TITLE: Algebraic $n$-torus and topological $n$-torus
QUESTION [7 upvotes]: Working with the field $\mathbb C$, one can find two different objects called "torus":
- Algebraic $n$-Torus that is $(\mathbb C^\ast)^n$
- Topological $n$-Torus defined (for example) as the quotient group $\mathbb C^n/\Lambda$ where $\Lambda$ is a lattice in $\mathbb C^n$.
What is the relation between these two objects? (Certainly the algebraic torus contains the topological torus.)

REPLY [9 votes]: They're homotopy equivalent. 
The latter can be thought of as the maximal compact subgroup of the former; in that sense they stand in a relationship analogous to the relationship between $\text{GL}_n(\mathbb{C})$ (algebraic) and $\text{U}(n)$ (topological), and indeed the algebraic torus is the subgroup of diagonal matrices in the former and the topological torus is the subgroup of diagonal matrices in the latter. 
They have the same representation theory in a suitable sense (which is also true for $\text{GL}_n(\mathbb{C})$ and $\text{U}(n)$). 
The Lie algebra of the former is the complexification of the Lie algebra of the latter.