GAP (CoReLG) - Chapter 3: Real Lie Algebras

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3 Real Lie Algebras
3.1 Construction of simple real Lie algebras
3.1-1 RealFormsInformation
3.1-2 NumberRealForms
3.1-3 RealFormById
3.1-4 IdRealForm
3.1-5 NameRealForm
3.1-6 AllRealForms
3.1-7 RealFormParameters
3.1-8 IsRealFormOfInnerType
3.1-9 IsRealification
3.1-10 CartanDecomposition
3.1-11 RealStructure
3.2 Maximal reductive subalgebras
3.2-1 MaximalReductiveSubalgebras
3.3 Isomorphisms
3.3-1 IsomorphismOfRealSemisimpleLieAlgebras
3.4 Cartan subalgebras and root systems
3.4-1 MaximallyCompactCartanSubalgebra
3.4-2 MaximallyNonCompactCartanSubalgebra
3.4-3 CompactDimensionOfCartanSubalgebra
3.4-4 CartanSubalgebrasOfRealForm
3.4-5 CartanSubspace
3.4-6 RootsystemOfCartanSubalgebra
3.4-7 ChevalleyBasis
3.5 Diagrams
3.5-1 VoganDiagram
3.5-2 SatakeDiagram

### 3 Real Lie Algebras

#### 3.1 Construction of simple real Lie algebras

A few functions print some information on what they are doing to the info class InfoCorelg.

##### 3.1-1 RealFormsInformation

`‣ RealFormsInformation`( type, rank ) | ( function )

This function displays information regarding the simple real Lie algebras that can be constructed from the complex Lie algebra of type type (which is a string) and rank rank (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.

    gap> RealFormsInformation( "A", 4 );

      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

    gap> RealFormsInformation( "E", 6 );

      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

    gap> NumberRealForms("D",10);
    12

##### 3.1-2 NumberRealForms

`‣ NumberRealForms`( type, rank ) | ( function )

This function returns the number of (isomorphism types of) all real forms of the simple complex Lie algebras of type type and rank rank.

##### 3.1-3 RealFormById

`‣ RealFormById`( type, rank, id ) | ( function )

`‣ RealFormById`( type, rank, id, F ) | ( function )

`‣ RealFormById`( [type, rank, id] ) | ( function )

`‣ RealFormById`( [type, rank, id, ]F ) | ( function )

`‣ RealFormById`( list ) | ( function )

`‣ RealFormById`( list, F ) | ( function )

Let \(L\) be the complex Lie algebra of type type and rank rank. This function constructs the real form of \(L\) with index id (see `RealFormsInformation` (3.1-1)). By default this Lie algebra is constructed over the field SqrtField. However, by adding as an optional fourth argument the field F, it is possible to construct the Lie algebra output by this function over F. It is required that the complex unit E(4) is contained in F. The function also accepts type, rank, and id in a list as an argument. Moreover, if list 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 ind is 0, then the realification of \(L\) is constructed, which, strictly speaking is not a real form of \(L\).

    gap> RealFormById( "A", 4, 2 );
    <Lie algebra of dimension 24 over SqrtField>
    gap> RealFormById( "A", 4, 2, CF(4) );
    <Lie algebra of dimension 24 over GaussianRationals>
    gap> RealFormById( [ ["A", 4, 2], ["D", 5, 2] ], SqrtField);
    <Lie algebra of dimension 69 over SqrtField>

##### 3.1-4 IdRealForm

`‣ IdRealForm`( L ) | ( function )

Let \(L\) 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).

    gap> L := RealFormById( [ ["A", 4, 2], ["D", 5, 2] ], SqrtField);;
    gap> IdRealForm( L );
    [ [ "A", 4, 2 ], [ "D", 5, 2 ] ]
    gap> K := RealFormById("A",5,2);;
    gap> IdRealForm( K );
    [ "A", 5, 2 ]

##### 3.1-5 NameRealForm

`‣ NameRealForm`( L ) | ( attribute )

Here L is a reductive real Lie algebra whose centre is stable under the Cartan involution of L. This function returns a string giving the names of the real forms of the simple components of the derived subalgebra of L, as well as the number of compact and non-compact dimensions of the centre of L. For the simple real Lie algebras we use the naming conventions as in [Kna02].

    gap> L := RealFormById( [ ["A", 4, 2], ["D", 5, 2] ], SqrtField);;
    gap> NameRealForm( L );
    "su(1,4)+so(2,8)"

##### 3.1-6 AllRealForms

`‣ AllRealForms`( type, rank ) | ( function )

This function returns all real forms of the simple complex Lie algebras of type type and rank rank up to isomorphism. In the same way as with `RealFormById` (3.1-3) it is possible to add the base field as an optional third argument.

