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MTS
MTS
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I understand that there is a universal property which tells me that given a Clifford algebra Cl(V$Cl(V, q)$, q). Then for a linear map f: V -> A$f: V \to A$ (A$A$ any associative algebra) satisfying f(v)f(v) = Q(v) then f satisfying $f(v)f(v) = Q(v)$, $f$ may be extended to a homomorphism between algebras Cl(V) -> A$Cl(V) \to A$.

I am wondering when A$A$ happens to be a Clifford algebra. Do all homomorphismhomomorphisms between Cl(V)$Cl(V)$ and A$A$ arise this way? My guess is not... If not, is there any general way one may check any given linear map between Clifford algebra is a homomorphism?

Thank you!

I understand that there is a universal property which tells me that given a Clifford algebra Cl(V, q). Then for a linear map f: V -> A (A any associative algebra) satisfying f(v)f(v) = Q(v) then f may be extended to a homomorphism between algebras Cl(V) -> A.

I am wondering when A happens to be a Clifford algebra. Do all homomorphism between Cl(V) and A arise this way? My guess is not... If not, is there any general way one may check any given linear map between Clifford algebra is homomorphism?

Thank you!

I understand that there is a universal property which tells me that given a Clifford algebra $Cl(V, q)$, for a linear map $f: V \to A$ ($A$ any associative algebra) satisfying $f(v)f(v) = Q(v)$, $f$ may be extended to a homomorphism between algebras $Cl(V) \to A$.

I am wondering when $A$ happens to be a Clifford algebra. Do all homomorphisms between $Cl(V)$ and $A$ arise this way? My guess is not... If not, is there any general way one may check any given linear map between Clifford algebra is a homomorphism?

Thank you!

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Clark Chong
Clark Chong
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General criterion for homomorphism between Clifford Algebras

I understand that there is a universal property which tells me that given a Clifford algebra Cl(V, q). Then for a linear map f: V -> A (A any associative algebra) satisfying f(v)f(v) = Q(v) then f may be extended to a homomorphism between algebras Cl(V) -> A.

I am wondering when A happens to be a Clifford algebra. Do all homomorphism between Cl(V) and A arise this way? My guess is not... If not, is there any general way one may check any given linear map between Clifford algebra is homomorphism?

Thank you!