The underlying variety of a linear elgebraic group (say, over an algebraically closed field) is affine, so doesn't have nontrivial (infinitesimal) deformations. I'm curious to know whether it's possible to deform the group structure on a fixed variety (that admits at least a structure of an algebraic group).
Edit: I should have added, though when I wrote the question I mistakenly didn't expect reductivity made any difference, that I was mostly interested in reductive groups (in which case the comment of user54268 shows my question was not that naive after all..).