Mixing
Constructing the Algebraic Dual Space as a K-Algebra
$begingroup$
Fix an arbitrary Field $K$ and suppose we are given a Vector Space $Vin{Obj(Vect_{K})}$, let me denote by $V^*:=Hom_{Vect_{K}}(V,K)$ the Algebraic Dual Space of $V$.
It's clear that
$[Vin{Obj(Vect_{K}})]implies[V^*in{Obj(Vect_{K})}]$
Now suppose $K_{Alg}$ is the Category of $K$ Algebras with $Ain{Obj(K_{Alg}})$, I wish to construct a $K$ - Algebra, $A^*$ having the same underlying Set as $V^*$, the trouble I'm having is defining an appropriate bilinear operation
$[•,•]:A^*times{A^*}rightarrow{A^*}$ satisfying the requirements of a $K$ Algebra, I know that $K$ has a natural multiplication defined on it making $(K, •)$ into an Ableian Group. So far I have worked out the properties that the multiplication in our "Dual Algebra" $A^*$ by the definition of a $K$ - Algebra must satisfy:
$forall{f_{1},f_{2},f_{3}in{A^*}}forall{alpha,betain{K}}$
$1)$Right Distributivity : $[(f_{1}+f_{2}),f_{3}]=[f_{1},f_{3}]+[f_{2},f_{3}]$
$2)$Left Distributivity : $[f_{1},(f_{2}+f_{3})] = [f_{1},f_{2}]+[f_{1},f_{3}]$
$3)$Scalar Compatibility : $[(alpha f_{1}),(beta f_{2})] = (alpha{beta})[f_{1},f_{2}]$
Other than that, it isn't immediately clear to me how to define a canonical multiplication on $A^*$ making it into a $K-algebra$, any help or hints are greatly appreciated.
dual-spaces algebras
asked Feb 2 at 19:24
user640930user640930
12
$endgroup$
add a comment |
$begingroup$
Fix an arbitrary Field $K$ and suppose we are given a Vector Space $Vin{Obj(Vect_{K})}$, let me denote by $V^*:=Hom_{Vect_{K}}(V,K)$ the Algebraic Dual Space of $V$.
It's clear that
$[Vin{Obj(Vect_{K}})]implies[V^*in{Obj(Vect_{K})}]$
Now suppose $K_{Alg}$ is the Category of $K$ Algebras with $Ain{Obj(K_{Alg}})$, I wish to construct a $K$ - Algebra, $A^*$ having the same underlying Set as $V^*$, the trouble I'm having is defining an appropriate bilinear operation
$[•,•]:A^*times{A^*}rightarrow{A^*}$ satisfying the requirements of a $K$ Algebra, I know that $K$ has a natural multiplication defined on it making $(K, •)$ into an Ableian Group. So far I have worked out the properties that the multiplication in our "Dual Algebra" $A^*$ by the definition of a $K$ - Algebra must satisfy:
$forall{f_{1},f_{2},f_{3}in{A^*}}forall{alpha,betain{K}}$
$1)$Right Distributivity : $[(f_{1}+f_{2}),f_{3}]=[f_{1},f_{3}]+[f_{2},f_{3}]$
$2)$Left Distributivity : $[f_{1},(f_{2}+f_{3})] = [f_{1},f_{2}]+[f_{1},f_{3}]$
$3)$Scalar Compatibility : $[(alpha f_{1}),(beta f_{2})] = (alpha{beta})[f_{1},f_{2}]$
Other than that, it isn't immediately clear to me how to define a canonical multiplication on $A^*$ making it into a $K-algebra$, any help or hints are greatly appreciated.
dual-spaces algebras
asked Feb 2 at 19:24
user640930user640930
12
$endgroup$
add a comment |
$begingroup$
Fix an arbitrary Field $K$ and suppose we are given a Vector Space $Vin{Obj(Vect_{K})}$, let me denote by $V^*:=Hom_{Vect_{K}}(V,K)$ the Algebraic Dual Space of $V$.
It's clear that
$[Vin{Obj(Vect_{K}})]implies[V^*in{Obj(Vect_{K})}]$
Now suppose $K_{Alg}$ is the Category of $K$ Algebras with $Ain{Obj(K_{Alg}})$, I wish to construct a $K$ - Algebra, $A^*$ having the same underlying Set as $V^*$, the trouble I'm having is defining an appropriate bilinear operation
$[•,•]:A^*times{A^*}rightarrow{A^*}$ satisfying the requirements of a $K$ Algebra, I know that $K$ has a natural multiplication defined on it making $(K, •)$ into an Ableian Group. So far I have worked out the properties that the multiplication in our "Dual Algebra" $A^*$ by the definition of a $K$ - Algebra must satisfy:
$forall{f_{1},f_{2},f_{3}in{A^*}}forall{alpha,betain{K}}$
$1)$Right Distributivity : $[(f_{1}+f_{2}),f_{3}]=[f_{1},f_{3}]+[f_{2},f_{3}]$
$2)$Left Distributivity : $[f_{1},(f_{2}+f_{3})] = [f_{1},f_{2}]+[f_{1},f_{3}]$
$3)$Scalar Compatibility : $[(alpha f_{1}),(beta f_{2})] = (alpha{beta})[f_{1},f_{2}]$
Other than that, it isn't immediately clear to me how to define a canonical multiplication on $A^*$ making it into a $K-algebra$, any help or hints are greatly appreciated.
dual-spaces algebras
asked Feb 2 at 19:24
user640930user640930
12
$endgroup$
Fix an arbitrary Field $K$ and suppose we are given a Vector Space $Vin{Obj(Vect_{K})}$, let me denote by $V^*:=Hom_{Vect_{K}}(V,K)$ the Algebraic Dual Space of $V$.
It's clear that
$[Vin{Obj(Vect_{K}})]implies[V^*in{Obj(Vect_{K})}]$
Now suppose $K_{Alg}$ is the Category of $K$ Algebras with $Ain{Obj(K_{Alg}})$, I wish to construct a $K$ - Algebra, $A^*$ having the same underlying Set as $V^*$, the trouble I'm having is defining an appropriate bilinear operation
$[•,•]:A^*times{A^*}rightarrow{A^*}$ satisfying the requirements of a $K$ Algebra, I know that $K$ has a natural multiplication defined on it making $(K, •)$ into an Ableian Group. So far I have worked out the properties that the multiplication in our "Dual Algebra" $A^*$ by the definition of a $K$ - Algebra must satisfy:
$forall{f_{1},f_{2},f_{3}in{A^*}}forall{alpha,betain{K}}$
$1)$Right Distributivity : $[(f_{1}+f_{2}),f_{3}]=[f_{1},f_{3}]+[f_{2},f_{3}]$
$2)$Left Distributivity : $[f_{1},(f_{2}+f_{3})] = [f_{1},f_{2}]+[f_{1},f_{3}]$
$3)$Scalar Compatibility : $[(alpha f_{1}),(beta f_{2})] = (alpha{beta})[f_{1},f_{2}]$
Other than that, it isn't immediately clear to me how to define a canonical multiplication on $A^*$ making it into a $K-algebra$, any help or hints are greatly appreciated.
dual-spaces algebras
dual-spaces algebras
asked Feb 2 at 19:24
user640930user640930
12
asked Feb 2 at 19:24
user640930user640930
12
edited Feb 2 at 19:29
user640930
asked Feb 2 at 19:24
user640930user640930
12
asked Feb 2 at 19:24
user640930user640930
12
asked Feb 2 at 19:24
user640930user640930
12
12
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