Talk:Pontryagin duality

Older comments

It would be nice to add the dual of the set of p-adic numbers in the section Examples [march, 10, 2009].

Done. Ringspectrum (talk) 08:42, 10 March 2009 (UTC)

Just a note here with a copy of a discussion on User talk:Charles Matthews outlining plans for three pages: Pontryagin duality, dual group and Harmonic analysis. This is just in case anyone else wants to do anything to these pages.

Hi Charles,

Pontryagin duality looks good, but I wondered why you created this instead of adding to and/or rewriting dual group. Both are about the same thing at different levels of sophistication. I threw dual group together the other day to expand an ugly stub -- essentially the first few lines describing characters. I have no problem with dual group 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 Pontryagin duality (a more accurate title) with dual group redirecting.

If a merge was to happen, dual group suffers from not having any reasons for wanting duality (which yours covers from a couple of angles) and Pontryagin duality 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, topology) from dual group and make dual group 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.

I have also since noticed some overlap with my version of dual group with the middle chunk of Harmonic analysis. 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 Fourier analysis) 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 Wikipedia:WikiProject Mathematics page only deals with things at the article level.

Cheers, AndrewKepert 01:21, 18 Nov 2003 (UTC)

Generally, I agree with your assessment. I wrote the PD page quickly once I'd actually found dual group (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).

There is no 'they' who sorts out policy-level stuff. We're on our own here ...

Charles Matthews 10:12, 18 Nov 2003 (UTC)

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!

I will go ahead with the merge as discussed (not today - too busy) and strip the corresponding section out of Harmonic analysis. I'll also tidy up notation - I prefer &gamma ∈ Γ = G^ to φ ∈ 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.

Cheers AndrewKepert 01:00, 19 Nov 2003 (UTC)

I've done the big move from dual group, and added some headers. This leaves detailed editing of the page.

Charles Matthews 08:47, 19 Nov 2003 (UTC)

Now moved the LCA stuff from harmonic analysis, and spliced it in. Leaves the page structure OK, I think - but needing a thorough edit on notation, excess links and so on.

Charles Matthews 09:30, 19 Nov 2003 (UTC)

What is it good for?

Can someone give some examples that show where Pontryagin duality can be used to prove something or understand something better? —The preceding unsigned comment was added by 217.189.228.213 (talk) 08:49, 3 May 2007 (UTC).

Dual of finite-dimensional vector space

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

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? 72.75.67.226 (talk) 01:53, 3 October 2009 (UTC)

It says they are not naturally isomorphic (a term from category theory), but they do become isomorphic once we pick a basis. See for example Natural_transformation#Further_examples. Terminus0 (talk) 05:50, 12 November 2009 (UTC)

Topological Groups

I think one should require the the topological group is also Hausdorff space. —Preceding unsigned comment added by 132.64.102.21 (talk) 13:12, 24 November 2009 (UTC)

What is it?

The article's first para gives a suitably gentle description of where Pontryagin duality might be useful, how, and why. But I don't anywhere find anything approaching a definition. The overall feeling I get from this article is similar to the effect I'd expect if the article on lion consisted of this text:

"Lions are native to Africa. The males have manes. Their complex behavior, including apex predation, gives them a central role in the culture of many indigenous social groups. See also: circus."

(Ummm, OK, but shouldn't we mention, for instance, that lion is a species (Panthera leo) of large cat?)

Can somebody please mend this shortcoming. I'd do it myself if I knew enough about the subject matter.—PaulTanenbaum (talk) 16:18, 1 March 2010 (UTC)

About non-commutative theories

Excuse me, this propositions is a little bit vague to me:

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 group.

Is it possible to reformulate it is such a way that it could be presented as a theorem?

