direct sums of xmod-algebras #91
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| Original file line number | Diff line number | Diff line change | ||||||
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| ############################################################################# | ||||||||
| ## | ||||||||
| #W dsum-xmod.gd The XMODALG package Zekeriya Arvasi | ||||||||
| #W & Alper Odabas | ||||||||
| #Y Copyright (C) 2014-2025, Zekeriya Arvasi & Alper Odabas, | ||||||||
| ## | ||||||||
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||||||||
| ############################################################################ | ||||||||
| ## | ||||||||
| ## 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 ] ); | ||||||||
|
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||||||||
| ## 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 ] ); | ||||||||
|
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||||||||
| ## 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 ] ); | ||||||||
|
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||||||||
| ## 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 ] ); | ||||||||
|
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||||||||
| ## 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> | ||||||||
|
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| ## <#/GAPDoc> | ||||||||
| ## | ||||||||
| DeclareOperation( "DirectSumOfXModAlgebras", | ||||||||
| [ IsXModAlgebra, IsXModAlgebra ] ); | ||||||||
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This should ideally avoid re-introducing the bad patterns fixed in PR #90