Talk:Weight (representation theory)

Checks needed

Please, could somebody check the information in this page? I'm not sure about the definition of G-integral weight for a Lie group G. How much important is that G is semisimple? Does the definition make reasonable sense for non-semisimple groups as well? Further, in the definition of a weight of a Lie group, is it reasonable to assume that H is a maximal commutative subgroup? Would it not be better to assume a maximal commutative torus (I think for G complex semisimple, it is the same)? Please, fix these subtle issues, if you are an expert! Thank u, Peter

Well, if G is compact one should take H to be a maximal torus (every maximal torus is a maximal abelian subgroup but not vice-versa). I'm not sure what the correct choice is for G noncompact. Maybe a maximal closed, connected abelian subgroup? I'll work on this page if time permits, but I'm not an expert either. -- Fropuff 22:40, 27 September 2006 (UTC)

I see no explanation here of how the roots are chosen for a given Lie algebra. The ideas of weights and roots need to be connected.Dewa 20:27, 24 February 2007 (UTC)

This seems to have been done now: it is stated that roots are the weights of the adjoint representation MorphismOfDoom (talk) 04:13, 11 July 2017 (UTC)

Merge weight space into weight (representation theory)

These are from the Talk:weight space page:

Weight space is a common term in representation theory, but maybe it should be merged with the article weight (representation theory), where all the information are contained. Franp9am 18:34, 8 November 2007 (UTC)

Thanks for cleaning up most of the simple issues with this article. I tried continuing your work, and I have to agree with the merger. The problems with this article are almost all already handled in the regular weight article. In fact, treating weights as generalized eigenvalues, this is just a separate article on eigenspaces. However, notice that eigenvector, eigenvalue, and eigenspace all redirect to the same article. Perhaps the same should be done here for weight space, weight vector, and weight (representation theory). I think weight (representation theory) is a nice name for the combined article, and it is already there.

If you would like, I can add the {{mergeto}} and {{mergefrom}} tags to more formally suggest the merger. I doubt anyone will have any objection, but its good to give some time for comments. JackSchmidt 20:06, 12 November 2007 (UTC)

I never heard back from Franp9am, so went ahead and added the tags. JackSchmidt (talk) 18:09, 19 November 2007 (UTC)

I added a few more very short articles on specific aspects of weights to the merge list. Most are already contained in this article. One might want to consider fundamental representation as well, though it is reasonably full and its short companion article fundamental weight is already merged into this article. JackSchmidt (talk) 01:11, 22 November 2007 (UTC)

Done without prejudice. My view is that it is better to merge first and consolidate, and then consider whether subarticles can be spun out. Geometry guy 22:24, 25 September 2008 (UTC)

Vacuum module

Vacuum module redirects to this page, but the phrase does not appear here (or anywhere else that I can find). Would someone like to add something to justify the redirect? Spectral sequence (talk) 18:23, 4 July 2013 (UTC)

Yeah, it seems to be a pretty bogus redirect, to me. I changed it to vertex operator algebra. Another candidate might be affine Lie algebra but there is no talk of modules on that page; maybe there is a Kac-Moody representation page somewhere??? The word "vacuum" is primarily used by physicists, so vertex algebras seems like the best choice (for now). (The vacuum is literally that state in which there are no particle excitations -- i.e. empty space. Non-trivial vacuum modules give you a hint of what the stucture of the vacuum might be. This is a major concern, since we know that the real physical vacuum is very complex.) 67.198.37.16 (talk) 04:35, 2 September 2015 (UTC)

Notes on diagonalization of commuting matrices in "Motivation and General Concept"

What is the point of note2 in this section, stating that "given a set of commuting matrices over an algebraically closed field, they are simultaneously triangularizable, without needing to assume that they are diagonalizable" ? Note1 has just stated that a set of matrices are commuting iff they are diagonalizable, and diagonalization is a special case of triangularization (and obviously so!), so note2 just makes a statement that obviously follows from note1, and is logically weaker than note1. Unless there is an objection, (e.g. explaining that my comment above is in error) I'm going to remove note2 in a few days.

Relatedly, should the "iff" statement in note 1 be qualified by requiring the matrices to be "over an algebraically closed field"? I will have a look at a relevant reference but perhaps someone knows offhand. MorphismOfDoom (talk) 03:16, 11 July 2017 (UTC)

The first note says that a set of commuting diagonalizable matrices is simultaneously diagonalizable (and vice versa), not that any set of commuting matrices is simultaneously diagonalizable. YohanN7 (talk) 12:41, 11 July 2017 (UTC)

Weights on Lie algebras

Re: "A weight on a Lie algebra g over a field F is a linear map λ: g → F with λ([x, y])=0 for all x, y in g" I have not seen this elsewhere, but perhaps I just haven't seen the relevant references or don't recall it. I have seen "infinitesimal character", which I think refers to such maps; the article says "character", but I guess these are the same. However, I guess a function with this property seems like it might be the same thing as a weight in the primary sense of the article, on the enveloping algebra (which is associative). I wonder if that is the context in which the term "weight" comes to be applied to such a function, and if that should be clarified in the article? Not a particularly high priority point to deal with, though. MorphismOfDoom (talk) 03:34, 11 July 2017 (UTC)

Also, the section "Highest weights" has a fair amount of redundancy. The discussion of ordering may be unnecessary since there is a section on ordering earlier. Is the notion of highest weight much used in the sense in which the highest weight can be nonunique? Shouldn't it be made clear that one sense is primary (presumably the sense usually used in defining "highest weight module"or "highest weight representation")? (I am pretty sure that uniqueness should be imposed in the case of irreducible representations, but for non-irreps I am not sure how things should go...) MorphismOfDoom (talk) 04:15, 11 July 2017 (UTC

In the section "Highest weight module", I presume "something more special than" means "a special case of"? In that case, I propose changing to the latter terminology, because it is clearer. Also, the section needs an example separating the notions of "highest weight module" and "g-module with a highest weight" because it is very natural to suppose these are the same thing. (Is this just the uniqueness issue again, or something else?) MorphismOfDoom (talk) 04:27, 11 July 2017 (UTC)

notation under Integral Element

The ⟨⋅,⋅⟩ operator is used without any definition. I am no expert so I didn't even know at first that this wasn't supposed to be an inner product. I found my way to the Root Systems page where that was mentioned, but the operator seems to be used differently in this page than on that page. It is using brackets here for what parentheses are used for there, with no clarification. Can a knowledgeable person please add the appropriate definitions and clarifications? Thx 24.56.247.67 (talk) 02:14, 21 July 2023 (UTC)