- #1
physicist_2be
- 1
- 0
Hello there,
Given a Lie Algebra ##\mathfrak{g}##, its Cartan Matrix ##A## and a finite representation ##R##, is there a way of determining its highest weight ##\Lambda## in a simple way?
In my course, we consider ##\mathfrak{g}=A_2= \mathfrak{L}_{\mathbb{C}}(SU(3))##. It is stated that the highest weight of the fundamental representation has Dynkin labels ##\Lambda = (1,0)## and the highest weight of the adjoint representation has Dynkin labels ##\Lambda = (1,1)##. Why is it so? From there, I can work out the other roots by removing weights given by the Cartan Matrix but it is of no use if I can't compute the highest weight in the first place.
Taking an example, let ##\mathfrak{g}=B_2= \mathfrak{L}_{\mathbb{C}}(SO(5))##. How do I work out the highest weight for the fundamental and adjoint representation?
Thanks in advance!
Given a Lie Algebra ##\mathfrak{g}##, its Cartan Matrix ##A## and a finite representation ##R##, is there a way of determining its highest weight ##\Lambda## in a simple way?
In my course, we consider ##\mathfrak{g}=A_2= \mathfrak{L}_{\mathbb{C}}(SU(3))##. It is stated that the highest weight of the fundamental representation has Dynkin labels ##\Lambda = (1,0)## and the highest weight of the adjoint representation has Dynkin labels ##\Lambda = (1,1)##. Why is it so? From there, I can work out the other roots by removing weights given by the Cartan Matrix but it is of no use if I can't compute the highest weight in the first place.
Taking an example, let ##\mathfrak{g}=B_2= \mathfrak{L}_{\mathbb{C}}(SO(5))##. How do I work out the highest weight for the fundamental and adjoint representation?
Thanks in advance!