Question about the "E" Lie algebras

[topsquark]

I'm in chapter 2 of my Lie algebra text. I have an interest in E_8 but I'm only going to work my way up to E_6 as it has fewer operators. Anyway for now I only have a quick question: Is E_6 defined by its Dynkin diagram or is E_6 defined by something else and the Dynkin diagram follows from that? I haven't been able to find a definition of what E_6 is in my text yet.

-Dan

 

[jvicens]

Hello Dan,

Thank you for your question. E_6 is indeed defined by its Dynkin diagram, which is a visual representation of the root system of the Lie algebra. In fact, all simple Lie algebras are classified by their Dynkin diagrams. However, the Dynkin diagram is not the only way to define E_6. It can also be defined algebraically, by specifying the structure constants and commutation relations of its generators.

In your text, E_6 may be introduced in different ways depending on the author's approach. Some texts may start with the Dynkin diagram, while others may start with the algebraic definition. Either way, the two definitions are equivalent and can be used interchangeably.

I would suggest looking for an algebraic definition of E_6 in your text, which will provide a more comprehensive understanding of the algebra. However, it is also important to understand the significance of the Dynkin diagram and how it relates to the algebraic definition.

I hope this helps answer your question. Best of luck with your studies in Lie algebras.