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 $(\mathbb{G}\_a)^\ell$.


Thanks for answering my question!



Structure of abelian connected complex linear algebraic groups?
A reference:
Lie Groups and Algebraic Groups (Springer Series in Soviet Mathematics) by Arkadij L. Onishchik, Ernest B. Vinberg and Dimitry A. Leites (new printing will appear on Amazon.com on Jul 31, 2012). See Ch. 3, Section 2.5, page 116, Corollary of Theorem 8: Any irreducible commutative (linear) algebraic group is the direct product of a torus and a vector group.


I highly recommend this book for beginners, and not only for beginners. It is mainly a collection of problems (with hints of proofs at the end of each section). 


