direct sums of xmod-algebras

.gitignore

@@ -21,15 +21,19 @@
 /doc/*.toc
 
 /doc/title.xml
+/doc/xmodalg.xml
 /doc/XModAlg.xml
 /doc/manual.pdf
 /doc/bib.xml.bib
 /doc/_*.xml
 
 /tmp
 
+/tst/algebra.g
 /tst/cat1.g
-/tst/xmod.g
+/tst/dsum-xmod.g
 /tst/convert.g
+/tst/module.g
+/tst/xmod.g
 
 /gh-pages/

CHANGES.md

@@ -1,6 +1,7 @@
 # CHANGES to the 'XModAlg' package
 
-## 1.32 -> 1.33 (27/08/25) 
+## 1.32 -> 1.33 (16/03/26)
+ * (18/12/25) added new material on direct sums
  * (27/08/25) added installation instructions to the manual introduction
               replaced \mathcal by \mathbb which suits MathJax better
 

PackageInfo.g

@@ -8,8 +8,8 @@ SetPackageInfo( rec(
 
 PackageName := "XModAlg",
 Subtitle := "Crossed Modules and Cat1-Algebras",
-Version := "1.32",
-Date := "11/04/2025", # dd/mm/yyyy format
+Version := "1.32-dev",
+Date := "17/03/2026", # dd/mm/yyyy format
 License := "GPL-2.0-or-later",
 
 Persons := [

doc/xmod.xml

@@ -166,6 +166,11 @@ endomorphisms of <C>Source(X)</C>.
 </Item>
 </List> 
 
+The action in <C>XIAk4</C> is straightforward multiplication,
+so <C>f1</C> and <C>f2</C> map the generating set <C>[1-f1,f1-f2,f2-f1*f2]</C> 
+to <C>[f1-1,1-f1*f2,f1*f2-f2]</C> and <C>[f2-f1*f2,f1*f2-1,1-f1]</C>
+respectively.
+<P/>
 The following standard &GAP; operations have special &XModAlg; implementations: 
 <List>
 <Item>

init.g

@@ -2,7 +2,7 @@
 ##
 #W  init.g                 The XMODALG package               Zekeriya Arvasi
 #W                                                            & Alper Odabas
-#Y  Copyright (C) 2014-2024, Zekeriya Arvasi & Alper Odabas,  
+#Y  Copyright (C) 2014-2025, Zekeriya Arvasi & Alper Odabas,  
 ##
 
 if not IsBound( PreImagesRepresentativeNC ) then 
@@ -14,3 +14,4 @@ ReadPackage( "xmodalg", "lib/algebra.gd" );
 ReadPackage( "xmodalg", "lib/alg2obj.gd" );  
 ReadPackage( "xmodalg", "lib/module.gd" );
 ReadPackage( "xmodalg", "lib/alg2map.gd" );  
+ReadPackage( "xmodalg", "lib/dsum-xmod.gd" );  

lib/alg2obj.gd

@@ -80,7 +80,7 @@ DeclareOperation( "PreXModAlgebraByBoundaryAndAction",
    [ IsAlgebraHomomorphism, IsAlgebraAction ] );
 DeclareOperation( "PreXModAlgebraByBoundaryAndActionNC",
    [ IsAlgebraHomomorphism, IsAlgebraAction ] );
- DeclareOperation( "XModAlgebraByBoundaryAndAction",
+DeclareOperation( "XModAlgebraByBoundaryAndAction",
    [ IsAlgebraHomomorphism, IsAlgebraAction ] );
 DeclareOperation( "XModAlgebraByBoundaryAndActionNC",
    [ IsAlgebraHomomorphism, IsAlgebraAction ] );

lib/algebra.gi

@@ -582,7 +582,9 @@ function ( hom )
     for k in vecK do 
         for a in vecA do 
             if not ( k*a = zA ) then 
-                Print( "kernel of hom is not in the annihilator of A\n" ); 
+                Print( "!!!\n" );
+                Print( "kernel of hom is not in the annihilator of A\n" );
+                Print( "!!!\n" );
                 return fail;  
             fi; 
         od; 

