Talk:Adjoint representation of a Lie algebra

From Wikipedia, the free encyclopedia
Jump to: navigation, search
WikiProject Mathematics (Rated Start-class, Mid-importance)
WikiProject Mathematics
This article is within the scope of WikiProject Mathematics, a collaborative effort to improve the coverage of Mathematics on Wikipedia. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks.
Mathematics rating:
Start Class
Mid Importance
 Field: Algebra

The relationship between ad and Ad should really be its own article, and that article should develop the vector fields as well. Unfortunately, the articles on differential forms and manifolds need to be brought up to snuff as well.

I don't agree that the relationship between Ad and ad should have separate articles. I think the adjoint rep of a group and the adjoint rep of an algebra should be in the same article, along with the relationship between the two. Lethe | Talk 03:05, Jan 31, 2005 (UTC)
Err, maybe .. as long as the article stays short, they can stay together. But I think one can say a lot more about Ad, and that would end up making the article long .. too long. Also, what we really really need is a good article on exp and Exp and all of that tangled relationship. Ten years ago, I ditched my book on Lie algebras, now I am sorry I did... linas 20:53, 7 Feb 2005 (UTC)

Another generic problem I'm running into is how to figure out how to say this so that it holds for arbitrary fields, supersymmetric crap, and also for categories, and I don;'t see a way out of that kind of mess.linas 02:02, 31 Jan 2005 (UTC)

notational problems[edit]

we have the problem in this article that we are using g for both an element of the Lie group G, and also for the corresponding Lie algebra. To avoid this confusion, I suggest all instances of the Lie algebra be changed to \mathfrak{g} which has the unfortunate consequence of requiring us to use extensive math tags inline with the text. -Lethe | Talk 02:48, Jan 31, 2005 (UTC)

Sounds good to me linas 20:41, 7 Feb 2005 (UTC)
Just some style remarks. You are inconsistent by using both g and mathfrac{g}.
I have hopefully removed the last of the inconsistencies now. -Lethe | Talk 07:56, Feb 9, 2005 (UTC)
Also, I have just rudimentary knowledge of Lie algebras and conventions surrounding them; but, just for style, why not use a nice italic A for the algebra? Oleg Alexandrov 05:16, 9 Feb 2005 (UTC)
It's really nice to use a name for the algebra which indicates that it is related to the group. A lot of texts forego the mathfrak notation, and just use italics, but we don't have that luxury here. Also, A would lead to notational overlap with the Lie algebras An. I'm sure there is a better notational solution to this problem, but I kinda like mathfrak, and I'm sick of waiting for browser support for mathml. Sick and tired! I'm going to go dunk my head in the sink now. -Lethe | Talk 07:56, Feb 9, 2005 (UTC)
PS Just wait until User:Michael Hardy will see you abusing inline PNG. :) Oleg Alexandrov 05:18, 9 Feb 2005 (UTC)
I won't tell if you won't (-: Lethe | Talk
I had to consult an online dictionary to understand what "snarky" means. I know I am snarky sometimes, so I will try to always be "nonsnarky" as you suggest; if no other reason, at least to keep things consistent. Oleg Alexandrov 17:44, 9 Feb 2005 (UTC)
Unfortunately, I can't make the same promise. (PS I liked some of the definitions at urbandictionary) Lethe | Talk 19:51, Feb 9, 2005 (UTC)
Got it now. You meant snarky in a positive way. Oleg Alexandrov 20:35, 9 Feb 2005 (UTC)

regular representation[edit]

Is the adjoint represetation really also called the regular representation? wouldn't this cause some confusion the rep over at regular representation? Is this usage in Fulton & Harris? -Lethe | Talk 03:01, Jan 31, 2005 (UTC)

My brain is off right now, but after skimming regular representation, it really sounds like the same thing as adjoint representation. Its the action of the group on itself. The article at regular representation is more general, whereas this article sticks to (and should stick to) Lie groups. linas 20:41, 7 Feb 2005 (UTC)
You think that the representation of a group acting by translations on the functional space on that group is the same thing as x\mapsto gxg^-1? I don't see anything in the article that could let you believe that. Lethe | Talk 21:16, Feb 7, 2005 (UTC)
I'm pretty sure that the regular representation (as defined here on wikipedia) is quite different from the adjoint representation. All I wanted to check with you is whether perhaps Fulton and Harris had given a different definition. Today I checked with them. They don't. I removed it. -Lethe | Talk 17:42, Feb 8, 2005 (UTC)
Lethe, you are confusing youself. And me too. Either that, or its late at night here, and I am confusing myself. Think about the adjoint rep of the algebra first ... its clearly just shoving vectors around. That is, a vector can act on another vector to generate a third vector. e.g. the adjoint rep of su(2) is so(3) which sounds like the regular representation over C to me. Now think "gosh, every vector in the algebra is a vector field on the manifold" ... you don't get just any representation, but a particular one, the one where the group acts on itself (which is why is SO(3)=S_3 and not SU(2)). For the Lie algebra, its the "topological group" case, where "translation on the functional space" means LX the lie derivative. and the lie derivative L_g is just x mapsto gxg^-1. Of course, I could be hallucinating too...
Ugh... I see, I think that actually that article is poorly written. I think the correct definition (for a finite group) is to take the set S to consist of the set of group elements. Then one builds a vector space from that set S over a field K. Then one asks "how does G act on that vector space". I think that is what the article tried to say, but that is not how that article is written.
Worse, I'll say that article is poorly written ... you won't get a clue as to what a regular representation is unless you happen to already know what it is. :( linas 08:35, 10 Feb 2005 (UTC)
Why not just ask User:Charles Matthews directly? He wrote the article ...linas 08:39, 10 Feb 2005 (UTC)
So you actually do think that the regular rep and the adjoint rep are the same thing? Could you provide some evidence? -Lethe | Talk 11:22, Feb 10, 2005 (UTC)

