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<div class="ChapSects"><a href="chap5.html#X812A5A097EADEB5E">5 Crossed products and their elements</a>
<div class="ContSect"> <a href="chap5.html#X79122C7F877430A7">5.1 Construction of crossed products</a>

<div class="ContSSBlock">
 <a href="chap5.html#X797F31EF7B51A4DF">5.1-1 CrossedProduct</a>
</div></div>
<div class="ContSect"> <a href="chap5.html#X8560A2F37B608A9F">5.2 Crossed product elements and their properties</a>

<div class="ContSSBlock">
 <a href="chap5.html#X7D2313AA82F1D5CC">5.2-1 ElementOfCrossedProduct</a>
</div></div>
</div>

<h3>5 Crossed products and their elements</h3>

The package Wedderga provides functions to construct crossed products over a group with coefficients in an associative ring with identity, and with the multiplication determined by a given action and twisting (see <a href="chap9.html#X7FB21779832CE1CB">9.6</a> for definitions). This can be done using the function <code class="func">CrossedProduct</code> (<a href="chap5.html#X797F31EF7B51A4DF">5.1-1</a>).

Note that this function does not check the associativity conditions, so in fact it is the NC-version of itself, and its output will be always assumed to be associative. For all crossed products that appear in Wedderga algorithms the associativity follows from theoretical arguments, so the usage of the NC-method in the package is safe. If the user will try to construct a crossed product with his own action and twisting, he/she should check the associativity conditions himself/herself to make sure that the result is correct.

<a id="X79122C7F877430A7" name="X79122C7F877430A7"></a>

<h4>5.1 Construction of crossed products</h4>

<a id="X797F31EF7B51A4DF" name="X797F31EF7B51A4DF"></a>

<h5>5.1-1 CrossedProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ CrossedProduct</code>( <var class="Arg">R</var>, <var class="Arg">G</var>, <var class="Arg">act</var>, <var class="Arg">twist</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
Returns: Ring in the category <code class="code">IsCrossedProduct</code>.

The input should be formed by:

<ul>
<li>an associative ring <var class="Arg">R</var>,

</li>
<li>a group <var class="Arg">G</var>,

</li>
<li>a function <var class="Arg">act(RG,g)</var> of two arguments: the crossed product <var class="Arg">RG</var> and an element <var class="Arg">g</var> in G. It must return a mapping from <var class="Arg">R</var> to <var class="Arg">R</var> which can be applied via the "\^" operation, and

</li>
<li>a function <var class="Arg">twist(RG,g,h)</var> of three arguments: the crossed product <var class="Arg">RG</var> and a pair of elements of <var class="Arg">G</var>. It must return an invertible element of <var class="Arg">R</var>.

</li>
</ul>
Returns the crossed product of <var class="Arg">G</var> over the ring <var class="Arg">R</var> with action <var class="Arg">act</var> and twisting <var class="Arg">twist</var>.

The resulting crossed product belongs to the category <code class="code">IsCrossedProduct</code>, which is defined as a subcategory of <code class="code">IsFLMLORWithOne</code>.

An example of the trivial action:

<div class="example"><pre>
act := function(RG,a)
 return IdentityMapping( LeftActingDomain( RG ) );
end;
</pre></div>

and the trivial twisting:

<div class="example"><pre>
twist := function( RG , g, h )
 return One( LeftActingDomain( RG ) );
end;
</pre></div>

Let n be a positive integer and ξ_n an n-th complex primitive root of unity. The natural action of the group of units of ℤ_n, the ring of integers modulo n, on ℚ (ξ_n) can be defined as follows:

<div class="example"><pre>
act := function(RG,a)
 return ANFAutomorhism( LeftActingDomain( RG ) , Int( a ) );
end;
</pre></div>

In the following example one constructs the Hamiltonian quaternion algebra over the rationals as a crossed product of the group of units of the cyclic group of order 2 over ℚ (i)=GaussianRationals. One realizes the cyclic group of order 2 as the group of units of ℤ / 4 ℤ and one uses the natural isomorphism ℤ / 4 ℤ → Gal( ℚ (i)/ ℚ ) to describe the action.

