Let $G$ be a connected algebraic group. Is it true that every $G$bundle on ${\mathbb A}^n$ is trivial? What is the reference? I am actually only interested in the case $n=2$.
In characteristic $p$ you can make an easy counterexample with $n=1$, right? An exact sequence of commutative algebraic groups $0\to E\to X\to \mathbb A\to 0$ with $E$ an elliptic curve. 


Over an algebraically closed field, for $G$ connected and reductive, every principal $G$bundle on ${\Bbb A}^n$ is trivial, also by a theorem of Raghunathan: "Principal bundles on affine space", in C. P. Ramanujam—a tribute, pp. 187–206, Tata Inst. Fund. Res. Studies in Math. 8 (1978). (Unfortunately I can't find this reference free online.) 


This is true for $G=GL(r)$, as shown by Quillen and Suslin. For arbitrary $G$ there are counterexamples. Quite surprisingly, even $G$bundles over $\textrm{Spec }k$ may not be trivial. See the paper M. S. Raghunathan "Principal bundles on affine space and bundles on the projective line", Mathematische Annalen Volume 285, Number 2, 309332 


$G$
is to be an affine algebraic group. The reductive case has been most studied, so it's also important to indicate if more generality is wanted. – Jim Humphreys Nov 15 '10 at 18:39