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5 Derivations and Sections

5.1 Whitehead Multiplication

5.1-1 IsDerivation

The Whitehead monoid Der(mathcalX) of mathcalX was defined in [Whi48] to be the monoid of all derivations from R to S, that is the set of all maps χ : R -> S, with Whitehead multiplication ⋆ (on the right) satisfying:

{\bf Der\ 1}: \chi(qr) ~=~ (\chi q)^{r} \; (\chi r), \qquad {\bf Der\ 2}: (\chi_1 \star \chi_2)(r) ~=~ (\chi_2 r)(\chi_1 r)(\chi_2 \partial \chi_1 r).

The zero map is the identity for this composition. Invertible elements in the monoid are called regular. The Whitehead group of mathcalX is the group of regular derivations in Der(mathcalX ). In the next chapter the actor of mathcalX is defined as a crossed module whose source and range are permutation representations of the Whitehead group and the automorphism group of mathcalX.

The construction for cat1-groups equivalent to the derivation of a crossed module is the section. The monoid of sections of mathcalC = (e;t,h : G -> R) is the set of group homomorphisms ξ : R -> G, with Whitehead multiplication ⋆ (on the right) satisfying:

{\bf Sect\ 1}: t \circ \xi ~=~ {\rm id}_R, \quad {\bf Sect\ 2}: (\xi_1 \star \xi_2)(r) ~=~ (\xi_1 r)(e h \xi_1 r)^{-1}(\xi_2 h \xi_1 r) ~=~ (\xi_2 h \xi_1 r)(e h \xi_1 r)^{-1}(\xi_1 r).

The embedding e is the identity for this composition, and h(ξ_1 ⋆ ξ_2) = (h ξ_1)(h ξ_2). A section is regular when h ξ is an automorphism, and the group of regular sections is isomorphic to the Whitehead group.

If ϵ denotes the inclusion of S = ker t in G then ∂ = h ϵ : S -> R and

\xi r ~=~ (e r)(e \chi r), \quad\mbox{which equals}\quad (r, \chi r) ~\in~ R \ltimes S,

determines a section ξ of mathcalC in terms of the corresponding derivation χ of mathcalX, and conversely.

5.1-2 DerivationByImages

Derivations are stored like group homomorphisms by specifying the images of a generating set. Images of the remaining elements may then be obtained using axiom Der 1. The function IsDerivation is automatically called to check that this procedure is well-defined.

In the following example a cat1-group C3 and the associated crossed module X3 are constructed, where X3 is isomorphic to the inclusion of the normal cyclic group c3 in the symmetric group s3.

gap> g18 := Group( (1,2,3), (4,5,6), (2,3)(5,6) );;
gap> SetName( g18, "g18" );
gap> gen18 := GeneratorsOfGroup( g18 );;
gap> g1 := gen18[1];;  g2 := gen18[2];;  g3 := gen18[3];;
gap> s3 := Subgroup( g18, gen18{[2..3]} );;
gap> SetName( s3, "s3" );;
gap> t := GroupHomomorphismByImages( g18, s3, gen18, [g2,g2,g3] );;
gap> h := GroupHomomorphismByImages( g18, s3, gen18, [(),g2,g3] );;
gap> e := GroupHomomorphismByImages( s3, g18, [g2,g3], [g2,g3] );;
gap> C3 := Cat1( t, h, e );
[g18=>s3]
gap> SetName( Kernel(t), "c3" );;
gap> X3 := XModOfCat1( C3 );;
gap> Display( X3 );
Crossed module [c3->s3] :- 
: Source group has generators:
  [ (1,2,3)(4,6,5) ]
: Range group has generators:
  [ (4,5,6), (2,3)(5,6) ]
: Boundary homomorphism maps source generators to:
  [ (4,6,5) ]
: Action homomorphism maps range generators to automorphisms:
  (4,5,6) --> { source gens --> [ (1,2,3)(4,6,5) ] }
  (2,3)(5,6) --> { source gens --> [ (1,3,2)(4,5,6) ] }
  These 2 automorphisms generate the group of automorphisms.
: associated cat1-group is [g18=>s3]

gap> imchi := [ (1,2,3)(4,6,5), (1,2,3)(4,6,5) ];;
gap> chi := DerivationByImages( X3, imchi );
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ],
[ (1,2,3)(4,6,5), (1,2,3)(4,6,5) ] )

5.1-3 SectionByImages

Sections are group homomorphisms, so do not need a special representation. Operations SectionByDerivation and DerivationBySection convert derivations to sections, and vice-versa, calling Cat1OfXMod and XModOfCat1 automatically.

Two strategies for calculating derivations and sections are implemented, see [AW00]. The default method for AllDerivations is to search for all possible sets of images using a backtracking procedure, and when all the derivations are found it is not known which are regular. In early versions of this package, the default method for AllSections( <C> ) was to compute all endomorphisms on the range group R of C as possibilities for the composite h ξ. A backtrack method then found possible images for such a section. In the current version the derivations of the associated crossed module are calculated, and these are all converted to sections using SectionByDerivation.

gap> xi := SectionByDerivation( chi );
SectionByImages( s3, g18, [ (4,5,6), (2,3)(5,6) ], [ (1,2,3), (1,2)(4,6) ] )

5.2 Whitehead Groups and Monoids

5.2-1 RegularDerivations

There are two functions to determine the elements of the Whitehead group and the Whitehead monoid of a crossed module, namely RegularDerivations and AllDerivations. (The functions RegularSections and AllSections perform corresponding tasks for a cat1-group.)

