Merge pull request #76 from cdwensley/manual

adjustments to the manual

CHANGES.md

@@ -1,5 +1,14 @@
 # CHANGES to the 'XModAlg' package
 
+## 1.25 -> 1.25dev (01/07/24) 
+ * (08/07/24) renamed the actions and algebras in the tests/examples as 
+              act1, act2, ,,, act6  and added direct sum operations
+              AlgebraActionOnDirectSum and DirectSumAlgebraActions
+ * (17/05/24) added operations for a module as an algebra,
+              revised XModAlgebraByModule
+ * (26/06/24) added AlgebraActionByHomomorphism,
+              changed the example of XModAlgebraByBoundaryAndAction
+
 ## 1.23 -> 1.25 (17/05/24) 
  * (17/05/24) just 5 PRs since 1.23 came out - mostly CI changes
 

PackageInfo.g

@@ -9,7 +9,7 @@ SetPackageInfo( rec(
 PackageName := "XModAlg",
 Subtitle := "Crossed Modules and Cat1-Algebras",
 Version := "1.25dev",
-Date := "17/06/2024", # dd/mm/yyyy format
+Date := "08/07/2024", # dd/mm/yyyy format
 License := "GPL-2.0-or-later",
 
 Persons := [

doc/algebra.xml

@@ -58,14 +58,16 @@ on an ideal <M>I</M> of <M>A</M>.
 
 <Example>
 <![CDATA[
-gap> A5c6 := GroupRing( GF(5), Group( (1,2,3,4,5,6) ) );;
-gap> vecA := BasisVectors( Basis( A5c6 ) );; 
-gap> v := vecA[1] + vecA[3] + vecA[5];
+gap> A1 := GroupRing( GF(5), Group( (1,2,3,4,5,6) ) );;
+gap> SetName( A1, "A1" );
+gap> BA1 := BasisVectors( Basis( A1 ) );; 
+gap> v := BA1[1] + BA1[3] + BA1[5];
 (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4)
-gap> I5c6 := Ideal( A5c6, [v] );; 
-gap> v2 := vecA[2];
+gap> I1 := Ideal( A1, [v] );; 
+gap> SetName( I1, "I1" );
+gap> v1 := BA1[2];
 (Z(5)^0)*(1,2,3,4,5,6)
-gap> m2 := RegularAlgebraMultiplier( A5c6, I5c6, v2 ); 
+gap> m1 := RegularAlgebraMultiplier( A1, I1, v1 ); 
 [ (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4), 
   (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) ] -> 
 [ (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2), 
@@ -75,7 +77,7 @@ gap> m2 := RegularAlgebraMultiplier( A5c6, I5c6, v2 );
 
 <ManSection>
    <Oper Name="IsAlgebraMultiplier"
-         Arg="m" />
+         Arg="mu" />
 <Description> 
 This function tests the condition <M>\mu(ab) = (\mu a)b = a(\mu b)</M> 
 for all <M>a,b</M> in the basis for <M>A</M>. 
@@ -84,16 +86,16 @@ for all <M>a,b</M> in the basis for <M>A</M>.
 
 <Example>
 <![CDATA[
-gap> IsAlgebraMultiplier( m2 ); 
+gap> IsAlgebraMultiplier( m1 ); 
 true
-gap> one := One( A5c6 );; 
-gap> L := List( vecA, v -> one );; 
-gap> m1 := LeftModuleHomomorphismByImages( A5c6, A5c6, vecA, L ); 
+gap> one := One( A1 );; 
+gap> L1 := List( BA1, v -> one );; 
+gap> h1 := LeftModuleHomomorphismByImages( A1, A1, BA1, L1 ); 
 [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), 
   (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2) 
  ] -> [ (Z(5)^0)*(), (Z(5)^0)*(), (Z(5)^0)*(), (Z(5)^0)*(), (Z(5)^0)*(), 
   (Z(5)^0)*() ]
-gap> IsAlgebraMultiplier( m1 );                                                
+gap> IsAlgebraMultiplier( h1 );                                                
 false
 ]]>
 </Example>
@@ -111,17 +113,16 @@ form an algebra with product <M>\mu_b \circ \mu_{b'} = \mu_{bb'}</M>.
 
