Non-uniqueness of group structure for affine algebraic groups We know that every abelian variety has a unique group structure, but in the affine case, is that every affine algebraic group has more than one (up to isomorphism) group structure?

Since a the affine variety underlying a non-trivial affine group always has non-identity automorphisms with no fixed points, you can always conjugate its group structure by one of them to get a different group structure.
Later. As for the more precise question, the answer is also no. Take an algebraically closed field. The only non-complete connected algebraic groups of dimension one are the multiplicative and the additive groups, and they are non-isomorphic as varieties. It follows that each of them is an algebraic group in exactly one way up to isomorphism.
