Lie Algebra states of a representation

[Silviu]

Hello! I am reading some representation theory/Lie algebra stuff and at a point the author says "the states of the adjoint representation correspond to generators". I am not sure I understand this. I thought that the states of a representation are the vectors in the vector space on which the representation acts. So for a 3D representation of SO(3), the states would be euclidian vectors. But in this case, the generators are forming the matrices that act on a given vector space (in the case of SO(3) they would be the $L_x$ matrices). So how can the generators, be also states of a representation, when they generate the representation? I assume I miss understood something, so I would really appreciate if someone can clarify this for me, or point me towards a good reading. Thank you!

 

[fresh_42]

The adjoint representation of the Lie group is a conjugation on the Lie algebra:
$$
\operatorname{Ad}(g)(X) = gXg^{-1}\quad (g\in G\; , \;X\in \mathfrak{g})
$$
The adjoint representation of the Lie algebra is the multiplication of the Lie algebra:
$$
\operatorname{ad}(Y)(X) = [Y,X] \quad (X,Y \in \mathfrak{g})
$$
So the vector space, on which Lie group or Lie algebra elements operate on, is in both cases the tangent space, i.e. the Lie algebra, which vectors are notoriously, and in my opinion unfortunately called "generators" by physicists. For a mathematical point of view you might have a look on this article:
https://www.physicsforums.com/insights/representations-precision-important/
The reason I dislike this wording can be found in a recent thread of yours about the same question. It illustrates very well, why this wording causes misunderstandings.
https://www.physicsforums.com/threads/su-2-generators.921303/

 

[Silviu]

Thank you so much for this reply. Your article was really helpful. I would appreciate if you can point me towards others of the same kind (where the concept itself is explained in great detail, as most of the book just delve into derivations). Also, what background should I have in order to understand Lie Algebra properly (for example I didn't know what a tangent space was before, so should I read a differential geometry book before?) or what good book would you recommend?

 

[fresh_42]

Thank you so much for this reply. Your article was really helpful. I would appreciate if you can point me towards others of the same kind (where the concept itself is explained in great detail, as most of the book just delve into derivations). Also, what background should I have in order to understand Lie Algebra properly (for example I didn't know what a tangent space was before, so should I read a differential geometry book before?) or what good book would you recommend?

This depends on where you want to arrive at. From a physical point of view, Lie theory is closely related to its (differential) geometry aspects and starts with Lie groups. The usual suspects are not very difficult groups, e.g. $\,SU(n)\, , \,O(n)\, , \,GL(n)$ or similar matrix groups. The more difficult part is their topological and analytical properties, which make them a Lie group. Their study is differential geometry and in its heart, the theorem of Noether which physicists use to describe conservation laws, is a theorem in Lie theory. In this context Lie algebras simply occur as tangent spaces which carry a Lie algebra structure. Their representations come into play, if their vectors are investigated as operators on certain vector spaces.

This is one side of it and the entry point for physicists. But you can also consider Lie algebras (and their representations) from a pure algebraic point of view, where the matrix groups are neglected and their Lie algebras, e.g. $\,\mathfrak{su}(n)\, , \,\mathfrak{o}(n)\, , \,\mathfrak{gl}(n)$ are considered in the first place. This is an easier approach as one doesn't need the mighty tools of analysis and topology, rather (usually finite dimensional) vector spaces instead. It's also a shorter way to understand what classical Lie groups / Lie algebras are, why they are important and what Killing-form and Dynkin diagrams have to do with it. In this context, representations are directly addressed and the case $\mathfrak{sl}(2) \cong \mathfrak{su}(2)$ can normally be found in every textbook about Lie algebras as the easiest example which representations can be fully developed in a few pages.

If we look at the number of pages my book on Lie groups has (430, [1]) and the one on Lie algebras (170, [2]), we'll get an impression of the differences between the two. Of course there are "thicker" books on Lie algebras (https://www.amazon.com/dp/0824767446/?tag=pfamazon01-20), so it still depends on what you want to learn on your way: differential geometry or algebra. I'm afraid the physical approach is the longer road.

 

[Silviu]

This depends on where you want to arrive at. From a physical point of view, Lie theory is closely related to its (differential) geometry aspects and starts with Lie groups. The usual suspects are not very difficult groups, e.g. $\,SU(n)\, , \,O(n)\, , \,GL(n)$ or similar matrix groups. The more difficult part is their topological and analytical properties, which make them a Lie group. Their study is differential geometry and in its heart, the theorem of Noether which physicists use to describe conservation laws, is a theorem in Lie theory. In this context Lie algebras simply occur as tangent spaces which carry a Lie algebra structure. Their representations come into play, if their vectors are investigated as operators on certain vector spaces.

This is one side of it and the entry point for physicists. But you can also consider Lie algebras (and their representations) from a pure algebraic point of view, where the matrix groups are neglected and their Lie algebras, e.g. $\,\mathfrak{su}(n)\, , \,\mathfrak{o}(n)\, , \,\mathfrak{gl}(n)$ are considered in the first place. This is an easier approach as one doesn't need the mighty tools of analysis and topology, rather (usually finite dimensional) vector spaces instead. It's also a shorter way to understand what classical Lie groups / Lie algebras are, why they are important and what Killing-form and Dynkin diagrams have to do with it. In this context, representations are directly addressed and the case $\mathfrak{sl}(2) \cong \mathfrak{su}(2)$ can normally be found in every textbook about Lie algebras as the easiest example which representations can be fully developed in a few pages.

If we look at the number of pages my book on Lie groups has (430, [1]) and the one on Lie algebras (170, [2]), we'll get an impression of the differences between the two. Of course there are "thicker" books on Lie algebras (https://www.amazon.com/dp/0824767446/?tag=pfamazon01-20), so it still depends on what you want to learn on your way: differential geometry or algebra. I'm afraid the physical approach is the longer road.

Thank you for your detailed answer! So I am an undergraduate, but I want to do a phD in theoretical physics. So what would you advice me to do? Like I assume that I need the physics approach at a point, but would it be easier to start with the mathematical one and turn later to the physics view?

 

[fresh_42]

Well as a theoretical physicist, you should probably learn whatever is possible in analysis, differential equations and differential geometry. A few basics about topology come automatically with it. With these foundations you should be able to learn Lie theory a bit easier than you probably would if you started with Lie groups. E.g. Humphreys book on Lie algebras (no Lie groups) can be read along these topics as a counterpoint to analysis. You won't necessarily have to learn the entire theory and may use it as a reference to look up things like weight spaces, roots, the four classical series of Lie groups / algebras or what this $E_8$ is which physicist talk about. Usually the Lie algebras in physics are semisimple, so you don't need to know how to prove, e.g. Engel's theorem. Just know what it says. (And before I get corrected, yes I know, Heisenberg and Poincaré algebras are not semisimple.)

Theoretical physics unfortunately means to learn at least three different dialects, if not languages: physics, physical math, mathematical math, and eventually English. This is often forgotten, but there are differences between the three. Generator is an example: change in movement, a differential, a tangent vector.

To summarize, my advice is to learn what you need anyway, which is calculus (analysis, differential equations, variations) and differential geometry. These are needed in basically all physical areas. With these, you could tackle Lie groups. The Lie algebra side of it could be done in portions alongside the other stuff as something differently, e.g. for Humphreys book within three parts: chapters I-III / VI / VII. It can be read with basic knowledge of linear algebra.