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Lie group representationsby spookyfish
Tags: representations 
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#1
May1114, 07:51 PM

P: 53

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 nonzero 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 


#2
May1214, 02:49 AM

P: 1,020

Which other representation of roots? Are you talking about the basis of the roots like Cartanweyl basis or Dynkin basis?



#3
May1214, 09:11 AM

P: 53

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 3dimensional and not the adjoint 8dimensional. 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
May1214, 09:50 AM

Sci Advisor
Thanks
P: 4,160

Lie group representations
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] 


#5
May1214, 10:03 AM

P: 53

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
May1214, 10:10 AM

Sci Advisor
Thanks
P: 4,160

Note that the root vectors are always ℓdimensional vectors, where ℓ is the rank of the group. They don't depend on the dimensionality of the representation. 


#7
May1214, 10:30 AM

P: 53

I see. Thank you



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