Automorphisms of Classical-type Reductive Algebras#

IdentityAutomorphism(L): AlgLie -> Map#: The trivial automorphism of the Lie algebra \(L\).

InnerAutomorphism(L, x): AlgLie, GrpLieElt -> Map#: The inner automorphism of the Lie algebra \(L\) induced by \(x\), where \(x\) is an element of the corresponding group of Lie type.

InnerAutomorphismGroup(L): AlgLie -> GrpLie, Map#: The group of Lie type \(G\) corresponding to the Lie algebra \(L\). The map \(G\to {\operatorname{Aut}}(L)\) is returned as second value.

DiagonalAutomorphism(L, v): AlgLie, ModTupRngElt -> Map#: The diagonal automorphism of the Lie algebra \(L\) induced by the vector \(v\).

GraphAutomorphism(L, p): AlgLie, GrpPermElt -> Map#

DiagramAutomorphism(L, p): AlgLie, GrpPermElt -> Map#

SimpleSigns: Any                    Default: 1

The graph automorphism of the Lie algebra \(L\) induced by the permutation \(p\). This must be either a permutation of the indices of the simple roots, or a permutation of the indices of all roots.

The optional parameter SimpleSigns can be used to specify the signs corresponding to each simple root. This should either be a sequence of integers \(\pm1\), or a single integer \(\pm1\).

Example: Graph Automorphism (ex-d6f1f7)#

We construct an automorphism of order three for the simple Lie algebra of type \(D_4\).

> DynkinDiagram( "D4" );

D4    3
     /
1 - 2
     \\
      4

> p:= Sym(4)!(1,3,4);
> L:= LieAlgebra( "D4", Rationals() );
> f:= GraphAutomorphism( L, p );
> f(L.3);
(0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
> f(L.4);
(0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)
> f(L.5);
(0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0)

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