Are Vectors Defined by Commutation Relations Always Roots in Any Representation?

In summary, the roots of the adjoint representation of a space are the same as the vectors defined by the commutation relations above.
  • #1
spookyfish
53
0
The vectors [itex]\vec{\alpha}=\{\alpha_1,\ldots\alpha_m \}[/itex] are defined by
[tex]
[H_i,E_\alpha]=\alpha_i E_\alpha
[/tex]
they are also known to be the non-zero weights, called the roots, in the adjoint representation. My question is - is this connection (that the vectors [itex]\vec{\alpha} [/itex] defined by the commutation relations above in some representation, are also the roots of the adjoint representation) is true only when [itex]\vec{\alpha} [/itex] is in the defining representation, or is it true for any representation?
I hope my question is clear
 
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  • #2
Which other representation of roots? Are you talking about the basis of the roots like Cartan-weyl basis or Dynkin basis?
 
  • #3
No. Sorry, I am talking about the representation in which [itex]\vec{\alpha} [/itex] is defined. for example, in su(3), the defining representation has 3 weights (because the space is 3 dimensional) and the vectors [itex]\vec{\alpha} [/itex] are the difference between these weights. The vectors [itex]\vec{\alpha} [/itex] also coincide with the roots of the adjoint representation.
Now, suppose I wanted to consider a different arbitrary representation of su(3), not the 3-dimensional and not the adjoint 8-dimensional. It would have a different number of weights. Would the root vectors (same roots vectors, defined in the adjoint representation) still carry me between the different weights, or will they now not coincide with the vectors [itex]\vec{\alpha} [/itex] satisfying
[tex]
[H_i,E_\alpha]=\alpha_i E_\alpha
[/tex]
 
  • #4
The root vectors are not defined in the adjoint representation, or any representation, they are defined in the Lie Algebra, by the formula you gave,
[tex]
[H_i, E_\alpha]=\alpha_i E_\alpha
[/tex]
spookyfish said:
Would the root vectors (same roots vectors, defined in the adjoint representation) still carry me between the different weights
Yes, this is easy to show. If m is an eikenket of H with weight m, namely H|m> = m |m>, then Eα|m> is an eigenket with weight m + α. This follows from the above commutator.
 
  • #5
I see. But since [itex]H_i [/itex] and [itex]E_\vec{\alpha} [/itex] are different in different representations, aren't the root vectors [itex]\vec{\alpha} [/itex] also different?

then, if they are, either they are equal to the weights in the adjoint representation, or not.
 
  • #6
spookyfish said:
I see. But since [itex]H_i [/itex] and [itex]E_\vec{\alpha} [/itex] are different in different representations, aren't the root vectors [itex]\vec{\alpha} [/itex] also different?
H and Eα are elements of the Lie algebra. In any representation there will be matrices that correspond to them, but the commutators of those matrices (and hence the root vectors) had better be the same, or else it would not be a representation!

Note that the root vectors are always ℓ-dimensional vectors, where ℓ is the rank of the group. They don't depend on the dimensionality of the representation.
 
  • #7
I see. Thank you
 

FAQ: Are Vectors Defined by Commutation Relations Always Roots in Any Representation?

1. What is a Lie group representation?

A Lie group representation is a mathematical concept that describes how a given Lie group (a type of mathematical group) can act on a vector space. It involves mapping elements of the Lie group to linear transformations on the vector space.

2. How are Lie group representations used in physics?

Lie group representations are used in physics to describe the symmetries and transformations of physical systems. For example, in quantum mechanics, the symmetries of a physical system can be represented by Lie groups and their corresponding representations.

3. What is the difference between a faithful and a unitary representation?

A faithful representation is one in which each element of the Lie group is uniquely represented by a linear transformation, while a unitary representation is one in which the linear transformations preserve the inner product of the vector space. All unitary representations are faithful, but not all faithful representations are unitary.

4. Can Lie group representations be reducible or irreducible?

Yes, Lie group representations can be reducible or irreducible. A reducible representation can be broken down into smaller, independent representations, while an irreducible representation cannot be further decomposed.

5. How are Lie group representations related to Lie algebras?

Lie group representations and Lie algebras are closely related. Lie algebras represent the infinitesimal generators of a Lie group, while Lie group representations describe how those generators act on a vector space. Essentially, Lie group representations provide a way to study the algebraic structure of a Lie group through its actions on a vector space.

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