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# 5 Actors of 2d-objects

### Sections

1. Actor of a Crossed Module

## 5.1 Actor of a Crossed Module

The actor of X is a crossed module (D : W(X) ® Aut(X)) which was shown by Lue and Norrie, in N2 and N1 to give the automorphism object of a crossed module X. In this implementation, the source of the actor is a permutation representation W of the Whitehead group of regular derivations, and the range is a permutation representation A of the automorphism group Aut(X) of X.

- `AutomorphismPermGroup(` xmod`) A`
- `WhiteheadXMod(` xmod`) A`
- `LueXMod(` xmod`) A`
- `NorrieXMod(` xmod`) A`
- `ActorXMod(` xmod`) A`

  An automorphism ( s, r) of `X` acts on the Whitehead monoid by c(s,r) = s°c°r-1, and this action determines the action for the actor. In fact the four groups R, S, W, A, the homomorphisms between them, and the various actions, give five crossed modules forming a crossed square:

  - X = (¶: S ® R), the initial crossed module, on the left,
  - W(X) = (h: S ® W), the Whitehead crossed module of X, at the top,
  - L(X) = (D°h = a°¶: S ® A), the Lue crossed module of X, along the top-left to bottom-right diagonal,
  - N(X) = (a: R ® A), the Norrie crossed module of X, at the bottom, and
  - Act(X) = ( D: W ® A), the actor crossed module of X, on the right.

- `InnerActor(` xmod`) A`
- `InnerMorphism(` xmod`) A`
- `Centre(` xmod`) A`

  Pairs of boundaries or identity mappings provide six morphisms of crossed modules. In particular, the boundaries of W(X) and N(X) form the inner morphism of X, mapping source elements to principal derivations and range elements to inner automorphisms. The image of X under this morphism is the inner actor of X, while the kernel is the centre of X. In the example which follows, using the crossed module `(X3 : c3 -> s3)` from the previous chapter, the inner morphism is an inclusion of crossed modules.

      gap> X3;
      [c3->s3]]
      gap> WGX3 := WhiteheadPermGroup( X3 );
      Group( [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ] )
      gap> APX3 := AutomorphismPermGroup( X3 );
      Group( [ (3,4,5), (1,2)(4,5) ] )
      gap> WX3 := WhiteheadXMod( X3 );;  Display( WX3 );
      Crossed module Whitehead[c3->s3] :-
      : Source group has generators:
        [ ( 1, 2, 3)( 4, 6, 5) ]
      : Range group has generators:
        [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ]
      : Boundary homomorphism maps source generators to:
        [ (1,3,2)(4,6,5) ]
      : Action homomorphism maps range generators to automorphisms:
        (1,2,3)(4,5,6) --> { source gens --> [ (1,2,3)(4,6,5) ] }
        (1,4)(2,6)(3,5) --> { source gens --> [ (1,3,2)(4,5,6) ] }
        These 2 automorphisms generate the group of automorphisms.
      gap> LX3 := LueXMod( X3 );
      Lue[c3->s3]
      gap> NX3 := NorrieXMod( X3 );
      Norrie[c3->s3]
      gap> AX3 := ActorXMod( X3 );;  Display( AX3);
      Crossed module Actor[c3->s3] :-
      : Source group has generators:
        [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ]
      : Range group has generators:
        [ (3,4,5), (1,2)(4,5) ]
      : Boundary homomorphism maps source generators to:
        [ (3,5,4), (1,2)(4,5) ]
      : Action homomorphism maps range generators to automorphisms:
        (3,4,5) --> { source gens --> [ (1,2,3)(4,5,6), (1,5)(2,4)(3,6) ] }
        (1,2)(4,5) --> { source gens --> [ (1,3,2)(4,6,5), (1,4)(2,6)(3,5) ] }
        These 2 automorphisms generate the group of automorphisms.
      gap> IAX3 := InnerActorXMod( X3 );;  Display( IAX3 );
      Crossed module InnerActor[c3->s3] :-
      : Source group has generators:
        [ (1,3,2)(4,6,5) ]
      : Range group has generators:
        [ (3,5,4), (1,2)(4,5) ]
      : Boundary homomorphism maps source generators to:
        [ (3,4,5) ]
      : Action homomorphism maps range generators to automorphisms:
        (3,5,4) --> { source gens --> [ (1,3,2)(4,6,5) ] }
        (1,2)(4,5) --> { source gens --> [ (1,2,3)(4,5,6) ] }
        These 2 automorphisms generate the group of automorphisms.
      gap> IMX3 := InnerMorphism( X3 );; Display( IMX3 );
      Morphism of crossed modules :-
      : Source = [c3->s3] with generating sets:
        [ ( 1, 2, 3)( 4, 6, 5) ]
        [ (4,5,6), (2,3)(5,6) ]
      :  Range = Actor[c3->s3] with generating sets:
        [ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ]
        [ (3,4,5), (1,2)(4,5) ]
      : Source Homomorphism maps source generators to:
        [ (1,3,2)(4,6,5) ]
      : Range Homomorphism maps range generators to:
        [ (3,5,4), (1,2)(4,5) ]
      gap> Centre( X3 );
      [Group( () )->Group( () )]

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