# Structure of abelian connected complex linear algebraic groups?

Let $G$ be an abelian connected complex linear algebraic group.

Is it true that $G$ is isomorphic to $(\mathbb{G}_m)^k\times (\mathbb{G}_a)^\ell$, where the nonnegative exponents denote repeated direct products of linear algebraic groups, $\mathbb{G}_m=\mathbb{C}^*$, and $\mathbb{G}_a=(\mathbb{C},+)$?

This seems like a very elementary statement, but I cannot find a good reference for it and it is the foundation for a paper that I'm writing. I've written down a proof using results from the 'Linear Algebraic Groups' books of Humphreys and Springer, but there must be a good citation (by someone who knows more about linear algebraic groups than I!). I can only speculate that finding a reference is difficult because these books wish to include statements that work over other fields, and the answer is much more complicated in these cases. (It would make a good exercise, though.)

My proof goes like this: use the Jordan decomposition so that $G=G_s\times G_u$; since $G$ is abelian and connected, this is the direct product of connected linear algebraic groups. Then $G_s$ is a 'd-group', so (since connected) is a torus. $G_u$ is 'elementary unipotent'. Over $\mathbb{C}$, these are vector groups isomorphic to $(\mathbb{G}_a)^\ell$.

• I voted to close, since this seems like homework for me. I would ask on math.exchange, if I have questions, which I assume are easy if you know the right tools. To the topic, I recall that every connected locally compact abelian group is isomorphic to a bunch of copies of $\mathbb{R}$ and circles. There is a book of Deitmar and Echterhoff "Principles of Harmonic analysis", which adresses this. – Marc Palm Jul 20 '12 at 15:52