lib/dsum-xmod.gd

@@ -0,0 +1,203 @@
+ #############################################################################
+##
+#W  dsum-xmod.gd               The XMODALG package            Zekeriya Arvasi
+#W                                                             & Alper Odabas
+#Y  Copyright (C) 2014-2025, Zekeriya Arvasi & Alper Odabas,  
+##
+
+############################################################################
+##
+## AlgebraActionByDirectSum( <act> <act> )
+##
+## <#GAPDoc Label="AlgebraActionByDirectSum"> 
+## <ManSection>
+## <Oper Name="AlgebraActionByDirectSum" Arg="act act" />
+##
+## <Description>
+## If <M>\alpha_1 : A_1 \to C_1</M> and <M>\alpha_2: A_2 \to C_2</M>
+## are two actions on algebra <M>B</M>
+## then the direct sum <M>A_1 \oplus A_2</M> acts on <M>B</M> by
+## <M>(a_1,a_2)\cdot b = a_1\cdot(a_2\cdot b) = a_2\cdot(a_1 \cdot b)</M>
+## provided the two actions commute.
+## <P/>
+## </Description>
+## </ManSection>
+## <Example>
+## <![CDATA[
+## gap> 
+## ]]>
+## </Example>
+## <#/GAPDoc>
+##
+DeclareOperation( "AlgebraActionByDirectSum", 
+    [ IsAlgebraAction, IsAlgebraAction ] );
+
+## section 2.4.3
+############################################################################
+##
+#O DirectSumOfAlgebraHomomorphisms( <hom> <hom> )
+#O AlgebraHomomorphismFromDirectSum( <hom> <hom> )
+##
+## <#GAPDoc Label="DirectSumOfAlgebraHomomorphisms"> 
+## <ManSection>
+## <Oper Name="DirectSumOfAlgebraHomomorphisms" Arg="hom1 hom2" />
+## <Oper Name="AlgebraHomomorphismFromDirectSum" Arg="hom1 hom2" />
+##
+## <Description>
+## Let <M>\theta_1 : B_1 \to A_1</M> and <M>\theta_2 : B_2 \to A_2</M>
+## be algebra homomorphisms. 
+## The first operation uses embeddings into <M>A = A_1 \oplus A_2</M> 
+## and <M>B = B_1 \oplus B_2</M> to construct
+## <M>\theta = \theta_1 \oplus \theta_2 : B \to A</M>
+## where <M>\theta(b_1,b_2) = (\theta_1b_1,\theta_2b_2)</M>.
+## <P/>
+## When <M>A_1=A_2</M> the second operation constructs
+## <M>\theta = \theta_1 \oplus \theta_2 : B \to A_1</M>
+## where <M>\theta(b_1,b_2) = \theta_1b_1 + \theta_2b_2</M>.
+## <P/>
+## The example uses the homomorphism <C>homg3</C> used in 
+## Section <Ref Sect="AlgebraActionByHomomorphism" />
+## <P/>
+## </Description>
+## </ManSection>
+## <Example>
+## <![CDATA[
+## gap> hom33a := DirectSumOfAlgebraHomomorphisms( homg3, homg3 );;
+## gap> Print( hom33a, "\n" );
+## AlgebraHomomorphismByImages( A3(+)A3, Algebra( Rationals, 
+## [ v.1, v.2, v.3, v.4, v.5, v.6 ] ), 
+## [ [ [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ], 
+##       [ 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0 ] ], 
+##   [ [ 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0 ], 
+##       [ 0, 0, 0, 0, 1, 0 ], [ 0, 0, 0, 0, 0, 1 ], [ 0, 0, 0, 1, 0, 0 ] ] ], 
+## [ v.1, v.4 ] )
+## ]]>
+## </Example>
+## <#/GAPDoc>
+##
+DeclareOperation( "DirectSumOfAlgebraHomomorphisms", 
+    [ IsAlgebraHomomorphism, IsAlgebraHomomorphism ] );
+DeclareOperation( "AlgebraHomomorphismFromDirectSum", 
+    [ IsAlgebraHomomorphism, IsAlgebraHomomorphism ] );
+
+## section 2.4.4
+############################################################################
+##
+## AlgebraActionOnDirectSum( <act> <act> )
+##
+## <#GAPDoc Label="AlgebraActionOnDirectSum"> 
+## <ManSection>
+## <Oper Name="AlgebraActionOnDirectSum" Arg="act act" />
+##
+## <Description>
+## If <M>\alpha_1 : A \to C_1</M> is an action on algebra <M>B_1</M>
+## and <M>\alpha_2 : A \to C_2</M> is an action on algebra <M>B_2</M>
+## by the same algebra <M>A</M>, 
+## then <M>A</M> acts on the direct sum <M>B_1 \oplus B_2</M>
+## by <M>a \cdot (b_1,b_2) = (a \cdot b_1, a \cdot b_2)</M>.
+## <P/>
+## In Section <Ref Sect="AlgebraActionByModule" /> there is created
+## an action <C>actB3</C> of <C>A3</C> on an isomorphic <C>B3</C>.
+## In the example here we construct <C>actA3</C>, with <C>A3</C> acting
