# Highest weight of representations of Lie Algebras

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?

fresh_42
Mentor
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?

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