 <Example>
 <![CDATA[
-gap> v3 := vecA[3];
+gap> u1 := BA1[3];
 (Z(5)^0)*(1,3,5)(2,4,6)
-gap> B5c3 := Subalgebra( A5c6, [ v3 ] );; 
-gap> M := MultiplierAlgebraOfIdealBySubalgebra( A5c6, I5c6, B5c3 );
+gap> S1 := Subalgebra( A3, [ u1 ] );; 
+gap> SetName( S1, "S1" );
+gap> MS1 := MultiplierAlgebraOfIdealBySubalgebra( A1, I1, S1 );
 <algebra of dimension 1 over GF(5)>
-gap> vecM := BasisVectors( Basis( M ) );; 
-gap> vecM[1];
-<linear mapping by matrix, 
-<two-sided ideal in <algebra-with-one of dimension 6 over GF(5)>, (dimension 2
- )> -> <two-sided ideal in <algebra-with-one of dimension 6 over GF(5)>, 
-  (dimension 2)>>
+gap> SetName( MS1, "MS1" );
+gap> BMS1 := BasisVectors( Basis( MS1 ) );; 
+gap> BMS1[1];
+<linear mapping by matrix, I1 -> I1>
 ]]>
 </Example>
 
@@ -138,10 +139,10 @@ This operation returns <C>MultiplierAlgebraOfIdealBySubalgebra(A,A,A);</C>.
 
 <Example>
 <![CDATA[
-gap> MA5c6 := RegularAlgebraMultiplier( A5c6 );
+gap> MA1 := MultiplierAlgebra( A1 );
 <algebra of dimension 6 over GF(5)>
-gap> vecM := BasisVectors( Basis( MA5c6 ) );; 
-gap> vecM[3];
+gap> BMA1 := BasisVectors( Basis( MA1 ) );; 
+gap> BMA1[3];
 <linear mapping by matrix, <algebra-with-one of dimension 
 6 over GF(5)> -> <algebra-with-one of dimension 6 over GF(5)>>
 ]]>
@@ -160,15 +161,14 @@ homomorphism from <M>B</M> to <M>M</M> mapping <M>b</M> to <M>\mu_b</M>.
 
 <Example>
 <![CDATA[
-gap> hom := MultiplierHomomorphism( MA5c6 );; 
-gap> ImageElm( hom, vecA[2] ); 
-Basis( <two-sided ideal in <algebra-with-one of dimension 6 over GF(5)>, 
-  (dimension 2)>, 
-[ (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4), 
-  (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) 
- ] ) -> 
-[ (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2), 
-  (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4) ]
+gap> hom1 := MultiplierHomomorphism( MA1 );;
+gap> ImageElm( hom1, BA1[2] ); 
+Basis( A1, [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2\
+,4,6), 
+  (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2) 
+ ] ) -> [ (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), 
+  (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2), 
+  (Z(5)^0)*() ]
 ]]>
 </Example> 
 
@@ -209,6 +209,20 @@ is a commutative action if and only if the following five axioms hold:
 for all <M>k \in </M><B>k</B>, <M>r,r' \in R</M>, and <M>s,s' \in S</M>. 
 <P/> 
 
