Why are the casimirs independent of the representation

In summary, the Casimir invariant is an element of the universal enveloping algebra of the Lie algebra, and its representation under a matrix is proportional to the identity matrix.
  • #1
Bobhawke
144
0
Question in the title, ie why is [tex] Tr(T_{a_1}T_{a_2}...T_{a_n}) [/tex] independent of which representation we choose, where the Ts are a matrix representation of the group generators.
 
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  • #2
That's not true.

For example, if you pick a positive integer m, there is a representation where every group element has trace m: this representation sends every group element to the identity matrix acting on an m-dimensional space.

And given any matrix representation [itex]\rho[/itex] of a group, one can construct a new representation [itex]\rho'[/itex] by

[tex]
\rho'(g) = \left[ \begin{matrix}{\rho(g) & 0 \\ 0 & \rho(g)} \end{matrix} \right]
[/tex]

and under this representation, [itex]Tr\, \rho'(g) = 2 Tr \, \rho(g)[/itex].



You're making some extra, relevant assumptions -- what are they?
 
  • #3
Ah I see. But couldn't we also have

[tex]
\rhosingle-quote(g) = \left[ \begin{matrix}{\rho(g) & 0 \\ 0 & 0} \end{matrix} \right]
[/tex]

as a higher dimensional rep with the same trace?

I think what I meant is that given a 2 different matrix reps of the same dimension, the casimirs are the same. But they can change when you change the dimension of the rep, thus they can be used to label reps. Thanks
 
  • #4
Different representations with the same dimension can still have that trace be different. For example, consider

[tex]\rho'(g) = \rho(g)^2[/tex]

and

[tex]\rho'(g) = (\det g)^{-1}\rho(g)[/tex]




Anyways, I went to look up Casimir on wikipedia; if that's what you're talking about, then I think you are misunderstanding things. The Casimir invariant is an element of the universal enveloping algebra of the Lie algebra, and its representation under [itex]\rho[/itex] is a matrix.

Wikipedia states that by Schur's lemma, for any irreducible representation, [itex]\rho(\Omega)[/itex] is proportional to the identity matrix. That constant of proportionality can be computed with a formula involving traces; maybe that's what you're thinking of?
 
  • #5
Yeah I think I am confusing things a little.

Casimirs can be constructed from the generators by:

[tex]
d_{a_1 a_2...a_n} = Tr(T_{a_1}T_{a_2}...T_{a_n})
[/tex]

But is is their eigenvalues that label the reps. So I think what I really meant to ask is why can the eigenvalues of the casimirs can be used to label reps of different dimension.

Sorry for my confusion
 

1. Why are the Casimirs important in representations?

The Casimirs are important because they are operators that measure the total energy and momentum of a system. They are used to classify and characterize different representations of a group, which is essential in understanding the symmetries and properties of a physical system.

2. What does it mean for the Casimirs to be independent of the representation?

It means that the values of the Casimirs do not change when the basis of the representation is changed. This is a fundamental property of Casimir operators, and it allows for a consistent and universal way of characterizing the different representations of a group.

3. How are the Casimirs related to the Lie algebra of a group?

The Casimir operators are constructed from the generators of the Lie algebra of a group. They are quadratic combinations of these generators and are used to label the different representations of a group. In this way, the Casimirs are intimately connected to the Lie algebra and are crucial in the study of group representations.

4. Can the Casimirs be used to determine the irreducible representations of a group?

Yes, the Casimir operators can be used to determine the irreducible representations of a group. Each irreducible representation has a unique set of Casimir values, which can be used to identify and classify the representations. This is a powerful tool in the study of group theory.

5. Are the Casimirs the only operators that are independent of the representation?

No, there are other operators that are also independent of the representation, such as the center of the group and the group's identity element. However, the Casimirs are the most commonly used and are essential in understanding the symmetries and properties of a physical system.

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