merge[edit]

this article should really be merged with adjoint representation, I think -Lethe | Talk 03:02, Jan 31, 2005 (UTC)

Yes, it should. I discovered that article after I got done writing this article, and was then hit with a bout of laziness. linas 20:41, 7 Feb 2005 (UTC)
No, actually, I take that back. This article is about algebras, and that article is about groups. I am somwhat concerned that by trying to talk about algebras and groups at the same time, the article will get unweildy, and too long ... the algebra and the group really do behave differently. However, this article should be renamed adjoint representation of a Lie algebra or something like that. linas 20:44, 7 Feb 2005 (UTC)

Requested move[edit]

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. Editors desiring to contest the closing decision should consider a move review. No further edits should be made to this section.

The result of the move request was: No Consensus to move.  Ronhjones  (Talk) 22:37, 20 February 2013 (UTC)


----

Adjoint endomorphismAdjoint representation – Lie algebra adjoint is in general not an endomorphism. This page should be called either adjoint representation or adjoint homomorphism. Relisted. BDD (talk) 00:34, 15 January 2013 (UTC) Ilyats (talk) 07:30, 7 January 2013 (UTC)

  • Query. Since Adjoint representation of a Lie group (still) exists, would Adjoint representation of a Lie algebra (still) be too wordy? CsDix (talk) 17:19, 12 February 2013 (UTC)
  • Right now, this is a WP:TWODABS situation with a large number of incoming links, which is unlikely to be the best solution. If a term has only two meanings, at least half the time the reader will be looking for one of those meanings. If that title is a disambiguation page, then the reader arriving on the page must figure out which meaning to go on to, and may very well need to visit both pages to figure out which one is the correct one. In other words, they would type or click the name and get taken to a page from which they need to make at least another click to get where they really want to go. Where there are only two possible meanings, having the more likely search target of those meanings at the undisambiguated title means that hopefully at least half the time, the reader landing on that page does not need to click any further, because they have found exactly what they are looking for. For the reader who is looking for the other meaning, it is right there in the hatnote, so they are not inconvenienced any more than they would be if they had gone to a disambiguation page. The basic question, of course, is which of the two pages should be at that title. In the interests of maximizing reader efficiency, I would urge that one topic or the other be moved to this title. bd2412 T 02:10, 13 February 2013 (UTC)
  • I'm not exactly sure what the nom intended with respect to Adjoint representation of a Lie group, but perhaps it is best to merge these two articles, which is a natural enough thing to do. They both reference each other in significant ways; and the merged article would fit well at adjoint representation. Mark M (talk) 09:43, 17 February 2013 (UTC)
  • The article title is definitely strange. Indeed, the article itself doesn't use the term. I like the idea of Mark. The two concepts are closely related and thus fits into one place. -- Taku (talk) 12:31, 17 February 2013 (UTC)
The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page or in a move review. No further edits should be made to this section.

Requested move II[edit]

The following discussion is an archived discussion of the proposal. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the proposal was moved by TakuyaMurata. --BDD (talk) 16:20, 30 April 2013 (UTC)

Adjoint endomorphismAdjoint representation of a Lie algebra – Basically because there is no rationale for the current article tittle. I myself have never seen it outside Wikipedia. Taku (talk) 17:16, 26 April 2013 (UTC)

Can we not just merge it into adjoint representation, and make that one article good? Mark M (talk) 07:52, 27 April 2013 (UTC)
Yes. That's what you suggested above and what I agreed. From the point of view of pure algebra (i.e., adjoint rep of an abstract Lie algebra), there is really not much to discuss. Lie algebra representation covers all of abstract parts, and that's enough.
I've have just boldly made a move. The article title was wrong, plain simple after all. I'm still with the merger and if there is no objectio,. I'm going to do it. -- Taku (talk) 11:14, 27 April 2013 (UTC)
The above discussion is preserved as an archive of the proposal. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.