Lie Algebras and Similarity Transformations

[topsquark]

Say we have an adjoint representation (specifically in n x n matrix format) of a Lie algebra. I have noted that we can create another n x n representation using a similarity transformation on the adjoint rep. I know I haven't discovered anything new but none of my sources mention this. Is there any advantage to using a similarity transformation rep over the adjoint rep?

-Dan

PS: I should perhaps also mention that the adjoint rep is a Cartan-Weyl basis and the similarity transformation rep is as well.

 

[Euge]

If I'm reading this correctly, what you've done is made an equivalent representation by changing basis from the Cartan-Weyl basis. What was the goal exactly? What kind of result are you looking for?

 

[topsquark]

If I'm reading this correctly, what you've done is made an equivalent representation by changing basis from the Cartan-Weyl basis. What was the goal exactly? What kind of result are you looking for?

A little while ago I was having some difficulty with basis changes for the generators of a Lie algebra. I finally figured out what I was doing wrong and in the process of fixing my flawed proofs and ideas I was working with the adjoint representation and discovered the similarity transformation property. For some reason I hadn't expected to find more than one n x n (nontrivial basis change) representation of the Lie algebra. So I was just checking to see if the similarity idea had any particular use.

Oddly enough I did find examples of this today in a Physics text and realized that one of my Lie algebra texts did actually make a brief mention of this for su(2), though in this particular example the structure factors do change under the similarity transformation. (The similarity transformation didn't use the adjoint rep for su(2) but was enough to satisfy my curiosity.)

-Dan