Understanding Equation XI.31 in Lie Algebras

In summary, the conversation is about the speaker's struggle to understand equation XI.31 in the lecture notes on Lie algebras. They are seeking guidance and may also post in a different forum for help. They also ask for clarification on the definition of unit length in the normalization procedure of the operator being used.
  • #1
vega12
11
0
I am currently trying to up my understanding of Lie algebras as the brief introductions I have had from various QFT textbooks feels insufficient, but have been stuck on one small point for a couple days now. I am reading through the lecture notes / book by Robert Cahn found here: http://theory.uchicago.edu/~sethi/Teaching/P385-W2011/texall.pdf. On page 97, I am having trouble understanding how equation XI.31 comes about. I think I get how [itex]Tr C = N_\Lambda \langle \Lambda, \Lambda + 2 \delta \rangle_2 [/itex] comes about, but don't see how I can use the statement regarding (( , )) yielding [itex]l_\phi ( , )_2[/itex] and how that directly leads to XI.31.

A bit of guidance here would be very much appreciated. Thanks.
 
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  • #2
I am very sorry for bumping my own thread with a double post. If it turns out that I cannot get a response here, could someone perhaps recommend a more appropriate subforum for me to post in? Would "general math" be more promising? Thanks.
 
  • #3
Hey vega12 and welcome to the forums.

With the normalization procedure, what is the definition of something with unit length in the context of the operator you are working with (the matrix)?
 

Related to Understanding Equation XI.31 in Lie Algebras

What is a simple Lie algebra?

A simple Lie algebra is a type of mathematical structure that is used to study symmetries in various areas of mathematics and physics. It is a type of algebraic structure that is defined by a set of generators and relations, and has important applications in areas such as group theory, differential geometry, and quantum mechanics.

How is a simple Lie algebra different from a general Lie algebra?

A simple Lie algebra is a special case of a general Lie algebra, which is a more broad and general type of algebraic structure. The main difference is that a simple Lie algebra cannot be decomposed into smaller subalgebras, whereas a general Lie algebra can. This makes simple Lie algebras particularly useful in studying symmetries, as they have a more streamlined and manageable structure.

What are the properties of a simple Lie algebra?

A simple Lie algebra has a few key properties that distinguish it from other types of algebras. These include being non-abelian (meaning the order of operations matters), being finite-dimensional, and having a non-degenerate bilinear form. Additionally, a simple Lie algebra is always semisimple, meaning it has no non-trivial ideals and is therefore completely reducible.

What are some applications of simple Lie algebras?

Simple Lie algebras have many important applications in mathematics and physics. In mathematics, they are used to study group theory, representation theory, and algebraic geometry. In physics, they are used to study symmetries in quantum mechanics and quantum field theory, as well as in the theory of special relativity. Simple Lie algebras also have applications in the study of differential equations and integrable systems.

How are simple Lie algebras classified?

Simple Lie algebras are classified according to their Dynkin diagrams, which are graphical representations of their structure. These diagrams are used to determine the type of Lie algebra, its dimension, and its properties. There are four infinite families of simple Lie algebras, known as types A, B, C, and D, as well as a few exceptional cases. This classification system helps to organize and understand the various types of simple Lie algebras and their properties.

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