About cartan and root generators of lie algebra

[quanti]

I am reading the text Group Theory A Physicist's Survey of Ramond, in particular chapter 7.

He explains classical lie algebra structure using cartan generators and root generators.

He sometimes uses reality condition of structure constant( i think he supposes that all generators are

hermitian)

But i think root generators are not hermitian in general and i can't using reality condition of structure constant.

Can you tell me what i missed?

 

[jvicens]

Hi there,

Thank you for bringing up this question. You are correct in noting that root generators are not necessarily Hermitian. In fact, the root generators in a Lie algebra are chosen to be anti-Hermitian. This is because the root generators are related to the structure constants of the Lie algebra, which are anti-symmetric matrices. Therefore, using the reality condition for Hermitian matrices would not be applicable in this case.

However, the reality condition can still be used in certain cases, such as when dealing with the Cartan generators. This is because the Cartan generators are chosen to be Hermitian, and they are related to the root generators through a linear combination. Therefore, the reality condition can be applied to the Cartan generators, which then gives information about the root generators.

I hope this helps clarify things for you. If you have any further questions, please don't hesitate to ask. Good luck with your studies!