<div class="example"><pre>

gap> R := GaussianRationals;
GaussianRationals
gap> G := Units( ZmodnZ(4) );; Size( G );
2
gap> act := function(RG,g)
> return ANFAutomorphism( LeftActingDomain(RG), Int(g) );
> end;
function( RG, g ) ... end
gap> twist1 := function( RG, g, h )
> if IsOne(g) or IsOne(h) then
> return One(LeftActingDomain(RG));
> else
> return -One(LeftActingDomain(RG));
> fi;
> end;
function( RG, g, h ) ... end
gap> RG := CrossedProduct( R, G, act, twist1 );
<crossed product over GaussianRationals of a group of size 2>
gap> i := E(4) * One(G)^Embedding(G,RG); 
(ZmodnZObj( 1, 4 ))*(E(4))
gap> j := ZmodnZObj(3,4)^Embedding(G,RG); 
(ZmodnZObj( 3, 4 ))*(1)
gap> i^2;
(ZmodnZObj( 1, 4 ))*(-1)
gap> j^2;
(ZmodnZObj( 1, 4 ))*(-1)
gap> i*j+j*i; 
<zero> of ...

</pre></div>

One can construct the following generalized quaternion algebra with the same action and a different twisting


ℚ (i,j|i^2=-1,j^2=-3,ji=-ij)


<div class="example"><pre>

gap> twist2:=function(RG,g,h)
> if IsOne(g) or IsOne(h) then
> return One(LeftActingDomain( RG ));
> else
> return -3*One(LeftActingDomain( RG ));
> fi;
> end;
function( RG, g, h ) ... end
gap> RG := CrossedProduct( R, G, act, twist2 ); 
<crossed product over GaussianRationals of a group of size 2>
gap> i := E(4) * One(G)^Embedding(G,RG); 
(ZmodnZObj( 1, 4 ))*(E(4))
gap> j := ZmodnZObj(3,4)^Embedding(G,RG); 
(ZmodnZObj( 3, 4 ))*(1)
gap> i^2; 
(ZmodnZObj( 1, 4 ))*(-1)
gap> j^2; 
(ZmodnZObj( 1, 4 ))*(-3)
gap> i*j+j*i; 
<zero> of ...

</pre></div>

The following example shows how to construct the Hamiltonian quaternion algebra over the rationals using the rationals as coefficient ring and the Klein group as the underlying group.

<div class="example"><pre>

gap> C2 := CyclicGroup(2);;
gap> G := DirectProduct(C2,C2);
<pc group of size 4 with 2 generators>
gap> act := function(RG,a)
> return IdentityMapping( LeftActingDomain(RG));
> end;
function( RG, a ) ... end
gap> twist := function( RG, g , h )
> local one,g1,g2,h1,h2,G;
> G := UnderlyingMagma( RG );
> one := One( C2 );
> g1 := Image( Projection(G,1), g );
> g2 := Image( Projection(G,2), g );
> h1 := Image( Projection(G,1), h );
> h2 := Image( Projection(G,2), h );
> if g = One( G ) or h = One( G ) then return 1;
> elif IsOne(g1) and not IsOne(g2) and not IsOne(h1) and not IsOne(h2)
> then return 1;
> elif not IsOne(g1) and IsOne(g2) and IsOne(h1) and not IsOne(h2)
> then return 1;
> elif not IsOne(g1) and not IsOne(g2) and not IsOne(h1) and IsOne(h2)
> then return 1;
> else return -1;
> fi;
> end;
function( RG, g, h ) ... end
gap> HQ := CrossedProduct( Rationals, G, act, twist );
<crossed product over Rationals of a group of size 4>

</pre></div>

Changing the rationals by the integers as coefficient ring one can construct the Hamiltonian quaternion ring.

