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In the paper Category Theory Applied to Pontryagin Duality by Roeder, the character group of an lca group is defined as the topological (under the compact-open topology) abelian group of continuous homomorphisms from $G$ to $\mathbb {T}=\mathbb R/\mathbb Z$. Why does the definition specifically involve $\mathbb{T}$? What is special about it and why is the class of continuous homomorphisms from some group to $\mathbb{T}$ more informative (if it is) than to some other group?

  • This has been asked before. You can actually take $H=\Bbb Q/\Bbb Z$. $H$ has elements of all orders, so there is always (by the structure theorem of PIDs) en element in $\hom_{\Bbb Z}(G,H)$ for any abelian group $G$, and in fact it is an injective $\Bbb Z$-module (since it is divisible and $\Bbb Z$ is a PID), so it is an injective cogenerator of $\Bbb Z$-mod. – Pedro Nov 04 '14 at 19:48
  • Note $\Bbb T$ is the circle group. A compact bounded abelian divisible group consisting of complex numbers which allow simple calculations. A better choice than $\Bbb R$. – Minimus Heximus Nov 04 '14 at 22:57
  • @user Okay, but surely there are many nice groups allowing for simple calculation.. Why pick specifically $\Bbb T$? –  Nov 04 '14 at 23:30
  • @Pedro could you link me to the other instance of my question? Also, thank you for you reply. I am new to group theory so it will take me some time to try and understand what you wrote. Is there a simpler, more intuitive explanation? –  Nov 04 '14 at 23:33
  • There may be no other reason except that the character group is considered in group theory and Pontryagin chose $\Bbb T$ to define duality. – Minimus Heximus Nov 04 '14 at 23:44
  • I think it's because a lot of the theory of locally compact groups and their representations is motivated by the groups $ \mathbb{R}, \mathbb{T} $ and Fourier theory on their functions spaces. I'm not familiar with the precise history, but I would guess, one tried to expand the theory of Fourier series/transform to more general groups. If you are familiar with those theories, you know that $ e^{2\pi i n x}, n \in \mathbb{Z} $ (for $ \mathbb{T} $) or $ e^{2\pi i \xi x}, \xi \in \mathbb{R} $ (for $ \mathbb{R} $) play an important role. But those functions are just characters to the circle group. – m.g. Nov 06 '14 at 08:57
  • @m.g. That makes sense, I'm just hoping for more precise motivating statements e.g a concrete property of $\Bbb T$ that suggests making this definition. –  Nov 06 '14 at 11:41
  • @Pedro It seems to me your analysis of the situation is not incorporating the topology that is present. – anon Nov 06 '14 at 22:11
  • @m.g., Exterior: indeed, the fact that nice functions decompose as a superposition of waves a la Fourier theory is a special case of a corollary to the Peter-Weyl theorem: irreducible characters form an orthonormal basis for the space of square-integrable class functions. One can determine the coefficients (if there are countably many irreps) by the usual method of taking the inner product $\langle f,\chi\rangle$. (There is a very nice question on the site here for this.) – anon Nov 06 '14 at 22:27
  • @whacka Yes, I only gave feedback from the algebraic point of view. – Pedro Nov 07 '14 at 00:50
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    @user153312 it may be a duplicate of my question here: https://math.stackexchange.com/q/124379/16490 – ziggurism Apr 04 '18 at 13:40

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These are precisely the unitary one-dimensional representations. Classifying all (complex) one-dimensional representations of a (locally compact) group is the obvious first step in classifying irreducible representations (the building blocks of all reps, in the semisimple situation).

There is another question on thise site asking why we focus on unitary representations, the answer likely being in its convenience: it allows us to generalize the averaging trick for finite groups and to construct the analogue of the group algebra.

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  • This seems like the sort of justification which can be given in retrospect, but was not the historical motivation. I don't know any representation theory yet, so I was hoping for something more obvious. Do you happen to know if this is the original motivation? Representation theory seems to share the categorical viewpoint of studying objects via maps. –  Nov 08 '14 at 10:51
  • @Exterior Your question says nothing about "historical" or "original" motivation, and to me this justification is obvious. By the way, what historical sources are you using to claim that this is not the historical or original motivation? – anon Nov 08 '14 at 17:21
  • I did not mean to assert it's not the historical motivation, I merely said it suspect it may not be; I could certainly be wrong! At any rate, perhaps I should just learn some representation theory to get a feel for this. –  Nov 09 '14 at 12:21
  • This answer https://math.stackexchange.com/questions/4027076/intuition-behind-pontryagin-duality?noredirect=1&lq=1 says pretty much the same thing, but gives another reason to focus on unitary representations: "The fundamental notion behind the theory of groups is the concept of symmetries and the best place to visualize symmetries is Euclidean space and, by extension, Hilbert's space. ... The group $\mathscr U(H)$, formed by all unitary operators on Hilbert's space is, according to this, the archetype of symmetry!" – D.R. May 05 '25 at 01:24