Hello,(adsbygoogle = window.adsbygoogle || []).push({});

it is known quaternions are isomorphic to [tex]\mathcal{C}\ell^{+}_{3,0}[/tex], which is theeven subalgebraof [tex]\mathcal{C}\ell_{3,0}[/tex]

Is it possible to find an isomorphism between [tex]\mathcal{C}\ell_{2,0}[/tex] and [tex]\mathbb{H} \cong \mathcal{C}\ell^{+}_{3,0}[/tex] ?

*** my attempt was: ***

Let's consider [tex]\{1,e_1, e_2, e_{12}\}[/tex] and the morphismfdefined as follows:

[tex]f(1)=1[/tex]

[tex]f(e_{32})=e_1[/tex]

[tex]f(e_{13})=e_2[/tex]

[tex]f(e_{21})=e_{21}[/tex]

This almost works, in fact:

[itex]f(xy)=f(x)f(y)[/itex] always holds with one exception:

[tex]f(e_1 e_2) = -e_{21} = f(e_2)f(e_1)[/tex]

So, in this case, the morphism f does not preserve well the geometric product.

Is it possible to make the whole thing work?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Quaternions in Clifford algebras

**Physics Forums | Science Articles, Homework Help, Discussion**