What qualities of an algebraic torus make it like a torus?

Over $\mathbb{C}$ the algebraic tori have the form $(\mathbb{C}^{\times})^n$, which has maximal compact subgroup an ordinary torus $T^n$. The inclusion of the maximal compact subgroup is a homotopy equivalence, and the two also have the same finite-dimensional representation theory (where for the algebraic torus we consider algebraic representations only). One can think of $(\mathbb{C}^{\times})^n$ as a complexification of $T^n$. The general case is by analogy with this one.

A major reason to work at this level of generality is to generalize the notion of a maximal torus from compact Lie groups to reductive algebraic groups. This can be done although I'm not the person to ask about how it works in detail.

This was a comment on Qiaochu’s answer, but it got too long so I thought I’d made it into an answer.

As Qiaochu said, one can define tori in several ways, but from a representation theoretic standpoint perhaps the most natural definition is the following:

Definition: A compact torus is a compact Lie group $\mathcal{T}$ isomorphic to $(S^1)^n$ for some $n$.

The real usage of such objects is in the theory of compact Lie groups where for a compact Lie group $\mathcal{G}$ one has a unique conjugacy class of maximal compact tori (i.e. there exists a compact torus $\mathcal{T}\subseteq\mathcal{G}$ which is maximal amongst compact tori contained in $\mathcal{G}$ and that such a maximal compact torus is unique up to conjugacy). The reason they are so useful is that representation theory of $\mathcal{G}$ is encapsulated entirely in terms of $\mathcal{T}$.

To make this precise, it’s useful to note denote by $\mathrm{Rep}(\mathcal{G})$ the representation ring of a compact Lie group $\mathcal{G}$. We then have the following:

Theorem (Highest weight theory, summarized): Let $\mathcal{G}$ be a connected compact Lie group and $\mathcal{T}$ a maximal torus. Then, the restriction map map $$\mathrm{Rep}(\mathcal{G})\to \mathrm{Rep}(\mathcal{T})$$ is an isomorphism of $\mathrm{Rep}(\mathcal{G})$ onto the subring $\mathrm{Rep}(\mathcal{T})^W$ of Weyl group invariant elements of $\mathrm{Rep}(\mathcal{T})$.

A very similar thing happens for tori in a reductive algebraic group over $k$ an algebraically closed field (one can extend this to the non-algebraically closed situation, but it’s not worth the effort here).

Definition: Let $k$ be a field. Then, an algebraic torus over $k$ is an algebraic group $T$ over $k$ such that $T_{\overline{k}}$ is isomorphic to $\mathbb{G}_{m,\overline{k}}^n$ for some $n$.

Again, it’s true that every (connected) reductive group $G$ over a field $k$ (reductive means, at least in characteristic $0$, that its category of algebraic representations is semisimple) has a maximal torus (defined in the same way) which is unique up to conjugacy if $k$ is algebraically closed. One also has a highest weight theory:

Theorem (Highest weight theory, summarized, [2, §22.b]): Let $k$ be an algebraically closed field and $G$ a (connected) reductive group over $k$. Then, the restriction map

$$\mathrm{Rep}(G)\to\mathrm{Rep}(T)$$

is an isomorphism of $\mathrm{Rep}(G)$ onto the subring $\mathrm{Rep}(T)^W$ of Weyl group invariant elements of $\mathrm{Rep}(T)$.

Thus, one sees that there is a strong analogy between compact tori and algebraic tori from the perspective of representation theory. But, in fact, one can make the connection more literal.

Namely, we have the following beautful theorem:

Theorem (Chevalley—Tanaka, [1, Theorem D.2.4]): The association $G\mapsto G(\mathbb{R})$ is an equivalence of categories $$\left\{\begin{matrix}\text{Connected reductive anisotropic groups}\\ \text{over }\mathbb{R}\end{matrix}\right\}\to\left\{\begin{matrix}\text{Connected compact Lie groups}\\ \text{over }\mathbb{R}\end{matrix}\right\}$$ Moreover, a connected reductive group $G$ is anisotropic if and only if $G(\mathbb{R})$ is compact.

Let us say briefly what the terminology `anisotropic’ means. Namely, let us call a torus $T$ over a field $k$ split if $T\cong \mathbb{G}_{m,k}^n$ (i.e. it’s isomorphic to powers of the multiplicative group not just over $\overline{k}$, but over $k$). A reductive group $G$ over $k$ is then called anisotropic it it contains no positive dimensional split tori. If $k=\mathbb{R}$ then this is equivalent to the claim that $G(\mathbb{R})$ is compact.

Let us obtain the following corollary of this result:

Corollary: Let $T$ be a reductive group over $\mathbb{R}$. Then, $T$ is an anisotropic torus if and only if $T(\mathbb{R})$ is a compact torus. Moreover, for an anisotropic (connected) reductive group $G$ over $\mathbb{R}$ the association $T\mapsto T(\mathbb{R})$ forms a bijection $$\left\{\begin{matrix}\text{Maximal algebraic tori}\\ T\subseteq G\end{matrix}\right\}\to \left\{\begin{matrix}\text{Maximal compact tori}\\ \mathcal{T}\subseteq G(\mathbb{R})\end{matrix}\right\}$$

Proof: It suffices to prove the first statement, since the second then follows immediately from the Tannaka—Chevalley theorem. Suppose first that $T$ is an anisotropic torus. Then, by the Tannaka—Chevalley theorem we have that $T(\mathbb{R})$ is a compact connected Lie group. But, since $T$ is abelian so is $T(\mathbb{R})$. Then, by standard Lie group theory this implies that $T$ is a compact torus. Conversely, suppose that $T(\mathbb{R})$ is a compact torus. By Tanaka—Chevalley we have that $T$ is a connected reductive anisotropic group over $\mathbb{R}$. We need to show that $T$ is abelian, since then $T$ is automatically a torus (e.g. see [2. Proposition 21.7]). But, since $T(\mathbb{R})$ is dense in $T$ (e.g. see [2, Theorem 17.93]) we see that the map $T\times T\to T$ given by $(t_1,t_2)\mapsto t_1t_2t_1^{-1}t_2^{-1}$ is trivial on a dense subset, so is trivial (since all schemes involved are affine). $\blacksquare$

From this, we see that compact tori can, in a very literal sense, be thought of as a special case of algebraic tori over $\mathbb{R}$.

References:

[1] Conrad, B., 2014. Reductive group schemes. Autour des schémas en groupes, 1, pp.93-444.

[2] Milne, J.S., 2017. Algebraic groups: The theory of group schemes of finite type over a field (Vol. 170). Cambridge University Press.