And if you don't mind, I would insert here a little review of non-commutative generalizations.Eozhik (talk) 08:39, 10 November 2012 (UTC)

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 Also:Peter-Weyl theorem. Maybe the main body of the article can say something about that. Mct mht (talk) 10:12, 10 November 2012 (UTC)

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 define an operation of multiplication such that the space of all irreducible unitary representrations becomes a group, is it possible? Eozhik (talk) 14:02, 10 November 2012 (UTC)

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 there is a functor 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 group and doing some casework. In particular, functoriality (which is really what we want) would require our multiplication on characters to satisfy (χf)•(ρf) = (χ•ρ)f where the dot is our supposed multiplication and juxtaposition is computation, for every homomorphism f : A -> A, A being the nonabelian group under consideration. Cyrapas (talk) 01:47, 21 January 2014 (UTC)

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, 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.) 50.132.4.93 (talk) 22:20, 17 November 2015 (UTC)

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, 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? Eozhik (talk) 14:54, 1 March 2019 (UTC)

I'll make some edits this week-end, if there will be questions, we can roll back the text and discuss. Eozhik (talk) 06:04, 2 March 2019 (UTC)

Invertibility of canonical map

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. Presumably this proof uses the fact that the groups are "small" because they are locally compact. Does anyone know it? Cyrapas (talk) 01:52, 21 January 2014 (UTC)

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 "finite-dimensional" here. 50.132.4.93 (talk) 22:57, 17 November 2015 (UTC)

Meaning of picture?

What are those patterns of colored hyperbolic tiles supposed to represent? 166.170.23.122 (talk) 04:57, 15 January 2016 (UTC)Collin237

Does Commons:File:2-adic_integers_with_dual_colorings.svg answer your question? Melchoir (talk) 04:59, 15 January 2016 (UTC)

Suitability of Picture

The 2-adic integers, with selected corresponding characters on their Pontryagin dual group

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 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.

Phrase about Plancherel measure

I must say that I don't understand the meaning of this phrase in the text:

but this diverges from the connection with harmonic analysis, which needs to tackle the question of the Plancherel measure on ${\widehat {G}}$.

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? Eozhik (talk) 10:52, 24 February 2019 (UTC)

• If there are no objections, I would like to move this subsection, Dualities for non-commutative topological groups, to the end, and to widen it. Eozhik (talk) 08:07, 1 March 2019 (UTC)

• 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. Eozhik (talk) 14:18, 1 March 2019 (UTC)

Pairing???

The section Pontrjagin duality and the Fourier transform has a subsection titled Fourier transform and Fourier inversion formula for L¹-functions.

In that subsection, this sentence appears: "As an example, suppose $G=\mathbb {R} ^{n}$, so we can think about ${\widehat {G}}$ as $\mathbb {R} ^{n}$ by the pairing $(\mathbf {v} ,\mathbf {w} )\mapsto e^{i\mathbf {v} \cdot \mathbf {w} }.$

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 think about ${\widehat {G}}$ by this pairing.

It would be very helpful if the article elaborated on how one is supposed to think of ${\widehat {G}}$ by this pairing, and why.50.205.142.50 (talk) 18:40, 25 March 2020 (UTC)

Good introduction, but not appropriate for the introduction

The introduction reads as follows:

"In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as $\mathbb {R}$, the circle, or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact abelian groups identify naturally with their bidual.

"The subject is named after Lev Semenovich Pontryagin 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 second-countable and either compact or discrete. This was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940."c

This is all certainly worthwhile information to include in the article.

But not in the introduction.

The introduction should inform the reader of what Pontrjagin duality is, at least roughly. That is much more important to include in the introduction than issues of how Pontrjagin duality fits with the rest of mathematics.

I hope someone knowledgeable about this subject can write an appropriate introduction. And perhaps move the current introduction to somewhere else in the article. 2601:200:C000:1A0:184C:28DD:BFD5:B103 (talk) 19:51, 17 April 2021 (UTC)

I agree with your remark. Many articles of mathematics have a similar issue because many editors forget that Wikipedia 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 encountered elsewhere.

I have rewritten the lead for fixing the issue. Be free to improve my version further. D.Lazard (talk) 11:52, 18 April 2021 (UTC)