Talk:Exceptional object

Sketch

Header added. —Nils von Barth (nbarth) (talk) 03:22, 1 December 2009 (UTC)

This is merely a sketch for an article. I'm hoping to encourage experts in the various fields to contribute to the relevant sections. Sigfpe 00:49, 25 May 2006 (UTC)

It’s certainly grown – thanks!

—Nils von Barth (nbarth) (talk) 03:22, 1 December 2009 (UTC)

Snarks: are they exceptional?

It's not clear from the [snark] page that snarks are indeed exceptional objects as described here. Perhaps more explanation is needed? 130.102.158.15 (talk) 00:37, 28 September 2010 (UTC)

I agree that snarks do not quite fit the article. I just put a citation-needed tag on the snark section, but given that the parent comment was made four years ago, I'll just be bold and move snarks to the 'see also' section. –Jérôme (talk) 18:58, 11 October 2014 (UTC)

Just moved snarks to the 'see also' section. I couldn't find references linking them to the other exceptional structures, and please also note that snarks are not related to sporadic groups / exceptional Lie groups, as do all other cases. –Jérôme (talk) 19:03, 11 October 2014 (UTC)

I removed snarks completely, because after researching a little bit more they do not represent anything exceptional (in the sense of the article) at all. –Jérôme (talk) 19:16, 11 October 2014 (UTC)

Ken Ono's partition formula

24 also appears in the recently-discovered finite algebraic formula for the partition numbers. I don't actually understand the subject, so I'll refrain from adding it in, but the formula is related to modular forms and the Dedekind eta function (and is rather important, I'm told) so it probably can be added to the 'connections' section.

Anyone know more about the topic? Vladimirdx (talk) 06:52, 2 April 2011 (UTC)

Open Issues

In the mind map, F4 is shown as "26-dimensional". The fundamental irreducible representation is 26-dimensional, but F4 itself is a 52-dimensional algebra. — Preceding unsigned comment added by 2A02:1205:C694:F520:36F3:9AFF:FE64:286E (talk) 22:26, 5 September 2020 (UTC)

Platonic manifold

This is very interesting material that does not waste much time with generally accessible discussion before launching into expertspace. That's OK in isolation, but I haven't been able to identify an elementary article that covers much of the lower-hanging fruit there is that interests generalists.

I identified an article whose basic structure seems to me that it could provide the scaffolding:

Matthias Goerner, 2017. A census of hyperbolic Platonic manifolds and augmented knotted trivalent graphs. New York Journal of Mathematics 23:527–553.

In case editors familiar with this article are unfamiliar with precedents around other mathematical topics, we have a gaggle of articles around Boolean algebra that fulfil the encyclopedic needs of classes of reader with quite different background. I'm hoping we can do something a bit like that here, and I wonder if Platonic manifold might be a good hook on which to hang such a more accessible article? — Charles Stewart (talk) 11:35, 18 August 2021 (UTC)

"Platonic manifold" would have to meet the wiki-notability standard to qualify for an article unto itself. It seems a bit too niche for that, but I could be mistaken. You could always try gathering a few more references and starting an article in draft space. On the whole, it seems difficult to write a generalists' introduction to the topic of "exceptional objects", since it concerns relations between multiple types of thing, all of which require considerable background to understand. XOR'easter (talk) 03:08, 20 August 2021 (UTC)

Is this true as stated?

The section Division algebras includes this passage:

"There are only three finite-dimensional associative division algebras over the reals — the real numbers, the complex numbers and the quaternions. The only non-associative division algebra is the algebra of octonions."

I know that it was proven about 1958 that any finite-dimensional real division algebra must have real dimension = 2^k ∈ {1,2,4,8}.

But I have never seen a reliable reference to the often-claimed result that the only finite-dimensional real division algebras are ℝ, ℂ, ℍ, 𝕆.

Nor does the article provide a reference for this claim.

I hope someone knowledgeable about this subject can either a) provide a reference (if there is one), or else b) fix this mistake (if it is one).

Addendum: The sentence:

"The only non-associative division algebra is the algebra of octonions."

is simply false. This can be seen from the answers online at

https://math.stackexchange.com/questions/2892818/finite-dimensional-division-algebras-over-the-reals-other-than-mathbbr-math

and following the links found there.

(Finite-dimensional division algebras over the reals have not been classified, and it is known that there are infinitely many non-isomorphic ones.)

I hope someone knowledgeable about this subject can fix this mistake. — Preceding unsigned comment added by 2601:204:F181:9410:A161:BD2A:AC22:E108 (talk) 18:48, 2 August 2024 (UTC)