+Notice in particular that, for fixed <M>r \in R</M>,
+the map <M>s \mapsto r \cdot s</M> is a vector space homomorphism,
+but not in general an algebra homomorphism.
+<P/>
+
+<ManSection>
+   <Func Name="AlgebraAction"
+         Arg="args" />
+<Description> 
+This global function calls one of the following operations, 
+depending on the arguments supplied.
+</Description>
+
+</ManSection>
 <ManSection>
    <Oper Name="AlgebraActionByMultipliers"
          Arg="A I B" />
@@ -223,20 +237,21 @@ over the field <M>GF(5)</M>.
 The ideal is generated by <M>v = () + (1,3,5)(2,4,6) + (1,5,3)(2,6,4)</M>. 
 The generator <M>r = (1,2,3,4,5,6)</M> acts on <M>v</M> 
 by multiplication to give the vector 
-<M>r \cdot v = (1,2,3,4,5,6) + (1,4)(2,5)(3,6) + (1,6,5,4,3,2)</M>. 
+<M>r \cdot v = (1,2,3,4,5,6) + (1,4)(2,5)(3,6) + (1,6,5,4,3,2)</M>,
+as shown in <Ref Oper="AlgebraActionByHomomorphism"/>
 </Description> 
 </ManSection> 
 
 <Example>
 <![CDATA[
-gap> A5c6 := GroupRing( GF(5), Group( (1,2,3,4,5,6) ) );;
-gap> vecA := BasisVectors( Basis( A5c6 ) );; 
-gap> v := vecA[1] + vecA[3] + vecA[5];
+gap> A1 := GroupRing( GF(5), Group( (1,2,3,4,5,6) ) );;
+gap> BA1 := BasisVectors( Basis( A1 ) );; 
+gap> v := BA1[1] + BA1[3] + BA1[5];
 (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4)
-gap> I5c6 := Ideal( A5c6, [v] );; 
-gap> actm := AlgebraActionByMultipliers( A5c6, I5c6, A5c6 );; 
-gap> actm2 := Image( actm, vecA[2] );; 
-gap> Image( actm2, v );
+gap> I1 := Ideal( A1, [v] );; 
+gap> act1 := AlgebraActionByMultipliers( A1, I1, A1 );; 
+gap> act12 := Image( act1, BA1[2] );; 
+gap> Image( act12, v );
 (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2)
 ]]>
 </Example>
@@ -245,66 +260,79 @@ gap> Image( actm2, v );
    <Oper Name="AlgebraActionBySurjection"
          Arg="hom" />
 <Description>
-Let <M>\theta : A \to B</M> be a surjective algebra homomorphism 
-such that <M>\ker\theta</M> is contained in the annihilator of <M>A</M>. 
-Then <M>B</M> acts on <M>A</M> by <M>b \cdot a = pa</M> 
-where <M>p \in (\theta^{-1}b)</M>. 
+Let <M>\theta : B \to A</M> be a surjective algebra homomorphism 
+such that <M>\ker\theta</M> is contained in the annihilator of <M>B</M>. 
+Then <M>A</M> acts on <M>B</M> by <M>a \cdot b = pb</M> 
+where <M>p \in (\theta^{-1}a)</M>. 
 Note that this action is well defined since 
-<M>\theta^{-1}b = \{ p+k ~|~ k \in \ker\theta \}</M> 
-and <M>(p+k)a = pa+ka = ca+0</M>. 
+<M>\theta^{-1}a = \{ p+k ~|~ k \in \ker\theta \}</M> 
+and <M>(p+k)b = pb+kb = pb+0</M>. 
 <P/>
 Continuing with the previous example, 
-we construct the quotient algebra <M>Q5c6 = A5c6/I5c6</M>, 
-and the natural homomorphism <M>\theta : A5c6 \to Q5c6</M>. 
-The kernel of <M>\theta</M> is not contained in 
-the annihilator of <M>A5c6</M>, so an attempt to form the action fails. 
+we construct the quotient algebra <M>Q1 = A1/I1</M>, 
+and the natural homomorphism <M>\theta_1 : A1 \to Q1</M>. 
+The kernel of <M>\theta</M> is not contained in the annihilator of <M>A1</M>,
+so an attempt to form the action fails. 
 <P/>
-An alternative example involves a single-generator matrix algebra.  
+An alternative example involves a matrix algebra <M>A_2</M>
+with generator <M>m_2</M>, basis <M>\{m_2,m_2^2,m_2^3\}</M>,
+and where <M>m_2^4=0</M>.
+The ideal <M>I_2</M> is generated by <M>m_2^3</M>
+and the quotient <M>Q_2</M> has basis <M>\{[m_2],[m_2^2]\}</M>.
 </Description> 
 </ManSection> 
 
