Insights Lie Algebras: A Walkthrough - The Basics

[fresh_42]

This article is meant to provide a quick reference guide to Lie algebras: the terminology, important theorems, and a brief overview of the subject. Physicists usually call the elements of Lie algebras generators, as for them they are merely differentials of trajectories, tangent vector fields generated by some operators. Thus the distinction between Lie groups and Lie algebras sometimes gets lost. It is the distinction between manifolds and their tangent spaces. If terms as commutator, adjoint, or representation, in general, are used, which apply to both, it is often unclear which of them is meant. The underlying connection is Noether's theorem, which establishes a correspondence between physical invariants and symmetric groups, Lie groups. The approximation of curved objects - the Lie group elements - by first-order approximations - the Lie algebra elements - is a standard procedure in physics, which might partially explain the neglect. However, the following lays the emphasis on the algebra part from a terminological point of view. The corresponding concept for groups will be named whenever there is an appropriate one. I cannot write another textbook about Lie algebras here, and there is no need to, as there are already many excellent ones! Instead, we will focus on the definitions and theorems, driven by the importance Lie algebras have to physics.

Lie algebras are algebras, are vector spaces. They have an internal multiplication, the commutators, as well as a scalar multiplication by elements of the underlying field - and right in the middle of some common misconceptions we are.

Continue reading ...

 

[Orodruin]

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).

 

[fresh_42]

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.

 

[Wrichik Basu]

What a timing! I was going to study lie algebras, when you posted this insight. That will be truly helpful.

 

[fresh_42]

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.

 

[Wrichik Basu]

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.

 

[George Jones]

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.

 

[martinbn]

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".

 

[FourEyedRaven]

I found a compact summary of Lie Groups and Lie Algebras by José Natário, a Portuguese mathematical physicist. I hope it helps.

https://www.math.tecnico.ulisboa.pt/~jnatar/GLAL-08/resumos.pdf