Lie Algebras: A Walkthrough the Basics - Comments

In summary, Greg Bernhardt submitted a new blog post on Lie algebras. He explains the difference between mathematicians and physicists' use of terminology, and provides a compact summary of Lie Groups and Lie Algebras by José Natário.
  • #1
fresh_42
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Greg Bernhardt submitted a new blog post

Lie Algebras: A Walkthrough the Basics
lie_algebra_basics.png


Continue reading the Original Blog Post.
 

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  • #2
I would not agree that physicists typically call the elements of a Lie algebra "generators". I have never encountered this although I am sure there are those who might do that. Typical physics jargon is that "generator" refers to a particular set of elements of the Lie algebra that forms a basis. For example, a physicist would say that the generators of SU(2) are the Pauli matrices (not the vector space spanned by them).
 
  • #3
Well, it was only the introduction. Engel called it "Berührungstransformation" (touching transformation) which I think is close to "generator". I only observed that the term is frequently used when actually a kind of tangent is meant. I admit that I never figured it out what exactly they mean, especially as it is referenced to a) the group and b) as in your example matrices which mathematically do not belong to the tangent space. Furthermore the term doesn't really fit mathematically. Humphreys defines generators as elements which generate a free Lie algebra, similar as it is used in group theory. This makes sense, the other usage is - as I assume - a historical leftover.
 
  • #4
What a timing! I was going to study lie algebras, when you posted this insight. That will be truly helpful.
 
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Wrichik Basu said:
What a timing! I was going to study lie algebras, when you posted this insight. That will be truly helpful.
There will be two other parts coming soon (I hope). But I just found by chance an old manuscript on the Cornell server which has incidentally the same structure as my insights will have. It's a pdf of 172 pages and for free: https://pi.math.cornell.edu/~hatcher/Other/Samelson-LieAlg.pdf

My article was mainly meant for physicists who have to deal with Lie algebras but don't want to attend a lecture, so I thought I'd write a summary.
 
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  • #6
fresh_42 said:
My article was mainly meant for physicists who have to deal with Lie algebras but don't want to attend a lecture, so I thought I'd write a summary.
Yes, that's why I am reading it. I am not much find of rigorous maths as of now.
 
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  • #7
Brings back memories ... so, some tangential personal reminisces (that are not in the tangent space).

I took a grad course that used Humphreys, and that was given by two of the authors of
https://arxiv.org/abs/1411.3788 ,

Dan Britten and Frank Lemire.

They also taught me a number of other algebra courses at the undergraduate and graduate levels, and Dan Britten was an external examiner on my Ph.D committee. Frank Lemire is referenced in Humphreys. They are largely responsible for whatever ability I have in following arguments in abstract algebra (and not responsible for my shortcomings in algebra).

These guys know some of the "good" stories from my wild university days. My wife only knows that the stories exist. After 20 years without any contact, a few years ago they tracked me down and emailed me, but I delayed, and then forgot about, responding.
 
  • #8
There is a difference how mathematicians and physicists use the terminology. For a mathematician a Lie algebra is what's written in the insight article i.e. a set with operations that... In physics books one can often find phrases as " E, F, and H satisfy the ##\mathfrak{sl}_2## Lie algebra".
 
  • #9

1. What is a Lie algebra?

A Lie algebra is a mathematical structure that studies the algebraic properties of vector fields, which represent infinitesimal transformations on a given space. It provides a way to understand the behavior of continuous symmetries in a mathematical setting.

2. What are some applications of Lie algebras?

Lie algebras have a wide range of applications in various fields, including physics, engineering, and mathematics. They are used to study the symmetries of physical systems, to classify and solve differential equations, and to understand the geometric properties of manifolds.

3. How are Lie algebras related to Lie groups?

Lie algebras and Lie groups are closely related mathematical structures. Lie algebras are the infinitesimal version of Lie groups, which are continuous groups of transformations. Essentially, a Lie group is a smooth manifold with a group structure, while a Lie algebra is the tangent space at the identity element of the Lie group.

4. What are the basic properties of Lie algebras?

Some of the basic properties of Lie algebras include closure under the Lie bracket operation, which measures the non-commutativity of the algebra, and the Jacobi identity, which ensures the associativity of the Lie bracket. Lie algebras are also equipped with a bilinear form called the Killing form, which measures the non-degeneracy of the algebra.

5. How can I learn more about Lie algebras?

There are many resources available for learning about Lie algebras, including textbooks, online courses, and research articles. It is recommended to have a strong background in linear algebra and abstract algebra before delving into the study of Lie algebras. Additionally, hands-on practice with solving problems and working with examples can greatly aid in understanding this complex mathematical structure.

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