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Here we describe the functions for working with the classification
of the semisimple subalgebras of the simple Lie algebras.
These subalgebras have been classified for the simple Lie algebras
over the complex numbers, of ranks up to 8. They have been classified
up to linear equivalence. Two subalgebras K1, K2 of a Lie algebra
L are linearly equivalent if for every representation of L the
induced representations of K1, K2 are equivalent.
The basic function for dealing with them returns a directed graph,
describing the inclusions among the subalgebras, and having the subalgebras
as labels of the vertices. (We refer to [dG11] for
the background details of this classification.)
Here t has to be a simple type of rank not exceeding 8.
This function returns a directed graph G. The vertices of this
graph are numbered from 1 to the number of semisimple subalgebras.
Furthermore, the last vertex is numbered 0. A vertex has a label that
is the semisimple subalgebra corresponding to it. The label of the
last vertex (numbered 0), has the Lie algebra L of type t as its
label. All other semisimple Lie algebras are subalgebras of this one.
The Lie algebra L (and its subalgebras) is defined over the rational numbers,
or over a cyclotomic field. It is sometimes necessary to take an extension,
because for some types not all subalgebras are defined over the
rationals.
Moreover, in G there is an edge from the vertex with label K1
to the vertex with label K2 if and only if K1 has a subalgebra
that is linearly equivalent (as subalgebra of L) to K2. We remark that
it does not mean that K2 is a subalgebra of K1 (rather that it is
linearly equivalent to a subalgebra of K1).
We consider the subalgebras of the Lie algebra of type C 3.
We compute the types of its maximal subalgebras.
> G:= SubalgebrasInclusionGraph( "C3" );
> G;
Digraph
Vertex Neighbours
1 ;
2 ;
3 ;
4 ;
5 ;
6 ;
7 ;
8 2 4 ;
9 5 10 ;
10 1 2 ;
11 1 2 3 ;
12 1 5 6 ;
13 3 4 6 ;
14 10 11 ;
15 9 12 14 ;
0 7 8 13 15 ;
> v:= Vertices(G);
> Label( v[10] );
Lie Algebra of dimension 6 with base ring Rational Field
> SemisimpleType( Label( v[7] ) );
A1
> SemisimpleType( Label( v[8] ) );
A2
> SemisimpleType( Label( v[13] ) );
A1 A1
> SemisimpleType( Label( v[15] ) );
A1 C2
Here G is a subalgebras inclusion graph of the simple Lie algebra L,
as output by the previous function, and k is a nonzero integer,
corresponding to a vertex. This function returns the restriction matrix
corresponding to L and the Lie algebra that is the label of the
k-th vertex of G. This restriction matrix maps weights in a
representation of L to weights of the subalgebra, and can be used
to decompose a representation of L, as a representation of the subalgebra.
We decompose the adjoint representation of the Lie algebra of type D 4,
when viewed as a representation of its subalgebra of type G 2.
> G:= SubalgebrasInclusionGraph( "D4" );
> v:= Vertices(G);
> tt:= [ SemisimpleType( Label(a) ) : a in v ];
> Index( tt, "G2" );
17
> M:= RestrictionMatrix( G, 17 );
> R:= RootDatum( "D4" : Isogeny:= "SC" );
> S:= RootDatum( "G2" : Isogeny:= "SC" );
> D:= AdjointRepresentationDecomposition(R);
> E:= Branch( S, D, M );
> WeightsAndMultiplicities(E);
[
(0 1),
(1 0)
]
[ 1, 2 ]
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