##### 3.1-7 RealFormParameters

`‣ RealFormParameters`( L ) | ( attribute )

For a real Lie algebra L constructed by the function `RealFormById` (3.1-3), this function returns a list of the parameters defining L as a real form of its complexification. The first element of the list is the type of L (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 1.1).

##### 3.1-8 IsRealFormOfInnerType

`‣ IsRealFormOfInnerType`( L ) | ( property )

Returns true if and only if the real form L is a defined by an inner involutive automorphism.

##### 3.1-9 IsRealification

`‣ IsRealification`( L ) | ( property )

Returns true if and only if the real form L is the realification of a complex simple Lie algebra.

##### 3.1-10 CartanDecomposition

`‣ CartanDecomposition`( L ) | ( attribute )

The Cartan decomposition of L as a record with entries K, P, and CartanInv, such that \(L=K\oplus P\) is the Cartan decomposition with corresponding Cartan involution CartanInv, which is defined as a function on L.

The Lie algebras constructed by `RealFormById` (3.1-3) 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 L, then the computation will not succeed.

    gap> L:= RealFormById( "A", 5, 3 );
    <Lie algebra of dimension 35 over SqrtField>
    gap> H := CartanSubalgebra(L);;
    gap> K:= LieCentralizer( L, Subalgebra( L, [Basis( H )[1]] ) );
    <Lie algebra of dimension 17 over SqrtField>
    gap> DK:= LieDerivedSubalgebra( K );
    <Lie algebra of dimension 15 over SqrtField>
    gap> CartanDecomposition( DK );
    rec( CartanInv := function( v ) ... end,
      K := <Lie algebra of dimension 15 over SqrtField>,
      P := <vector space of dimension 0 over SqrtField> )
    gap> # We see that the semisimple subalgebra DK is compact.

##### 3.1-11 RealStructure

`‣ RealStructure`( L ) | ( attribute )

`‣ RealStructure`( L: basis := B ) | ( attribute )

The real structure of the real form L is the (complex) conjugation with respect to L, that is, the function which maps an element in L to the element constructed as follows: write it as a linear combination of the basis elements of L and replace each coefficient by its complex conjugate. If the optional argument basis:=B is given, then B has to be a basis whose span contains L (which is not checked by the code); in this case the linear combination is done with respect to B. The latter construction is important when one considers a subalgebra M of a real form L; here one could either do Realstructure(M:basis:=Basis(L)) or SetRealStructure(M,RealStructure(L)).

#### 3.2 Maximal reductive subalgebras

##### 3.2-1 MaximalReductiveSubalgebras

`‣ MaximalReductiveSubalgebras`( type, rank, id ) | ( operation )

Here the input parameters are as in `RealFormById` (3.1-3), and rank is between 1 and 8. These parameters correspond to a real simple Lie algebra \(L\). This function returns a list of maximal reductive subalgebras of \(L\). More precisely, it returns a record with two fields, liealg and subalgs. The first field contains the Lie algebra \(L\), 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 \(L\). 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.

    gap> r:= MaximalReductiveSubalgebras("F",4,3);;
    gap> NameRealForm( r.liealg );
    "F4(-20)"
    gap> for K in r.subalgs do Print( NameRealForm(K), "\n" ); od;
    su(1,2)+su(3)
    su(2)+sp(1,2)
    so(8,1)
    so(9)
    sl(2,R)+G2c

#### 3.3 Isomorphisms

##### 3.3-1 IsomorphismOfRealSemisimpleLieAlgebras

`‣ IsomorphismOfRealSemisimpleLieAlgebras`( K, L ) | ( function )

Here K, L are two real forms of a semisimple complex Lie algebra. This function returns an isomorphism if one exists. Otherwise false is returned.

    gap> L:=RealFormById("E",6,3);;
    gap> H:=CartanSubalgebra(L);;
    gap> K:=LieCentralizer(L,Subalgebra(L,Basis(H){[1,2,4]}));;
    gap> DK:=LieDerivedSubalgebra(K);
    <Lie algebra of dimension 8 over SqrtField>
    gap> IdRealForm(DK);
    [ "A", 2, 2 ]
    gap> M:=RealFormById("A",2,2);
    <Lie algebra of dimension 8 over SqrtField>
    gap> IsomorphismOfRealSemisimpleLieAlgebras(DK,M);
    <Lie algebra isomorphism between Lie algebras of dimension 8 over SqrtField>

#### 3.4 Cartan subalgebras and root systems

##### 3.4-1 MaximallyCompactCartanSubalgebra

`‣ MaximallyCompactCartanSubalgebra`( L ) | ( attribute )

Here L is a real semisimple Lie algebra. This function returns a maximally compact Cartan subalgebra of L.