 <Example>
 <![CDATA[
-gap> theta := NaturalHomomorphismByIdeal( A5c6, I5c6 );
+gap> theta1 := NaturalHomomorphismByIdeal( A1, I1 );
 <linear mapping by matrix, <algebra-with-one of dimension 
 6 over GF(5)> -> <algebra of dimension 4 over GF(5)>>
-gap> List( vecA, v -> ImageElm( theta, v ) ); 
+gap> List( BA1, v -> ImageElm( theta1, v ) ); 
 [ v.1, v.2, v.3, v.4, (Z(5)^2)*v.1+(Z(5)^2)*v.3, (Z(5)^2)*v.2+(Z(5)^2)*v.4 ]
-gap> actp := AlgebraActionBySurjection( theta );
+gap> AlgebraActionBySurjection( theta1 );
 kernel of hom is not in the annihilator of A
 fail
 gap> ## an example which does not fail: 
-gap> m := [ [0,1,2,3], [0,0,1,2], [0,0,0,1], [0,0,0,0] ];; 
-gap> m^2;
+gap> m2 := [ [0,1,2,3], [0,0,1,2], [0,0,0,1], [0,0,0,0] ];; 
+gap> m2^2;
 [ [ 0, 0, 1, 4 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ]
-gap> m^3;
+gap> m2^3;
 [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ]
-gap> A1 := Algebra( Rationals, [m] );;
-gap> SetName( A1, "A1" );
-gap> A3 := Subalgebra( A1, [m^3] );; 
-gap> nat3 := NaturalHomomorphismByIdeal( A1, A3 ); 
-<linear mapping by matrix, A1 -> <algebra of dimension 2 over Ration\
+gap> A2 := Algebra( Rationals, [m2] );;
+gap> SetName( A2, "A2" );
+gap> S2 := Subalgebra( A2, [m2^3] );; 
+gap> SetName( S2, "S2" );
+gap> nat2 := NaturalHomomorphismByIdeal( A2, S2 ); 
+<linear mapping by matrix, A2 -> <algebra of dimension 2 over Ration\
 als>>
-gap> Q13 := Image( nat3 );;
-gap> SetName( Q13, "Q13" );
-gap> Display(nat3);
-LeftModuleHomomorphismByMatrix( Basis( A1, 
+gap> Q2 := Image( nat2 );;
+gap> SetName( Q2, "Q2" );
+gap> Display( nat2 );
+LeftModuleHomomorphismByMatrix( Basis( A2, 
 [ [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ], 
   [ [ 0, 1, 2, 3 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ] ], 
   [ [ 0, 0, 1, 4 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] ] ), 
-[ [ 0, 0 ], [ 1, 0 ], [ 0, 1 ] ], CanonicalBasis( Q13 ) )
-gap> act3 := AlgebraActionBySurjection( nat3 );; 
-gap> B3 := Image( act3 );;
-gap> bvB3 := BasisVectors( Basis( B3 ) );;
-gap> b1 := bvB3[1];;  b2 := bvB3[2];;
-gap> [ Image(b1,m)=m^2, Image(b1,m^2)=m^3, Image(b1,m^3)=Zero(A1) ];
+[ [ 0, 0 ], [ 1, 0 ], [ 0, 1 ] ], CanonicalBasis( Q2 ) )
+gap> act2 := AlgebraActionBySurjection( nat2 );; 
+gap> C2 := Image( act2 );;
+gap> BC2 := BasisVectors( Basis( C2 ) );;
+gap> b1 := BC2[1];;  b2 := BC2[2];;
+gap> [ Image(b1,m2)=m2^2, Image(b1,m2^2)=m2^3, Image(b1,m2^3)=Zero(A2) ];
 [ true, true, true ]
-gap> [ Image(b2,m)=m^3, b2=b1^2 ];
+gap> [ Image(b2,m2)=m2^3, b2=b1^2 ];
 [true, true ]
 ]]>
 </Example>
 
