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%W  actor2.tex                 XMOD documentation               Chris Wensley
%W                                                                & Murat Alp
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%H  $Id: actor2.tex,v 2.006 2004/09/06 gap Exp $
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%Y  Copyright (C) 1999,2001 University of St. Andrews, North Haugh,
%Y                          St. Andrews, Fife KY16 9SS, Scotland
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\Chapter{Actors of 2d-objects}

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\Section{Actor of a Crossed Module}

The *actor* of ${\cal X}$  is a crossed module
$(\Delta ~:~ {\cal W}({\cal X}) \to {\rm Aut}({\cal X}))$
which was shown by Lue and Norrie, in \cite{N2} and \cite{N1} to give
the automorphism object of a crossed module ${\cal X}$.
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 
${\rm Aut}({\cal X})$ of ${\cal X}$.

\> AutomorphismPermGroup( <xmod> ) A
\> WhiteheadXMod( <xmod> ) A
\> LueXMod( <xmod> ) A
\> NorrieXMod( <xmod> ) A
\> ActorXMod( <xmod> ) A

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

\beginlist%unordered
\item{$\bullet$}
${\cal X} = (\partial : S \to R)$,~  
the initial crossed module, on the left,
\item{$\bullet$}
${\cal W(X)} = (\eta : S \to W)$,~  
the Whitehead crossed module of ${\cal X}$, at the top,
\item{$\bullet$}
${\cal L(X)} = (\Delta\circ\eta = \alpha\circ\partial : S \to A)$,~  
the Lue crossed module of ${\cal X}$, 
along the top-left to bottom-right diagonal,
\item{$\bullet$}
${\cal N(X)} = (\alpha : R \to A)$,~  
the Norrie crossed module of ${\cal X}$, at the bottom, and 
\item{$\bullet$}
${\rm Act}({\cal X}) = ( \Delta : W \to A)$,~  
the actor crossed module of ${\cal X}$, on the right.
\endlist

\> 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 ${\cal W(X)}$ and ${\cal N(X)}$ 
form the *inner morphism* of ${\cal X}$, 
mapping source elements to principal derivations
and range elements to inner automorphisms.  
The image of ${\cal X}$ under this morphism is the *inner actor* of 
${\cal X}$, while the kernel is the *centre* of ${\cal X}$.
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.

\beginexample
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( () )]
\endexample

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