# 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`

```magma
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 $D_4$.

```magma
> 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)

```