<div class="example"><pre>

gap> HZ := CrossedProduct( Integers, G, act, twist );
<crossed product over Integers of a group of size 4>
gap> i := GeneratorsOfGroup(G)[1]^Embedding(G,HZ); 
(f1)*(1)
gap> j := GeneratorsOfGroup(G)[2]^Embedding(G,HZ);
(f2)*(1)
gap> i^2;
(<identity> of ...)*(-1)
gap> j^2; 
(<identity> of ...)*(-1)
gap> i*j+j*i; 
<zero> of ...

</pre></div>

One can extract the arguments used for the construction of the crossed product using the following attributes:

* <code class="code">LeftActingDomain</code> for the coefficient ring.

* <code class="code">UnderlyingMagma</code> for the underlying group.

* <code class="code">ActionForCrossedProduct</code> for the action.

* <code class="code">TwistingForCrossedProduct</code> for the twisting.

<div class="example"><pre>

gap> LeftActingDomain(HZ);
Integers
gap> G:=UnderlyingMagma(HZ);
<pc group of size 4 with 2 generators>
gap> ac := ActionForCrossedProduct(HZ);
function( RG, a ) ... end
gap> List( G , x -> ac( HZ, x ) );
[ IdentityMapping( Integers ), IdentityMapping( Integers ),
 IdentityMapping( Integers ), IdentityMapping( Integers ) ]
gap> tw := TwistingForCrossedProduct( HZ );
function( RG, g, h ) ... end
gap> List( G, x -> List( G , y -> tw( HZ, x, y ) ) );
[ [ 1, 1, 1, 1 ], [ 1, -1, -1, 1 ], [ 1, 1, -1, -1 ], [ 1, -1, 1, -1 ] ]

</pre></div>

Some more examples of crossed products arise from the Wedderburn decomposition (<a href="chap9.html#X84BB4A6081EAE905">9.3</a>) of group algebras.

<div class="example"><pre>

gap> G := SmallGroup(32,50);
<pc group of size 32 with 5 generators>
gap> A := SimpleAlgebraByCharacter( GroupRing(Rationals,G), Irr(G)[17]) ;
( <crossed product with center Rationals over GaussianRationals of a group of \
size 2>^[ 2, 2 ] )
gap> SimpleAlgebraByCharacterInfo( GroupRing(Rationals,G), Irr(G)[17]) ;
[ 2, Rationals, 4, [ 2, 3, 2 ] ]
gap> B := LeftActingDomain(A);
<crossed product with center Rationals over GaussianRationals of a group of si\
ze 2>
gap> L := LeftActingDomain(B);
GaussianRationals
gap> H := UnderlyingMagma( B );
<group of size 2 with 2 generators>
gap> Elements(H);
[ ZmodnZObj( 1, 4 ), ZmodnZObj( 3, 4 ) ]
gap> i := E(4) * One(H)^Embedding(H,B);
(ZmodnZObj( 1, 4 ))*(E(4))
gap> j := ZmodnZObj(3,4)^Embedding(H,B);
(ZmodnZObj( 3, 4 ))*(1)
gap> i^2;
(ZmodnZObj( 1, 4 ))*(-1)
gap> j^2;
(ZmodnZObj( 1, 4 ))*(-1)
gap> i*j+j*i;
<zero> of ...
gap> ac := ActionForCrossedProduct( B );
function( RG, a ) ... end
gap> tw := TwistingForCrossedProduct( B );
function( RG, a, b ) ... end
gap> List( H , x -> ac( B, x ) );
[ IdentityMapping( GaussianRationals ), ANFAutomorphism( GaussianRationals,
 3 ) ]
gap> List( H , x -> List( H , y -> tw( B, x, y ) ) );
[ [ 1, 1 ], [ 1, -1 ] ]

