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but since dimensions multiply under tensoring, it has no chance of being invertible in general, so we would generally just have a monoid. Still, the classical definition of the Brauer group for CSA's naively has the same issue--tensoring only gives a monoid structure--which gets resolved by weakening the notion of equivalence slightly. The quoted text is vague about why one should never expect such a theory to exist, and it's also vague about its own abstract underpinnings. I like it, I just wish it were expanded, clarified, and referenced. (I know nothing about this subject.)
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those who have a working familiarity with the subject, this fact may not be immediately apparent. However, a failure to understand is obviously not sufficient reason to remove information (image or otherwise) from a page. It would perhaps be best for the editor who added the picture to remove it themselves, however, if I remain convinced in another few months, I will probably cut to the chase and remove it myself. Or not? Please chime in. Ciao.
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it is not clear how to introduce multiplication on the dual object if it is supposed to be the set of irreducible unitary representatioons, and it is even not clear whether the set of irrducuble unitary representations is indeed a good choice for the status of the dual object for the group. So the task of constructing duality in this case requires complete rethinking. Would anybody mind if I write something like this?
343:. I'll also tidy up notation - I prefer &gamma â Î = G^ to Ď â G' myself, but will change additive notation in the group to multiplicative. I think the merge will work well, given the complementary POV of the two pages. Then I will start thinking about how HA should be structured and fleshing it out. Hopefully whatever I do will offend sufficiently many people, thus attracting interest and contributions.
305:) should really be a high-level page with a general flavour of what harmonic analysis is about, some history and many links to the disparate fields that come under or are related to HA - from the abstract to the applications (classical things like PDE solutions through to wavelets and whatever else). Do you know of anyone who has an interest in (or oversees/coordinates) this higher-level organisation of WP? The
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323:(I've thought subsequently of adding a bit more, on discrete and compact R-modules, and End(G) = opposite ring of End (G^)). It could indeed be good to merge the two under PD as main title, putting the dual group material first in the article (and moving across from HA anything that is really about LCA in general as you suggest).
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I must say that I don't like this vagueness. It seems to me it would be more honest to write that in the noncommutative case the classical construction stops working for various reasons, in particular, because characters stop separating points, irreducible representations cease to be one-dimensional,
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I am considering removing the picture at the top of the page, though I thought it prudent to begin a discussion first, giving those who may disagree the opportunity to articulate their reasons; mine are straight forward enough: It is too technical to be understood by a lay audience. I believe for
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In the section about the duality theorem and the canonical map, the canonical map is defined, but it is not shown to be an isomorphism. This, to me, seems a rather important step, since in other contexts (e.g. infinite-dimensional vector spaces) analogous canonical maps can fail to be isomorphisms.
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In a very naive sense, and assuming choice, yes. The axiom of choice implies that every nonempty set carries some group structure, so we can pick one essentially at random for each of our sets of representations. Of course, this will not have any nice properties; the interesting question is whether
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suffers from not actually saying what the dual group is (in terms that an undergrad maths student can handle). Your account of the development of the theory is also better. Maybe a solution is to import and tidy up some of the material on the concrete construction of the dual group (characters,
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I interpreted that sentence to mean that in a more abstract treatment of G^, we would find that its elements are precisely the 1-dimensional representations (up to isomorphism) and the group operation is the tensor product. This has a chance of being invertible for one-dimensional representations,
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between appropriate categories that we can define. Honestly, the statement above confuses me, too: what isomorphism classes are we taking? where did the notion of dimension come in? I suspect that there is not such a functor; perhaps one could prove this by taking a small (i.e. finite) nonabelian
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I agree with your remark. Many articles of mathematics have a similar issue because many editors forget that
Knowledge is not a text book, and thus that, generally, readers do not come to an article for learning why the subject is important, but for learning more on a subject that they have
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Not sure what you want in a Wiki article--a sketch of a proof? The explicit map was probably only included because it's so easy to define. Any textbook presentation of this material will include a proof that it's in fact an isomorphism. Of course "locally compact" is standing in for
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group and doing some casework. In particular, functoriality (which is really what we want) would require our multiplication on characters to satisfy (Ďf)â˘(Ďf) = (Ďâ˘Ď)f where the dot is our supposed multiplication and juxtaposition is computation, for every homomorphism f : A -: -->
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Seems to be all that statement says is, for non-abelian groups, one cannot multiply irreducible unitary representations to get another one unlike the characters in the abelian case. That said, clarification/elaboration is always good. The simplest nonabelian case is listed in See
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Such a theory cannot exist in the same form for non-commutative groups G, since in that case the appropriate dual object G^ of isomorphism classes of representations cannot only contain one-dimensional representations, and will fail to be a
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Thanks
Charles - yes I know Wiki is anarchic. I also thought that informing somebody(s) who might consider the higher-level organisation as their patch might bring in a few tips and avoid subsequent angst when somebody reverted the lot!
