# Highest weight of representations of Lie Algebras

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• physicist_2be
In summary, determining the highest weight of a finite representation of a Lie Algebra ##\mathfrak{g}## with Cartan Matrix ##A## can be done by constructing the root spaces from the Dynkin diagram or Cartan matrix. However, for more complex Lie algebras, such as ##\mathfrak{sl}(2)## with multiple strains of roots, it is recommended to consult books such as Humphreys for more detailed explanations.
physicist_2be
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?

physicist_2be said:
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?

You need to say semisimple or simple Lie algebra!
Here's an example of how to construct the root spaces from the Dynkin diagram, or the Cartan matrix:
https://www.physicsforums.com/insights/lie-algebras-a-walkthrough-the-structures/

I think highest weight is in general a bit more complicated. ##\mathfrak{sl}(2)## is easy, but with more than one strain of roots it becomes more complex. I would look it up in books, e.g. Humphreys.

physicist_2be

## 1. What is the highest weight of a representation of a Lie algebra?

The highest weight of a representation of a Lie algebra is the weight with the highest value in the weight space. It is used to classify and label representations of Lie algebras.

## 2. How is the highest weight of a representation determined?

The highest weight of a representation is determined by finding the highest weight vector, which is an eigenvector of the Cartan subalgebra with the highest eigenvalue. This vector then generates the entire representation.

## 3. What is the significance of the highest weight in representation theory?

The highest weight is significant because it allows for the classification of representations of Lie algebras, which are important mathematical objects in the study of symmetry and group theory. It also plays a crucial role in the construction of highest weight modules.

## 4. Can the highest weight of a representation change?

No, the highest weight of a representation is an intrinsic property of that representation and cannot be changed by any transformation or change of basis. However, different representations of the same Lie algebra may have different highest weights.

## 5. How is the highest weight used in the study of Lie algebras?

The highest weight is used to classify representations of Lie algebras and to determine their properties. It is also used in the construction of highest weight modules, which are important tools in the study of Lie algebras and their representations.

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