Talk:Adjoint endomorphism

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Mathematics rating: Start Class Mid Priority  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)

[edit] notational problems

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)

[edit] regular representation

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)

[edit] merge

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)