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301:. I think it doesn't really belong there, as harmonic analysis on LCA groups is only a part of HA, whereas it reads as though it is the most important idea. My feeling is that HA (also served by a redirect from
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And to tell the truth, I want to delete this phrase about the
Plancherel measure. It is too vague, in my opinion. At least for some groups the theorem remains true as this is written in A.A.Kirillov's book.
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for that content in the latter page, and it must not be deleted so long as the latter page exists. Please leave this template in place to link the article histories and preserve this attribution.
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who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the group being
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a redirect? Possibly keep the classical examples as well, as they tie it to the
Fourier transform/series articles. My attempt at referencing was deliberately lazy, so could be improved.
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I would like to ask its author to clarify this. The duality theories for non commutative groups are constructed. What is the reason to mention the
Plancherel measure in this connection?
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being improved -- such is the way of the wiki. It suffers from the its history and while trying to be consistent with related pages. The best solution may be a merged page under
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The article's first para gives a suitably gentle description of where
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I hope someone knowledgeable about this subject can write an appropriate introduction. And perhaps move the current introduction to somewhere else in the article.
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It is analogous to the dual vector space of a finite-dimensional vector space: a vector space V and its dual vector space V * are not naturally isomorphic
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But if we do not see how to multiply representations, I suppose, this doesn't mean that such a multiplication does'n t exist... Perhaps one can
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This is unclear, but seems to say that a finite-dimensional vector space is not isomorphic to its dual. That's not right, is it?
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Can someone give some examples that show where
Pontryagin duality can be used to prove something or understand something better?
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an operation of multiplication such that the space of all irreducible unitary representrations becomes a group, is it possible?
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together the other day to expand an ugly stub -- essentially the first few lines describing characters. I have no problem with
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But this is the first use of the word "pairing" in the article, so the reader does not know why or how one is supposed to
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Presumably this proof uses the fact that the groups are "small" because they are locally compact. Does anyone know it?
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I will go ahead with the merge as discussed (not today - too busy) and strip the corresponding section out of
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and either compact or discrete. This was improved to cover the general locally compact abelian groups by
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suffers from not having any reasons for wanting duality (which yours covers from a couple of angles) and
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to include in the introduction than issues of how
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I'll make some edits this week-end, if there will be questions, we can roll back the text and discuss.
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Can somebody please mend this shortcoming. I'd do it myself if I knew enough about the subject matter.â
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Knowledge. If you would like to participate, please visit the project page, where you can join
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And if you don't mind, I would insert here a little review of non-commutative generalizations.
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Is it possible to reformulate it is such a way that it could be presented as a theorem?
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Examples .
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I have rewritten the lead for fixing the issue. Be free to improve my version further.
1086:{\displaystyle (\mathbf {v} ,\mathbf {w} )\mapsto e^{i\mathbf {v} \cdot \mathbf {w} }.}
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I think one should require the the topological group is also Hausdorff space.
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There is no 'they' who sorts out policy-level stuff. We're on our own here ...
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This is all certainly worthwhile information to include in the article.
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If there are no objections, I would like to move this subsection,
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181:this version
177:19 June 2019
137:Mid-priority
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62:Midâpriority
40:WikiProjects
743:50.132.4.93
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636:there is a
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112:Mathematics
103:mathematics
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1338:Categories
1274:AndrĂŠ Weil
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