SU(2) as representation of SO(3)

[maxverywell]

The SU(2) and SO(3) groups are homomorphic groups. Can we say that the SU(2) group is representation of SO(3) and vice versa (SU(2) representation of SO(3))?

Is a representation R of some group G a group too? If so, is it true that G is representation of R?

 

[haushofer]

In the case of matrix Lie groups, the fundamental representation is the group itself, so then you could say, in an abuse of language, that "the representation is a group". But in general the group is an abstract set with four rules, and can be represented in many ways.

But be careful: not every representation of SU(2) is a representation of SO(3)! SU(2) is the double cover of SO(3), and SU(2) is isomorphic to the coset SO(3)/Z2.

 

[maxverywell]

In the case of matrix Lie groups, the fundamental representation is the group itself, so then you could say, in an abuse of language, that "the representation is a group". But in general the group is an abstract set with four rules, and can be represented in many ways.

But be careful: not every representation of SU(2) is a representation of SO(3)! SU(2) is the double cover of SO(3), and SU(2) is isomorphic to the coset SO(3)/Z2.

Oh I see... I think I get the point now, thnx!

Btw, what is a fundamental representation and a double cover?

 

[arkajad]

Btw, what is a fundamental representation and a double cover?

Elements of $$SU(2)$$ are 2x2 complex matrices. If to each matrix $$A\in SU(2)$$ you assing the transformation $$x\mapsto Ax$$ of $$\mathbf{C}^2$$ - then you have the fundamental represantation of $$SU(2)$$

There is a very nice a natural group homomorphism, call it $$\rho$$,

$$\rho: SU(2)\rightarrow SO(3)$$.

It has the property

$$\rho(A)=\rho(-A)$$.

Matrices $$A$$ and $$-A$$ are mapped to the same element of $$SO(3)$$. Thus the name "double cover".

 

[Fredrik]

The SU(2) and SO(3) groups are homomorphic groups. Can we say that the SU(2) group is representation of SO(3) and vice versa (SU(2) representation of SO(3))?

Is a representation R of some group G a group too? If so, is it true that G is representation of R?

R isn't a group. It's a group homomorphism from G into GL(V) (the group of invertible linear operators on a vector space V).

 

[maxverywell]

R isn't a group. It's a group homomorphism from G into GL(V) (the group of invertible linear operators on a vector space V).

R is a set of matrices which with matrix multiplication forms a group.

 

[arkajad]

R is a set of matrices which with matrix multiplication forms a group.

Set is not a representation. Representation is a map from one set to another, with particular properties.

 

[maxverywell]

Set is not a representation. Representation is a map from one set to another, with particular properties.

I said set with multiplication. So a group representation is a group whose elements are matrices.

 

[arkajad]

No. Please, check the definition from a good book. Well, I will do it for you:

From H. Jones, "Groups, Representations and Physics", p. 37:

Definition

A representation of dimension n of the abstract group G is defined as a
homomorphism D: G -> GL(n, C), the group of non-singular nxn
matrices with complex entries.

More generally, you can replace GL(n,C) by L(V,K). But the important thing is that it is homomorphism, that is a map with appropriate properties, not a "set with appropriate properties".

 

[maxverywell]

I know that it's a homomorphism etc. but it's a group as I said. Take for example the group of order 2: G={e,a} and its regular representation D={D(e),D(a)}
were

$$D(e)=\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$$ and $$D(a)=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$

Now, this two matrices form a group (which is actually the same as G, they are isomorphic because it's regular representation).

 

[arkajad]

I know that it's a homomorphism etc. but it's a group as I said.

Well, you may like to learn how to distinguish between objects and arrows that connect objects. You never know, one day this ability may come handy...

You may find some info here: http://en.wikipedia.org/wiki/Category_theory"

 

[maxverywell]

You are right, sorry. Indeed, a representation is an arrow from one group to another (group of matrices).

 

[arkajad]

So, for instance, we have representations:

$$SU(2)\rightarrow GL(2,C)$$

$$SU(2)\rightarrow SO(3)\subset GL(3,R)$$

Physicists also sometimes say that there is a "double-valued representation"
$$SO(3)\rightarrow SU(2)$$
but this must considered with great care, and mathematicians do not like it all. Instead mathematicians prefer to discuss "projective representations", "multipliers", "cocycles" etc. And they are right. But this is an "advanced subject".