</pre></div>

<div class="example"><pre>

gap> QG:=GroupRing( Rationals, SmallGroup(24,3) );;
gap> WedderburnDecomposition(QG);
[ Rationals, CF(3), ( Rationals^[ 3, 3 ] ),
 <crossed product with center Rationals over GaussianRationals of a group of \
size 2>, <crossed product with center CF(3) over AsField( CF(3), CF(
 12) ) of a group of size 2> ]
gap> R:=WedderburnDecomposition( QG )[4];
<crossed product with center Rationals over GaussianRationals of a group of si\
ze 2>
gap> IsCrossedProduct(R);
true
gap> IsAlgebra(R);
true
gap> IsRing(R); 
true
gap> LeftActingDomain( R );
GaussianRationals
gap> AsList( UnderlyingMagma( R ) );
[ ZmodnZObj( 1, 4 ), ZmodnZObj( 3, 4 ) ]
gap> Print( ActionForCrossedProduct( R ) ); Print("\n");
function ( RG, a )
 local cond, redu;
 cond := OperationRecord( RG ).cond;
 redu := OperationRecord( RG ).redu;
 return
 ANFAutomorphism( CF( cond ), Int( PreImagesRepresentativeNC( redu, a ) ) );
end
gap> Print( TwistingForCrossedProduct( R ) ); Print("\n"); 
function ( RG, a, b )
 local orderroot, cocycle;
 orderroot := OperationRecord( RG ).orderroot;
 cocycle := OperationRecord( RG ).cocycle;
 return E( orderroot ) ^ Int( cocycle( a, b ) );
end
gap> IsAssociative(R);
true
gap> IsFinite(R); 
false
gap> IsFiniteDimensional(R);
true
gap> AsList(Basis(R));
[ (ZmodnZObj( 1, 4 ))*(1), (ZmodnZObj( 3, 4 ))*(1) ]
gap> GeneratorsOfLeftOperatorRingWithOne(R);
[ (ZmodnZObj( 1, 4 ))*(1), (ZmodnZObj( 3, 4 ))*(1) ]
gap> One(R);
(ZmodnZObj( 1, 4 ))*(1)
gap> Zero(R);
<zero> of ...
gap> Characteristic(R);
0
gap> CenterOfCrossedProduct(R);
Rationals

</pre></div>

The next example shows how one can use <code class="func">CrossedProduct</code> to produce generalized quaternion algebras. Note that one can construct quaternion algebras using the GAP function <code class="code">QuaternionAlgebra</code>.

<div class="example"><pre>

gap> Quat := function(R,a,b)
> local G,act,twist;
> if not(a in R and b in R and a <> Zero(R) and b <> Zero(R) ) then
> Error("<a> and must be non zero elements of <R>!!!");
> fi;
> G := SmallGroup(4,2);
> act := function(RG,a)
> return IdentityMapping( LeftActingDomain(RG));
> end;
> twist := function( RG, g , h )
> local one,g1,g2;
> one := One(G);
> g1 := G.1;
> g2 := G.2;
> if g = one or h = one then
> return One(R);
> elif g = g1 then
> if h = g2 then
> return One(R);
> else
> return a;
> fi;
> elif g = g2 then
> if h = g1 then
> return -One(R);
> elif h=g2 then
> return b;
> else
> return -b;
> fi;
> else
> if h = g1 then
> return -b;
> elif h=g2 then
> return b;
> else
> return -a*b;
> fi;
> fi;
> end;
> return CrossedProduct(R,G,act,twist);
> end;
function( R, a, b ) ... end
gap> HQ := Quat(Rationals,2,3);
<crossed product over Rationals of a group of size 4>
gap> G := UnderlyingMagma(HQ);
<pc group of size 4 with 2 generators>
gap> tw := TwistingForCrossedProduct( HQ );
function( RG, g, h ) ... end
gap> List( G, x -> List( G, y -> tw( HQ, x, y ) ) );
[ [ 1, 1, 1, 1 ], [ 1, 3, -1, -3 ], [ 1, 1, 2, 2 ], [ 1, 3, -3, -6 ] ]