##### 3.4-2 MaximallyNonCompactCartanSubalgebra

`‣ MaximallyNonCompactCartanSubalgebra`( L ) | ( attribute )

Here L is a real semisimple Lie algebra. This function returns a maximally non-compact Cartan subalgebra of L.

##### 3.4-3 CompactDimensionOfCartanSubalgebra

`‣ CompactDimensionOfCartanSubalgebra`( L ) | ( function )

`‣ CompactDimensionOfCartanSubalgebra`( L, H ) | ( function )

Here L is a real semisimple Lie algebra. This function returns the compact dimension of the Cartan subalgebra H. If H is not given, then CartanSubalgebra(L) 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.

##### 3.4-4 CartanSubalgebrasOfRealForm

`‣ CartanSubalgebrasOfRealForm`( L ) | ( attribute )

Here L is a real form of a complex semisimple Lie algebra. This function returns a list of Cartan subalgebras of L. They are representatives of all classes of conjugate (by the adjoint group) Cartan subalgebras of L.

##### 3.4-5 CartanSubspace

`‣ CartanSubspace`( L ) | ( attribute )

Here L is a real semisimple Lie algebra. This function returns a Cartan subspace of L. That is a maximal abelian subspace of the subspace P given in the `CartanDecomposition` (3.1-10) of L.

##### 3.4-6 RootsystemOfCartanSubalgebra

`‣ RootsystemOfCartanSubalgebra`( L ) | ( operation )

`‣ RootsystemOfCartanSubalgebra`( L, H ) | ( operation )

Here L is a semisimple Lie algebra, and H is a Cartan subalgebra. (If H is not given, then CartanSubalgebra(L) will be taken.) This function returns the root system of L with respect to H. It is necessary that the eigenvalues of the adjoint maps corresponding to all elements of H lie in the ground field of L. However, even if they do, it is not guaranteed that this function succeeds, as it may happen that GAP has no polynomial factorisation algorithm over the ground field.

The root system is stored in H, so that a new call to this function, with the same input, will return the same root system.

##### 3.4-7 ChevalleyBasis

`‣ ChevalleyBasis`( R ) | ( attribute )

Here R is a root system of a semisimple Lie algebra L. This function returns a Chevalley basis of L, consisting of root vectors of R.

#### 3.5 Diagrams

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, Display 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.

##### 3.5-1 VoganDiagram

`‣ VoganDiagram`( L ) | ( attribute )

Here L is a real semisimple Lie algebra. This function returns the Vogan diagram of L.

    gap> L:= RealFormById( [["E", 6, 3],["A", 3, 2]] );;
    gap>  K:= LieCentralizer( L, Subalgebra( L, Basis( CartanSubalgebra(L) ){[1]} ) );
    <Lie algebra of dimension 51 over SqrtField>
    gap>  DK:= LieDerivedSubalgebra( K );
    <Lie algebra of dimension 50 over SqrtField>
    gap> vd:= VoganDiagram(DK);
    <Vogan diagram in Lie algebra of type A3+A5>
    gap>  Display( vd );
    A3:  (1)---2---3
    A5:  4---(5)---6---7---8
    Involution: ()
    Types of direct summands:
    [ [ "A", 3, 2 ], [ "A", 5, 3 ] ]

##### 3.5-2 SatakeDiagram

`‣ SatakeDiagram`( L ) | ( attribute )

Here L is a real semisimple Lie algebra. This function returns the Satake diagram of L.

    gap> L:= RealFormById( [["E", 6, 3],["A", 3, 2]] );;
    gap> K:= LieCentralizer( L, Subalgebra( L, Basis( CartanSubalgebra(L) ){[1]} ) );
    <Lie algebra of dimension 51 over SqrtField>
    gap>  DK:= LieDerivedSubalgebra( K );
    <Lie algebra of dimension 50 over SqrtField>
    gap> sd:= SatakeDiagram( DK );
    <Satake diagram in Lie algebra of type A5xA3>
    gap> Display( sd );
    A5:  1---2---(3)---4---5
    A3:  6---(7)---8
    Involution:  (1,5)(2,4)(6,8)

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