 
+<#Include Label="AlgebraActionByHomomorphism">
+
+</Section>
+
+<Section>
+
+<Heading>Algebra modules</Heading> 
+
 <#Include Label="AlgebraModules">
 
 <#Include Label="ModuleAsAlgebra">
@@ -315,6 +343,27 @@ gap> [ Image(b2,m)=m^3, b2=b1^2 ];
 
 <#Include Label="AlgebraActionByModule">
 
+</Section>
+
+<Section>
+
+<Heading>Actions on direct sums of algebras</Heading>
+
+<#Include Label="DirectSumOfAlgebrasInfo">
+
+<#Include Label="EmbeddingForDirectSumOfAlgebras">
+
+<#Include Label="DirectSumOfAlgebraHomomorphisms">
+
+<#Include Label="AlgebraActionOnDirectSum">
+
+<#Include Label="DirectSumOfAlgebraActions">
+
+</Section>
+
+<Section>
+
+<Heading>Other operations on algebras</Heading>
 
 <ManSection>
    <Oper Name="SemidirectProductOfAlgebras"
@@ -352,26 +401,26 @@ Continuing the example above,
 
 <Example>
 <![CDATA[
-gap> P := SemidirectProductOfAlgebras( A5c6, actm, I5c6 ); 
+gap> P1 := SemidirectProductOfAlgebras( A1, act1, I1 ); 
 <algebra of dimension 8 over GF(5)>
-gap> Embedding( P, 1 );
+gap> Embedding( P1, 1 );
 [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), 
   (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2) 
  ] -> [ v.1, v.2, v.3, v.4, v.5, v.6 ]
-gap> Embedding( P, 2 );
+gap> Embedding( P1, 2 );
 [ (Z(5)^0)*()+(Z(5)^0)*(1,3,5)(2,4,6)+(Z(5)^0)*(1,5,3)(2,6,4), 
   (Z(5)^0)*(1,2,3,4,5,6)+(Z(5)^0)*(1,4)(2,5)(3,6)+(Z(5)^0)*(1,6,5,4,3,2) ] -> 
 [ v.7, v.8 ]
-gap> Projection( P, 1 );
+gap> Projection( P1, 1 );
 [ v.1, v.2, v.3, v.4, v.5, v.6, v.7, v.8 ] -> 
 [ (Z(5)^0)*(), (Z(5)^0)*(1,2,3,4,5,6), (Z(5)^0)*(1,3,5)(2,4,6), 
   (Z(5)^0)*(1,4)(2,5)(3,6), (Z(5)^0)*(1,5,3)(2,6,4), (Z(5)^0)*(1,6,5,4,3,2), 
   <zero> of ..., <zero> of ... ]
-gap> P3 := SemidirectProductOfAlgebras( Q13, act3, A1 );
-<algebra of dimension 5 over Rationals>
-gap> Embedding( P3, 1 );
+gap> P2 := SemidirectProductOfAlgebras( Q2, act2, A2 );
+Q2 |X A2
+gap> Embedding( P2, 1 );
 [ v.1, v.2 ] -> [ v.1, v.2 ]
-gap> Embedding( P3, 2 );
+gap> Embedding( P2, 2 );
 [ [ [ 0, 1, 2, 3 ], [ 0, 0, 1, 2 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ] ], 
   [ [ 0, 0, 1, 4 ], [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ], 
   [ [ 0, 0, 0, 1 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ], [ 0, 0, 0, 0 ] ] ] ->