</pre></div>

<a id="X8560A2F37B608A9F" name="X8560A2F37B608A9F"></a>

<h4>5.2 Crossed product elements and their properties</h4>

<a id="X7D2313AA82F1D5CC" name="X7D2313AA82F1D5CC"></a>

<h5>5.2-1 ElementOfCrossedProduct</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ElementOfCrossedProduct</code>( <var class="Arg">Fam</var>, <var class="Arg">zerocoeff</var>, <var class="Arg">coeffs</var>, <var class="Arg">elts</var> )</td><td class="tdright">( property )</td></tr></table></div>
Returns the element m_1*c_1 + ... + m_n*c_n of a crossed product, where <var class="Arg">elts</var> = [ m_1, m_2, ..., m_n ] is a list of magma elements, <var class="Arg">coeffs</var> = [ c_1, c_2, ..., c_n ] is a list of coefficients. The output belongs to the crossed product whose elements lie in the family <var class="Arg">Fam</var>. The second argument <var class="Arg">zerocoeff</var> must be the zero element of the coefficient ring containing coefficients c_i, and will be stored in the attribute <code class="code">ZeroCoefficient</code> of the crossed product element.

The output will be in the category <code class="code">IsElementOfCrossedProduct</code>, which is a subcategory of <code class="code">IsRingElementWithInverse</code>. It will have the presentation <code class="code">IsCrossedProductObjDefaultRep</code>.

Similarly to magma rings, one can obtain the list of coefficients and elements with <code class="code">CoefficientsAndMagmaElements</code> .

Also note from the example below and several other examples in this chapter that instead of <code class="code">ElementOfCrossedProduct</code> one can use <code class="code">Embedding</code> to embed elements of the coefficient ring and of the underlying magma into the crossed product.

<div class="example"><pre>

gap> QG := GroupRing( Rationals, SmallGroup(24,3) );
<algebra-with-one over Rationals, with 4 generators>
gap> R := WedderburnDecomposition( QG )[4];
<crossed product with center Rationals over GaussianRationals of a group of si\
ze 2>
gap> H := UnderlyingMagma( R );;
gap> fam := ElementsFamily( FamilyObj( R ) );;
gap> g := ElementOfCrossedProduct( fam, 0, [ 1, E(4) ], AsList(H) );
(ZmodnZObj( 1, 4 ))*(1)+(ZmodnZObj( 3, 4 ))*(E(4))
gap> CoefficientsAndMagmaElements( g ); 
[ ZmodnZObj( 1, 4 ), 1, ZmodnZObj( 3, 4 ), E(4) ]
gap> t := List( H, x -> x^Embedding( H, R ) );
[ (ZmodnZObj( 1, 4 ))*(1), (ZmodnZObj( 3, 4 ))*(1) ]
gap> t[1] + t[2]*E(4); 
(ZmodnZObj( 1, 4 ))*(1)+(ZmodnZObj( 3, 4 ))*(E(4))
gap> g = t[1] + E(4)*t[2];
false
gap> g = t[1] + t[2]*E(4);
true
gap> h := ElementOfCrossedProduct( fam, 0, [ E(4), 1 ], AsList(H) ); 
(ZmodnZObj( 1, 4 ))*(E(4))+(ZmodnZObj( 3, 4 ))*(1)
gap> g+h;
(ZmodnZObj( 1, 4 ))*(1+E(4))+(ZmodnZObj( 3, 4 ))*(1+E(4))
gap> g*E(4);
(ZmodnZObj( 1, 4 ))*(E(4))+(ZmodnZObj( 3, 4 ))*(-1)
gap> E(4)*g; 
(ZmodnZObj( 1, 4 ))*(E(4))+(ZmodnZObj( 3, 4 ))*(1)
gap> g*h;
(ZmodnZObj( 1, 4 ))*(2*E(4))

</pre></div>

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