- #1

Korybut

- 64

- 3

- TL;DR Summary
- Algebra isomorphism

Hello!

Reading book o Clifford algebra authors claim that ##\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}## as algebras. Unfortunately proof is absent and provided hint is pretty misleading

As vector spaces they are obviously isomorphic since

##\dim_{\mathbb{R}} \mathbb{C}\oplus\mathbb{C}=\dim_{\mathbb{R}} \mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}=4##.

Product in ##\mathbb{C}\otimes_{\mathbb{R}} \mathbb{C}## looks as follows

##(z_1\otimes z_2) (z_3\otimes z_4)=z_1 z_3 \otimes z_2 z_4##

Product in ##\mathbb{C}\oplus\mathbb{C}## looks as follows

##(z_1,z_2) (z_3,z_4)=(z_1 z_3,z_2 z_4)##

From that perspective it is quite tempting to identify ##z_1 \otimes z_2## with ##(z_1,z_2)## however ##z_1\otimes z_2## and ##\lambda z_1 \otimes \frac{1}{\lambda} z_2## (for some real ##\lambda##) are the same elements in ##\mathbb{C}\otimes \mathbb{C}## but they will be mapped to different elements of ##\mathbb{C}\oplus\mathbb{C}##. Obviously I need better map if this isomorphism indeed take place. But which one?

Many thanks in advance.

Reading book o Clifford algebra authors claim that ##\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}## as algebras. Unfortunately proof is absent and provided hint is pretty misleading

As vector spaces they are obviously isomorphic since

##\dim_{\mathbb{R}} \mathbb{C}\oplus\mathbb{C}=\dim_{\mathbb{R}} \mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}=4##.

Product in ##\mathbb{C}\otimes_{\mathbb{R}} \mathbb{C}## looks as follows

##(z_1\otimes z_2) (z_3\otimes z_4)=z_1 z_3 \otimes z_2 z_4##

Product in ##\mathbb{C}\oplus\mathbb{C}## looks as follows

##(z_1,z_2) (z_3,z_4)=(z_1 z_3,z_2 z_4)##

From that perspective it is quite tempting to identify ##z_1 \otimes z_2## with ##(z_1,z_2)## however ##z_1\otimes z_2## and ##\lambda z_1 \otimes \frac{1}{\lambda} z_2## (for some real ##\lambda##) are the same elements in ##\mathbb{C}\otimes \mathbb{C}## but they will be mapped to different elements of ##\mathbb{C}\oplus\mathbb{C}##. Obviously I need better map if this isomorphism indeed take place. But which one?

Many thanks in advance.