a real formof its complexification. The first element of the list is the type of <var class="Arg">L</var> (given by a string), the second element is its rank, the third and fourth elements are the list of signs and the permutation defining the Cartan involution (see Section <a href="chap1.html#X7A0F3100829CD1E1"><span class="RefLink">1.1</span></a>).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRealFormOfInnerType</code>( <var class="Arg">L</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns <var class="Arg">true</var> if and only if the real form <var class="Arg">L</var> is a defined by an inner involutive automorphism.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsRealification</code>( <var class="Arg">L</var> )</td><td class="tdright">( property )</td></tr></table></div>
<p>Returns <var class="Arg">true</var> if and only if the real form <var class="Arg">L</var> is the realification of a complex simple Lie algebra.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CartanDecomposition</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The Cartan decomposition of <var class="Arg">L</var> as a record with entries <var class="Arg">K</var>, <var class="Arg">P</var>, and <var class="Arg">CartanInv</var>, such that <span class="SimpleMath">L=K⊕ P</span> is the Cartan decomposition with corresponding Cartan involution <var class="Arg">CartanInv</var>, which is defined as a function on <var class="Arg">L</var>.</p>
<p>The Lie algebras constructed by <code class="func">RealFormById</code> (<a href="chap3.html#X8443E03C868CA7D3"><span class="RefLink">3.1-3</span></a>) have this attribute stored. For other semisimple real Lie algebras it is computed. However, we do remark that the in the computation the root system is computed with respect to a Cartan subalgebra. If the program does not succeed in splitting the Cartan subalgebra over the base field of <var class="Arg">L</var>, then the computation will not succeed.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= RealFormById( "A", 5, 3 );</span>
<Lie algebra of dimension 35 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput">H := CartanSubalgebra(L);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:= LieCentralizer( L, Subalgebra( L, [Basis( H )[1]] ) );</span>
<Lie algebra of dimension 17 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput">DK:= LieDerivedSubalgebra( K );</span>
<Lie algebra of dimension 15 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput">CartanDecomposition( DK );</span>
rec( CartanInv := function( v ) ... end,
K := <Lie algebra of dimension 15 over SqrtField>,
P := <vector space of dimension 0 over SqrtField> )
<span class="GAPprompt">gap></span> <span class="GAPinput"># We see that the semisimple subalgebra DK is compact. </span>
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealStructure</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RealStructure</code>( <var class="Arg">L:</var> <var class="Arg">basis</var> <var class="Arg">:=</var> <var class="Arg">B</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>The real structure of the real form <var class="Arg">L</var> is the (complex) conjugation with respect to <var class="Arg">L</var>, that is, the function which maps an element in <var class="Arg">L</var> to the element constructed as follows: write it as a linear combination of the basis elements of <var class="Arg">L</var> and replace each coefficient by its complex conjugate. If the optional argument <var class="Arg">basis:=B</var> is given, then <var class="Arg">B</var> has to be a basis whose span contains <var class="Arg">L</var> (which is not checked by the code); in this case the linear combination is done with respect to <var class="Arg">B</var>. The latter construction is important when one considers a subalgebra <var class="Arg">M</var> of a real form <var class="Arg">L</var>; here one could either do <var class="Arg">Realstructure(M:basis:=Basis(L))</var> or <var class="Arg">SetRealStructure(M,RealStructure(L))</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximalReductiveSubalgebras</code>( <var class="Arg">type</var>, <var class="Arg">rank</var>, <var class="Arg">id</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here the input parameters are as in <code class="func">RealFormById</code> (<a href="chap3.html#X8443E03C868CA7D3"><span class="RefLink">3.1-3</span></a>), and <var class="Arg">rank</var> is between 1 and 8. These parameters correspond to a real simple Lie algebra <span class="SimpleMath">L</span>. This function returns a list of maximal reductive subalgebras of <span class="SimpleMath">L</span>. More precisely, it returns a record with two fields, <var class="Arg">liealg</var> and <var class="Arg">subalgs</var>. The first field contains the Lie algebra <span class="SimpleMath">L</span>, and the second field contains a list of its maximal reductive subalgebras. Isomorphic copies of the regular semisimple Lie algebras can occur more than once. If that happens then those copies are not conjugate under the adjoint group of <span class="SimpleMath">L</span>. There are no isomorphisms between the non-reductive subalgebras. There can be non-conjugate copies of those as well, but the database does not contain these.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsomorphismOfRealSemisimpleLieAlgebras</code>( <var class="Arg">K</var>, <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Here <var class="Arg">K</var>, <var class="Arg">L</var> are two real forms of a semisimple complex Lie algebra. This function returns an isomorphism if one exists. Otherwise <var class="Arg">false</var> is returned.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:=RealFormById("E",6,3);; </span>
<span class="GAPprompt">gap></span> <span class="GAPinput">H:=CartanSubalgebra(L);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:=LieCentralizer(L,Subalgebra(L,Basis(H){[1,2,4]}));;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">DK:=LieDerivedSubalgebra(K);</span>
<Lie algebra of dimension 8 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput">IdRealForm(DK); </span>
[ "A", 2, 2 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">M:=RealFormById("A",2,2);</span>
