Product Groups and their dimensions

[Kontilera]

My understanding was that the product of two groups A and B will yield a group C for which the dimension of C is dim(A)*dim(B).
Now however, the author I'm reading defines the group product multiplication as:

(a1, b1) * (a2, b2) = (a1*a2, b1*b2), for a1,a2 in A and b1, b2 in B.

Does this give the same result for the dimension??



Lets take the example with A and B corresponding to the real numbers with addition as a group operation. This gives:

(x1, y1) + (x2, y2) = (x1+x2, y1+y2)... It looks like we have created the two dimensional plane! But dim(R)*dim(R) = 1 * 1 = 1. C should have dimension 1 not 2. :/


Please help me!

 

[DonAntonio]

My understanding was that the product of two groups A and B will yield a group C for which the dimension of C is dim(A)*dim(B).
Now however, the author I'm reading defines the group product multiplication as:

(a1, b1) * (a2, b2) = (a1*a2, b1*b2), for a1,a2 in A and b1, b2 in B.

Does this give the same result for the dimension??



Lets take the example with A and B corresponding to the real numbers with addition as a group operation. This gives:

(x1, y1) + (x2, y2) = (x1+x2, y1+y2)... It looks like we have created the two dimensional plane! But dim(R)*dim(R) = 1 * 1 = 1. C should have dimension 1 not 2. :/


Please help me!

What do you mean by "dimension of a group"? I think there's a huge confusion here...

DonAntonio

 

[Kontilera]

By the dimension of the group I mean the parameters we need to describe the group.. For the real numbers and U(1) this is 1. For SU(3) this is 3, etc.

 

[Dickfore]

I think the "dimension" of the direct product of two groups is the sum of the corresponding dimensions.

This is because of the exponentiation of Lie groups.

Edit:
After reviewing what you wrote, I am puzzled as to how you found the dimension of SU(3) to be 3?

 

[Kontilera]

Gah! hahah, confusing.
I think I am confusing the tensor product which I have encountered in the representations theory of groups and the group product.. For the tensor product the dimension of the representations is clearly multiplicative.. then I assumed that the same would be true for the group product.
It should of course be dimension 2 for SU(3)- it is topologically equivalent to the 2-sphere. :)

 

[Dickfore]

It should of course be dimension 2 for SU(3)- it is topologically equivalent to the 2-sphere. :)

These are incorrect statements.

 

[Kontilera]

Lets skip the SO(3) for the moment, and focus on the topic. What is the relation between the tensor product between two representation and the correponding product group? :/
Can anything be said here?

 

[Dickfore]

Yes, since representations are nothing more but matrices, and the dimension of the Kronecker product is the product of the dimensions, the dimension of the direct tensor product representation would be a product of the dimensions of the representations of each group.