Using our example X3 we find that there are just nine derivations. (In fact, six of them regular, as we shall see in the next section. The associated group is isomorphic to the symmetric group s3.)

gap> all3 := AllDerivations( X3 );;
gap> imall3 := ImagesList( all3 );; 
gap> PrintListOneItemPerLine( imall3 );
[ [ (), () ],
  [ (), (1,3,2)(4,5,6) ],
  [ (), (1,2,3)(4,6,5) ],
  [ (1,3,2)(4,5,6), () ],
  [ (1,3,2)(4,5,6), (1,3,2)(4,5,6) ],
  [ (1,3,2)(4,5,6), (1,2,3)(4,6,5) ],
  [ (1,2,3)(4,6,5), () ],
  [ (1,2,3)(4,6,5), (1,3,2)(4,5,6) ],
  [ (1,2,3)(4,6,5), (1,2,3)(4,6,5) ]
  ]
gap> KnownAttributesOfObject( all3 );
[ "Object2d", "ImagesList", "AllOrRegular", "ImagesTable" ]
gap> PrintListOneItemPerLine( ImagesTable( all3 ) );
[ [ 1, 1, 1, 1, 1, 1 ],
  [ 1, 1, 1, 3, 3, 3 ],
  [ 1, 1, 1, 2, 2, 2 ],
  [ 1, 3, 2, 1, 3, 2 ],
  [ 1, 3, 2, 3, 2, 1 ],
  [ 1, 3, 2, 2, 1, 3 ],
  [ 1, 2, 3, 1, 2, 3 ],
  [ 1, 2, 3, 3, 1, 2 ],
  [ 1, 2, 3, 2, 3, 1 ]
  ]

5.2-2 CompositeDerivation

The Whitehead product χ_1 ⋆ χ_2 is implemented as CompositeDerivation(<chi1>,<chi2>). The composite of two sections is just the composite of homomorphisms.

gap> reg3 := RegularDerivations( X3 );;
gap> imder3 := ImagesList( reg3 );;  Length( imder3 );
6
gap> chi4 := DerivationByImages( X3, imder3[4] );
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ (1,3,2)(4,5,6), () ] )
gap> chi5 := DerivationByImages( X3, imder3[5] );
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], 
[ (1,3,2)(4,5,6), (1,3,2)(4,5,6) ] )
gap> im4 := UpImagePositions( chi4 );
[ 1, 3, 2, 1, 3, 2 ]
gap> im5 := UpImagePositions( chi5 );
[ 1, 3, 2, 3, 2, 1 ]
gap> chi45 := chi4 * chi5;
DerivationByImages( s3, c3, [ (4,5,6), (2,3)(5,6) ], [ (), (1,3,2)(4,5,6) ] )
gap> im45 := UpImagePositions( chi45 );
[ 1, 1, 1, 3, 3, 3 ]
gap> Position( imder3, UpGeneratorImages( chi45 ) );
2

5.2-3 WhiteheadGroupTable

Multiplication tables for the Whitehead group or monoid enable the construction of permutation or transformation representations.

gap> wgt3 := WhiteheadGroupTable( X3 );; 
gap> PrintListOneItemPerLine( wgt3 );
[ [ 1, 2, 3, 4, 5, 6 ],
  [ 2, 3, 1, 5, 6, 4 ],
  [ 3, 1, 2, 6, 4, 5 ],
  [ 4, 6, 5, 1, 3, 2 ],
  [ 5, 4, 6, 2, 1, 3 ],
  [ 6, 5, 4, 3, 2, 1 ]
  ]
gap> wpg3 := WhiteheadPermGroup( X3 );
Group([ (1,2,3)(4,5,6), (1,4)(2,6)(3,5) ])
gap> wmt3 := WhiteheadMonoidTable( X3 );; 
gap> PrintListOneItemPerLine( wmt3 );
[ [ 1, 2, 3, 4, 5, 6, 7, 8, 9 ],
  [ 2, 3, 1, 5, 6, 4, 8, 9, 7 ],
  [ 3, 1, 2, 6, 4, 5, 9, 7, 8 ],
  [ 4, 6, 5, 1, 3, 2, 7, 9, 8 ],
  [ 5, 4, 6, 2, 1, 3, 8, 7, 9 ],
  [ 6, 5, 4, 3, 2, 1, 9, 8, 7 ],
  [ 7, 7, 7, 7, 7, 7, 7, 7, 7 ],
  [ 8, 8, 8, 8, 8, 8, 8, 8, 8 ],
  [ 9, 9, 9, 9, 9, 9, 9, 9, 9 ]
  ]
gap> wtm3 := WhiteheadTransMonoid( X3 );
<transformation monoid on 9 pts with 3 generators>
gap> GeneratorsOfMonoid( wtm3 ); 
[ Transformation( [ 2, 3, 1, 5, 6, 4, 8, 9, 7 ] ), 
  Transformation( [ 4, 6, 5, 1, 3, 2, 7, 9, 8 ] ), 
  Transformation( [ 7, 7, 7, 7, 7, 7, 7, 7, 7 ] ) ]

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