  We study Hom-bialgebras and objects admitting coactions by Hom-bialgebras. In
particular, we construct a Hom-bialgebra M representing the functor of
2x2-matrices on Hom-associative algebras. Then we construct a Hom-algebra
analogue of the affine plane and show that it is a comodule Hom-algebra over M
in a suitable sense. It is also shown that the enveloping Hom-associative
algebra of a Hom-Lie algebra is naturally a Hom-bialgebra.
