<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <!-- --> <!-- gp2act.xml XMOD documentation Chris Wensley --> <!-- & Murat Alp --> <!-- --> <!-- $Id: gp2act.xml,v 2.11 2008/04/30 gap Exp $ --> <!-- --> <!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <?xml version="1.0" encoding="ISO-8859-15"?> <!-- <M>Id: gp2act.xml,v 2.11 Exp <M> --> <Chapter Label="chap-actor2"> <Heading>Actors of 2d-objects</Heading> <Section><Heading>Actor of a crossed module</Heading> <Index>actor</Index> The <E>actor</E> of <M>{\cal X}</M> is a crossed module <M>(\Delta ~:~ {\cal W}({\cal X}) \to {\rm Aut}({\cal X}))</M> which was shown by Lue and Norrie, in \cite{N2} and \cite{N1} to give the automorphism object of a crossed module <M>{\cal X}</M>. In this implementation, the source of the actor is a permutation representation <M>W</M> of the Whitehead group of regular derivations, and the range is a permutation representation <M>A</M> of the automorphism group <M>{\rm Aut}({\cal X})</M> of <M>{\cal X}</M>. <ManSection> <Attr Name="WhiteheadXMod" Arg="xmod" /> <Attr Name="LueXMod" Arg="xmod" /> <Attr Name="NorrieXMod" Arg="xmod" /> <Attr Name="ActorXMod" Arg="xmod" /> <Attr Name="AutomorphismPermGroup" Arg="xmod" /> <Description> An automorphism <M>( \sigma, \rho )</M> of <C>X</C> acts on the Whitehead monoid by <M>\chi^{(\sigma,\rho)} = \sigma \circ \chi \circ \rho^{-1}</M>, and this action determines the action for the actor. In fact the four groups <M>R, S, W, A</M>, the homomorphisms between them, and the various actions, give five crossed modules forming a <E>crossed square</E>: <Index>crossed square</Index> <List> <Item> <M>{\cal X} = (\partial : S \to R)</M>,~ the initial crossed module, on the left, </Item> <Item> <M>{\cal W(X)} = (\eta : S \to W)</M>,~ the Whitehead crossed module of <M>{\cal X}</M>, at the top, </Item> <Item> <M>{\cal L(X)} = (\Delta\circ\eta = \alpha\circ\partial : S \to A)</M>,~ the Lue crossed module of <M>{\cal X}</M>, along the top-left to bottom-right diagonal, </Item> <Item> <M>{\cal N(X)} = (\alpha : R \to A)</M>,~ the Norrie crossed module of <M>{\cal X}</M>, at the bottom, and </Item> <Item> <M>{\rm Act}({\cal X}) = ( \Delta : W \to A)</M>,~ the actor crossed module of <M>{\cal X}</M>, on the right. </Item> </List> </Description> </ManSection> <ManSection> <Attr Name="Centre" Arg="xmod" /> <Attr Name="InnerActor" Arg="xmod" /> <Attr Name="InnerMorphism" Arg="xmod" /> <Description> Pairs of boundaries or identity mappings provide six morphisms of crossed modules. In particular, the boundaries of <M>\mathcal{W}(\mathcal{X})</M> and <M>\mathcal{N}(\mathcal{X})</M> form the <E>inner morphism</E> of <M>\mathcal{X}</M>, mapping source elements to principal derivations and range elements to inner automorphisms. The image of <M>\mathcal{X}</M> under this morphism is the <E>inner actor</E> of <M>\mathcal{X}</M>, while the kernel is the <E>centre</E> of <M>\mathcal{X}</M>. In the example which follows, using the crossed module <C>(X3 : c3 -> s3)</C> from Chapter <Ref Chap="chap-up2" />, the inner morphism is an inclusion of crossed modules. </Description> </ManSection> <Example> <![CDATA[ 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( () )] ]]> </Example> </Section> </Chapter>