##\mathbb{C}\oplus\mathbb{C}\cong\mathbb{C}\otimes\mathbb{C}##

[Korybut]

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.

 

[martinbn]

##z_1\otimes z_2 \mapsto (z_1z_2, z_1\bar{z}_2)##

 

[martinbn]

This one shows up at the bottom of the page
https://www.physicsforums.com/threads/tensor-product-of-c-with-itself-over-r.675424/
but it should read as follow

##\mathbb C \otimes \mathbb C = \mathbb C \otimes \mathbb R[T]/<T^2+1> = \mathbb C[T]/<T^2+1>=\mathbb C[T]/<(T-i)(T+i)> = \mathbb C \oplus \mathbb C##

 

[Korybut]

##z_1\otimes z_2 \mapsto (z_1z_2, z_1\bar{z}_2)##

I've seen this formula before and indeed this identification doesn't suffer from the problem I mentioned earlier however I don't get why this is isomorphism.
If I consider this map
##i: \mathbb{C}\otimes \mathbb{C}\rightarrow \mathbb{C}\oplus\mathbb{C}##
then I can look at element which are mapped to zero in ##\mathbb{C}\oplus\mathbb{C}## and there are way more than one.

 

[martinbn]

I've seen this formula before and indeed this identification doesn't suffer from the problem I mentioned earlier however I don't get why this is isomorphism.
If I consider this map
##i: \mathbb{C}\otimes \mathbb{C}\rightarrow \mathbb{C}\oplus\mathbb{C}##
then I can look at element which are mapped to zero in ##\mathbb{C}\oplus\mathbb{C}## and there are way more than one.

How? If it is mapped to ##0##, then either ##z_1## or ##z_2## is ##0##, so ##z_1\otimes z_2## is also ##0##.

 

[Korybut]

This one shows up at the bottom of the page
https://www.physicsforums.com/threads/tensor-product-of-c-with-itself-over-r.675424/
but it should read as follow

##\mathbb C \otimes \mathbb C = \mathbb C \otimes \mathbb R[T]/<T^2+1> = \mathbb C[T]/<T^2+1>=\mathbb C[T]/<(T-i)(T+i)> = \mathbb C \oplus \mathbb C##

I wish someone can explain notation in this formula. What is ##T##? What does ##/<...> means?

 

[Korybut]

How? If it is mapped to ##0##, then either ##z_1## or ##z_2## is ##0##, so ##z_1\otimes z_2## is also ##0##.

Sorry! My Bad! Indeed! Thanks for help!

 

[martinbn]

I wish someone can explain notation in this formula. What is ##T##? What does ##/<...> means?

##T## is a variable, you are looking at polynomials. ##/<...>## means the quotient by the ideal.

 

[martinbn]

ps By the way which book is that?

 

[Korybut]

ps By the way which book is that?

Lawson and Michelson "Spin Geometry". They suggested the following
##(1,0)\rightarrow \frac{1}{2}(1\otimes 1+i\otimes i)##
##(0,1)\rightarrow \frac{1}{2}(1\otimes 1 -i \otimes i)##
And I don't get how I proceed with the full proof with just that

 

[martinbn]

Lawson and Michelson "Spin Geometry". They suggested the following
##(1,0)\rightarrow \frac{1}{2}(1\otimes 1+i\otimes i)##
##(0,1)\rightarrow \frac{1}{2}(1\otimes 1 -i \otimes i)##
And I don't get how I proceed with the full proof with just that

This is meant to be as ##\mathbb C##-algebras, where the structure on the tensor product is through the first argument. If you solve it for the inverse map it sends ##1\otimes 1## to ##(1,1)## and ##i\otimes i## to ##(1,-1)##. As real algebras it will map as follows

##1\otimes 1## to ##(1,1)##
##i\otimes 1## to ##(i,i)##
##i\otimes i## to ##(1,-1)##
##-1\otimes i## to ##(i,-i)##

which is the map ##a\otimes b \mapsto (a\bar{b}, ab)## in that basis.

 

[Korybut]

This is meant to be as ##\mathbb C##-algebras, where the structure on the tensor product is through the first argument.

Thanks for another clarification. I wish those lines were in the book)

 

[fresh_42]

This is one of the cases that show why I dislike the tensor notation without mentioning the scalar field. We have ##\mathbb{C}\otimes_\mathbb{R} \mathbb{C}\cong \mathbb{C}^2.##

I first read it as ##\mathbb{C}\otimes_\mathbb{C} \mathbb{C}## which would be isomorphic to ##\mathbb{C}^1.##

 

[martinbn]

This is one of the cases that show why I dislike the tensor notation without mentioning the scalar field. We have ##\mathbb{C}\otimes_\mathbb{R} \mathbb{C}\cong \mathbb{C}^2.##

I first read it as ##\mathbb{C}\otimes_\mathbb{C} \mathbb{C}## which would be isomorphic to ##\mathbb{C}^1.##

But he did write it indicating it is over the
reals. See the first post.