73.113 About actors

The emphactor 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 xmodN2 and xmodN1 to give the automorphism object of a crossed module {cal X}. 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}.

An automorphism ( sigma, rho ) of X acts on the Whitehead monoid by chi^{(sigma,rho)} = sigma^{-1} chi rho, 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, form five crossed modules:

{cal X} &:& S to R & the initial crossed module,
{cal W(X)} &:& S to W & the Whitehead crossed module of {cal X},
{cal L(X)} &:& S to A & the Lue crossed module of {cal X},
{cal N(X)} &:& R to A & the Norrie crossed module of {cal X}, and
{rm Act}({cal X}) &:& W to A & the actor crossed module of {cal X}.

These 5 crossed modules, together with the evaluation W times R to S, (chi,r) mapsto chi r, form a crossed square:

                        S ------ WX ------> W
                        :  \                :
                        :     \             :
                        X        LX        ActX
                        :           \       :
                        :              \    :
                        V                 \ V
                        R ------ NX ------> A  

in which pairs of boundaries or identity mappings provide six morphisms of crossed modules. In particular, the boundaries of WX and NX form the emphinner morphism of X, mapping source elements to inner derivations and range elements to inner automorphisms. The image of X under this morphism is the emphinner actor of X, while the kernel is the emphcentre of X.

In the example which follows, using the usual (X1 : c5 -> Aut(c5)), Act(X1) is isomorphic to X1 and to LX1 while the Whitehead and Norrie boundaries are identity homomorphisms.

    gap> X1;
    Crossed module [c5->PermAut(c5)]
    gap> WGX1 := WhiteheadPermGroup( X1 );
    WG([c5->PermAut(c5)])
    gap> WGX1.generators;
    [ (1,2,3,4,5) ]
    gap> AX1 := AutomorphismPermGroup( X1 );
    PermAut([c5->PermAut(c5)])
    gap> AX1.generators;
    [ (1,2,4,3) ]
    gap> XModMorphismAutoPerm( X1, AX1.generators[1] );
    Morphism of crossed modules <[c5->PermAut(c5)] >-> [c5->PermAut(c5)]>

    gap> WX1 := Whitehead( X1 );
    Crossed module Whitehead[c5->PermAut(c5)]
    gap> NX1 := Norrie( X1 );
    Crossed module Norrie[c5->PermAut(c5)]
    gap> LX1 := Lue( X1 );
    Crossed module Lue[c5->PermAut(c5)]
    gap> ActX1 := Actor( X1 );;
    gap> XModPrint( ActX1);
    Crossed module Actor[c5->PermAut(c5)] :- 
    : Source group WG([c5->PermAut(c5)]) has generators:
      [ (1,2,3,4,5) ]
    : Range group has parent ( PermAut(c5)xPermAut(PermAut(c5)) )
        and has generators:  [ (1,2,4,3) ]
    : Boundary homomorphism maps source generators to:
      [ () ]
    : Action homomorphism maps range generators to automorphisms:
      (1,2,4,3) --> { source gens --> [ (1,3,5,2,4) ] }
      This automorphism generates the group of automorphisms.

    gap> InActX1 := InnerActor( X1 );
    Crossed module Actor[c5->PermAut(c5)]
    gap> InActX1 = ActX1;
    true
    gap> InnerMorphism( X1 );
    Morphism of crossed modules
        <[c5->PermAut(c5)] >-> Actor[c5->PermAut(c5)]>
    gap> Centre( X1 );
    Crossed module Centre[c5->PermAut(c5)]  

All of these constructions are stored in a sub-record X1.actorSquare.

Previous Up Top Next
Index