<Lie algebra of dimension 8 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsomorphismOfRealSemisimpleLieAlgebras(DK,M);</span>
<Lie algebra isomorphism between Lie algebras of dimension 8 over SqrtField>
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximallyCompactCartanSubalgebra</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a real semisimple Lie algebra. This function returns a maximally compact Cartan subalgebra of <var class="Arg">L</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ MaximallyNonCompactCartanSubalgebra</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a real semisimple Lie algebra. This function returns a maximally non-compact Cartan subalgebra of <var class="Arg">L</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompactDimensionOfCartanSubalgebra</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CompactDimensionOfCartanSubalgebra</code>( <var class="Arg">L</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a real semisimple Lie algebra. This function returns the compact dimension of the Cartan subalgebra <var class="Arg">H</var>. If <var class="Arg">H</var> is not given, then <var class="Arg">CartanSubalgebra(L)</var> will be taken. The compact dimension will be stored in the Cartan subalgebra, so that a new call to this function, with the same input, will return the compact dimension immediately.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CartanSubalgebrasOfRealForm</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a real form of a complex semisimple Lie algebra. This function returns a list of Cartan subalgebras of <var class="Arg">L</var>. They are representatives of all classes of conjugate (by the adjoint group) Cartan subalgebras of <var class="Arg">L</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CartanSubspace</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a real semisimple Lie algebra. This function returns a Cartan subspace of <var class="Arg">L</var>. That is a maximal abelian subspace of the subspace <var class="Arg">P</var> given in the <code class="func">CartanDecomposition</code> (<a href="chap3.html#X81E1C65282CE3130"><span class="RefLink">3.1-10</span></a>) of <var class="Arg">L</var>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ RootsystemOfCartanSubalgebra</code>( <var class="Arg">L</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">‣ RootsystemOfCartanSubalgebra</code>( <var class="Arg">L</var>, <var class="Arg">H</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a semisimple Lie algebra, and <var class="Arg">H</var> is a Cartan subalgebra. (If <var class="Arg">H</var> is not given, then <var class="Arg">CartanSubalgebra(L)</var> will be taken.) This function returns the root system of <var class="Arg">L</var> with respect to <var class="Arg">H</var>. It is necessary that the eigenvalues of the adjoint maps corresponding to all elements of <var class="Arg">H</var> lie in the ground field of <var class="Arg">L</var>. However, even if they do, it is not guaranteed that this function succeeds, as it may happen that <strong class="pkg">GAP</strong> has no polynomial factorisation algorithm over the ground field.</p>
<p>The root system is stored in <var class="Arg">H</var>, so that a new call to this function, with the same input, will return the same root system.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ChevalleyBasis</code>( <var class="Arg">R</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">R</var> is a root system of a semisimple Lie algebra <var class="Arg">L</var>. This function returns a Chevalley basis of <var class="Arg">L</var>, consisting of root vectors of <var class="Arg">R</var>.</p>
<p>In this section we document the functionality for computing the Satake and Vogan diagrams of a real semisimple Lie algebra. In both cases the relevant function computes an object, which, when printed, does not reveal much information. However, <var class="Arg">Display</var> with as input such an object, displays the diagram. Here we use the convention that every node is represented by an integer; nodes that are painted black are represented by integers in brackets; and the involution (i.e., the arrows in the diagram) are represented by a permutation of the nodes, printed on a line below the diagram.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ VoganDiagram</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a real semisimple Lie algebra. This function returns the Vogan diagram of <var class="Arg">L</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= RealFormById( [["E", 6, 3],["A", 3, 2]] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput"> K:= LieCentralizer( L, Subalgebra( L, Basis( CartanSubalgebra(L) ){[1]} ) );</span>
<Lie algebra of dimension 51 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput"> DK:= LieDerivedSubalgebra( K );</span>
<Lie algebra of dimension 50 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput">vd:= VoganDiagram(DK);</span>
<Vogan diagram in Lie algebra of type A3+A5>
<span class="GAPprompt">gap></span> <span class="GAPinput"> Display( vd );</span>
A3: (1)---2---3
A5: 4---(5)---6---7---8
Involution: ()
Types of direct summands:
[ [ "A", 3, 2 ], [ "A", 5, 3 ] ]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ SatakeDiagram</code>( <var class="Arg">L</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p>Here <var class="Arg">L</var> is a real semisimple Lie algebra. This function returns the Satake diagram of <var class="Arg">L</var>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">L:= RealFormById( [["E", 6, 3],["A", 3, 2]] );;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">K:= LieCentralizer( L, Subalgebra( L, Basis( CartanSubalgebra(L) ){[1]} ) );</span>
<Lie algebra of dimension 51 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput"> DK:= LieDerivedSubalgebra( K );</span>
<Lie algebra of dimension 50 over SqrtField>
<span class="GAPprompt">gap></span> <span class="GAPinput">sd:= SatakeDiagram( DK );</span>
<Satake diagram in Lie algebra of type A5xA3>
<span class="GAPprompt">gap></span> <span class="GAPinput">Display( sd );</span>
A5: 1---2---(3)---4---5
A3: 6---(7)---8
Involution: (1,5)(2,4)(6,8)
</pre></div>
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