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6 Actors of 2d-groups
 6.1 Actor of a crossed module

6 Actors of 2d-groups

6.1 Actor of a crossed module

The actor of \(\mathcal{X}\) is a crossed module \(\Act(\mathcal{X}) = (\Delta : \mathcal{W}(\mathcal{X}) \to \Aut(\mathcal{X}))\) which was shown by Lue and Norrie, in [Nor87] and [Nor90] to give the automorphism object of a crossed module \(\mathcal{X}\). In this implementation, the source of the actor is a permutation representation \(W\) of the Whitehead group of regular derivations, and the range of the actor is a permutation representation \(A\) of the automorphism group \(\Aut(\mathcal{X})\) of \(\mathcal{X}\).

6.1-1 AutomorphismPermGroup
‣ AutomorphismPermGroup( xmod )( attribute )
‣ GeneratingAutomorphisms( xmod )( attribute )
‣ PermAutomorphismAsXModMorphism( xmod, perm )( operation )

The automorphisms \(( \sigma, \rho )\) of \(\mathcal{X}\) form a group \(\Aut(\mathcal{X})\) of crossed module isomorphisms. The function AutomorphismPermGroup finds a set of GeneratingAutomorphisms for \(\Aut(\mathcal{X})\), and then constructs a permutation representation of this group, which is used as the range of the actor crossed module of \(\mathcal{X}\). The individual automorphisms can be constructed from the permutation group using the function PermAutomorphismAsXModMorphism. The example below uses the crossed module X3=[c3->s3] constructed in section 5.1.


gap> APX3 := AutomorphismPermGroup( X3 );
Group([ (5,7,6), (1,2)(3,4)(6,7) ])
gap> Size( APX3 );
6
gap> genX3 := GeneratingAutomorphisms( X3 );    
[ [[c3->s3] => [c3->s3]], [[c3->s3] => [c3->s3]] ]
gap> e6 := Elements( APX3 )[6];
(1,2)(3,4)(5,7)
gap> m6 := PermAutomorphismAsXModMorphism( X3, e6 );;
gap> Display( m6 );
Morphism of crossed modules :- 
: Source = [c3->s3] with generating sets:
  [ (1,2,3)(4,6,5) ]
  [ (4,5,6), (2,3)(5,6) ]
: Range = Source
: Source Homomorphism maps source generators to:
  [ (1,3,2)(4,5,6) ]
: Range Homomorphism maps range generators to:
  [ (4,6,5), (2,3)(4,5) ]

6.1-2 WhiteheadXMod
‣ WhiteheadXMod( xmod )( attribute )
‣ LueXMod( xmod )( attribute )
‣ NorrieXMod( xmod )( attribute )
‣ ActorXMod( xmod )( attribute )

An automorphism \(( \sigma, \rho )\) of X acts on the Whitehead monoid by \(\chi^{(\sigma,\rho)} = \sigma \circ \chi \circ \rho^{-1}\), and this 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:


gap> WGX3 := WhiteheadPermGroup( X3 );
Group([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ])
gap> WX3 := WhiteheadXMod( X3 );; 
gap> 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:
  [ (1,2,3)(4,5,6) ]
: 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 );;
gap> Display( LX3 );
Crossed module Lue[c3->s3] :- 
: Source group has generators:
  [ (1,2,3)(4,6,5) ]
: Range group has generators:
  [ (5,7,6), (1,2)(3,4)(6,7) ]
: Boundary homomorphism maps source generators to:
  [ (5,7,6) ]
: Action homomorphism maps range generators to automorphisms:
  (5,7,6) --> { source gens --> [ (1,2,3)(4,6,5) ] }
  (1,2)(3,4)(6,7) --> { source gens --> [ (1,3,2)(4,5,6) ] }
  These 2 automorphisms generate the group of automorphisms.
gap> NX3 := NorrieXMod( X3 );; 
gap> Display( NX3 );
Crossed module Norrie[c3->s3] :- 
: Source group has generators:
  [ (4,5,6), (2,3)(5,6) ]
: Range group has generators:
  [ (5,7,6), (1,2)(3,4)(6,7) ]
: Boundary homomorphism maps source generators to:
  [ (5,6,7), (1,2)(3,4)(6,7) ]
: Action homomorphism maps range generators to automorphisms:
  (5,7,6) --> { source gens --> [ (4,5,6), (2,3)(4,5) ] }
  (1,2)(3,4)(6,7) --> { source gens --> [ (4,6,5), (2,3)(5,6) ] }
  These 2 automorphisms generate the group of automorphisms.
gap> AX3 := ActorXMod( X3 );; 
gap> 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:
  [ (5,7,6), (1,2)(3,4)(6,7) ]
: Boundary homomorphism maps source generators to:
  [ (5,7,6), (1,2)(3,4)(6,7) ]
: Action homomorphism maps range generators to automorphisms:
  (5,7,6) --> { source gens --> [ (1,2,3)(4,5,6), (1,6)(2,5)(3,4) ] }
  (1,2)(3,4)(6,7) --> { 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 );;  
gap> Display( IAX3 );
Crossed module InnerActor[c3->s3] :- 
: Source group has generators:
  [ (1,2,3)(4,5,6) ]
: Range group has generators:
  [ (5,6,7), (1,2)(3,4)(6,7) ]
: Boundary homomorphism maps source generators to:
  [ (5,7,6) ]
: Action homomorphism maps range generators to automorphisms:
  (5,6,7) --> { source gens --> [ (1,2,3)(4,5,6) ] }
  (1,2)(3,4)(6,7) --> { source gens --> [ (1,3,2)(4,6,5) ] }
  These 2 automorphisms generate the group of automorphisms.

6.1-3 XModCentre
‣ XModCentre( xmod )( attribute )
‣ InnerActorXMod( xmod )( attribute )
‣ InnerMorphism( xmod )( attribute )

Pairs of boundaries or identity mappings provide six morphisms of crossed modules. In particular, the boundaries of \(\mathcal{W}(\mathcal{X})\) and \(\mathcal{N}(\mathcal{X})\) form the inner morphism of \(\mathcal{X}\), mapping source elements to principal derivations and range elements to inner automorphisms. The image of \(\mathcal{X}\) under this morphism is the inner actor of \(\mathcal{X}\), while the kernel is the centre of \(\mathcal{X}\). In the example which follows, the inner morphism of X3=(c3->s3), from Chapter 5, is an inclusion of crossed modules.

Note that we appear to have defined two sorts of centre for a crossed module: XModCentre here, and CentreXMod (4.1-7) in the chapter on isoclinism. We suspect that these two definitions give the same answer, but this remains to be resolved.


gap> IMX3 := InnerMorphism( X3 );; 
gap> 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) ]
  [ (5,7,6), (1,2)(3,4)(6,7) ]
: Source Homomorphism maps source generators to:
  [ (1,2,3)(4,5,6) ]
: Range Homomorphism maps range generators to:
  [ (5,6,7), (1,2)(3,4)(6,7) ]
gap> IsInjective( IMX3 );
true
gap> ZX3 := XModCentre( X3 ); 
[Group( () )->Group( () )]

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