+## on itself, formed using <C>AlgebraActionByMultipliers</C>.
+## Then we construct the direcct sum of these actions.
+## </Description>
+## </ManSection>
+## <Example>
+## <![CDATA[
+## gap> actMA3 := AlgebraActionByMultipliers( A3, A3, A3 );;
+## gap> act4 := AlgebraActionOnDirectSum( actMA3, actg3 );
+## [ [ [ 0, 1, 0 ], [ 0, 0, 1 ], [ 1, 0, 0 ] ], 
+##   [ [ 0, 0, 1 ], [ 1, 0, 0 ], [ 0, 1, 0 ] ], 
+##   [ [ 1, 0, 0 ], [ 0, 1, 0 ], [ 0, 0, 1 ] ] ] -> 
+## [ [ v.1, v.2, v.3, v.4, v.5, v.6 ] -> [ v.2, v.3, v.1, v.5, v.6, v.4 ], 
+##   [ v.1, v.2, v.3, v.4, v.5, v.6 ] -> [ v.3, v.1, v.2, v.6, v.4, v.5 ], 
+##   [ v.1, v.2, v.3, v.4, v.5, v.6 ] -> [ v.1, v.2, v.3, v.4, v.5, v.6 ] ]
+## ]]>
+## </Example>
+## <#/GAPDoc>
+##
+DeclareOperation( "AlgebraActionOnDirectSum", 
+    [ IsAlgebraAction, IsAlgebraAction ] );
+
+## section 2.4.5
+############################################################################
+##
+## DirectSumOfAlgebraActions( <act> <act> )
+##
+## <#GAPDoc Label="DirectSumOfAlgebraActions"> 
+## <ManSection>
+## <Oper Name="DirectSumOfAlgebraActions" Arg="act act" />
+##
+## <Description>
+## Let 
+## If <M>\alpha_1 : A_1 \to C_1</M> is an action on algebra <M>B_1</M>,
+## and <M>\alpha_2 : A_2 \to C_2</M> is an action on algebra <M>B_2</M>, 
+## then <M>A_1 \oplus A_2</M> acts on the direct sum <M>B_1 \oplus B_2</M>
+## by <M>(a_1,a_2) \cdot (b_1,b_2) = (a_1 \cdot b_1, a_2 \cdot b_2)</M>.
+## The example forms the direct sum of the actions constructed in
+## sections <Ref Sect="AlgebraActionBySurjection" /> 
+## and <Ref Sect="AlgebraActionByHomomorphism" />
+## <P/>
+## </Description>
+## </ManSection>
+## <Example>
+## <![CDATA[
+## gap> act5 := DirectSumOfAlgebraActions( actg3, act3 );;
+## gap> A5 := Source( act5 );
+## A3(+)A3
+## gap> B5 := AlgebraActedOn( act5 );
+## GR(c3)(+)A(M3)
+## gap> em3 := ImageElm( Embedding( A5, 1 ), m3 ); 
+## [ [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ], [ 1, 0, 0, 0, 0, 0 ], 
+##   [ 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0 ] ]
+## gap> ImageElm( act5, em3 );                     
+## Basis( GR(c3)(+)A(M3), [ v.1, v.2, v.3, v.4, v.5, v.6 ] ) -> 
+## [ v.2, v.3, v.1, 0*v.1, 0*v.1, 0*v.1 ]
+## gap> ea3 := ImageElm( Embedding( A5, 2 ), a3 );
+## [ [ 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 0 ], 
+##   [ 0, 0, 0, 0, 2, 3 ], [ 0, 0, 0, 3, 0, 2 ], [ 0, 0, 0, 2, 3, 0 ] ]
+### gap> ImageElm( act5, ea3 );
+## Basis( GR(c3)(+)A(M3), [ v.1, v.2, v.3, v.4, v.5, v.6 ] ) -> 
+## [ 0*v.1, 0*v.1, 0*v.1, (3)*v.5+(2)*v.6, (2)*v.4+(3)*v.6, (3)*v.4+(2)*v.5 ]
+## ]]>
+## </Example>
+## <#/GAPDoc>
+##
+DeclareOperation( "DirectSumOfAlgebraActions", 
+    [ IsAlgebraAction, IsAlgebraAction ] );
+
+## section 4.1.9
+############################################################################
+##
+## DirectSumOfXModAlgebras( <X1> <X2> )
+##
+## <#GAPDoc Label="DirectSumOfXModAlgebras"> 
+## <ManSection>
+## <Oper Name="DirectSumOfXModAlgebras" Arg="X1 X2" />
+##
+## <Description>
+## In Sections <Ref Sect="DirectSumOfAlgebraHomomorphisms" />
+## and <Ref Sect="DirectSumOfAlgebraActions" />
+## we constructed direct sums of algebra homomorphisms and algebra actions.
+## So, given two crossed modules of algebras <M>X_1, X_2</M>,
+## we can form the direct sums of their boundaries and actions
+## to form a direct sum <M>X_1 \oplus X_2</M> of crossed modules.
+## <P/>
+## In the example we combine crossed modules <C>X3</C> from section
+## <Ref Sect="XModAlgebraByBoundaryAndAction" /> and <C>X4</C>
+## from section <Ref Sect="XModAlgebraByModule" />.
+## <P/>
+## </Description>
+## </ManSection>
+## <Example>
+## <![CDATA[
+## gap> XY3 := DirectSumOfXModAlgebras( X3, Y3 );
+## [ GR(c3)(+)A(M3) -> A3(+)A3 ]
+## ]]>
+## </Example>
+## <#/GAPDoc>
+##
+DeclareOperation( "DirectSumOfXModAlgebras", 
+    [ IsXModAlgebra, IsXModAlgebra ] );
+