Roots and Weights in Lie algebras

• Hymne
In summary, the conversation discusses the subject of Lie algebras and their representation in SU(2). The author is trying to understand this topic and asks for a short introduction to clarify their confusion. They mention that the weights in SU(2) are just vectors of eigenvalues and the roots are the eigenvalues in the adjoint representation. The conversation also touches on the use of E-operators in SU(2) and the concept of simple positive roots being a total number of m. The conversation ends with a mention of the masterformula and its purpose in calculating all other roots.
Hymne
Hello! I am trying to understand this subject but its not simple..
I will ask some question but if anybody wants to write a short introduction which explains my confusions in a continiuous text, that would be awesome as well. :)

I think I got a good view of what our weigts are.. just vectors of our eigenvalues.
And our roots are just the eigenvalues in the adjoint representation.

When the author (Im currently reading Giorgi's book) takes this to SU(2) he talks about either using E_\alpha or E_\beta... but SU(2) just got one pair of creation- and destruction operators right? I think of the "E-operators" as creation and destruction but is there really anything to choose from in SU(2)?

Futhermore, the simple positive roots are always a total number of m, because this are the number of cartan generators, i.e. the observable we have been able to diagonalize simultanious. So each simple positive root is a creation operator for an observable which can be measured at the same time as the other observables/operators in our cartan subalgebra?

Lastly I´ve seen the masterformula and the derivation but when is it needed? Am I right if the main point is that if we have the dimension of our representation and one root, we can calculate all other roots? :/ Is there much more to it?

Thanks really much!
Best regards
Hymne

1. What are roots in Lie algebras?

Roots in Lie algebras are elements that are used to define the structure of the algebra. They are vectors in a vector space that satisfy certain properties and are used to generate the entire algebra. They are crucial in the study of Lie algebras as they provide a way to classify and understand the algebra's structure.

2. How are roots related to weights in Lie algebras?

Roots and weights are closely related in Lie algebras. The roots are used to define the weights of the algebra, which are eigenvalues of a certain linear transformation. The weights provide a way to classify the representations of the algebra, and they are important in understanding its structure and properties.

3. What is the significance of the root system in Lie algebras?

The root system is a set of vectors that satisfies certain properties and is used to classify and understand the structure of a Lie algebra. It provides a way to determine the algebra's properties, such as its dimensions, representations, and symmetries. The root system is a fundamental concept in the study of Lie algebras.

4. How do roots and weights relate to the Cartan matrix?

The Cartan matrix is a matrix that is constructed using the roots of a Lie algebra. It encodes information about the structure of the algebra, such as its root system and the relationships between its roots. The entries of the Cartan matrix are related to the weights of the algebra, providing a way to understand the algebra's representation theory.

5. Can the root system of a Lie algebra be visualized?

Yes, the root system of a Lie algebra can be visualized using a root diagram. This is a graphical representation of the root system, where the roots are represented by points in a plane or space, and the relationships between the roots are shown by lines connecting them. This diagram is a useful tool for understanding the structure of a Lie algebra and its properties.

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