  In this article we use a parametrized version of the FRT construction to
construct two new coquasitriangular Hopf algebras. The first one,
$\widehat{SL_q(2)}$, is a quantization of the coordinate ring on affine
$SL(2)$. We show that there is a duality relation between this object and the
more well-known $U_q(\widehat{\mathfrak{sl}_2})$. We then build certain
irreducible comodules of this Hopf algebra and prove an irreducibility
criterion for their tensor product in the spirit of Chari and Pressley.
  The second object is built from a solution of the parametrized Yang-Baxter
equation with parameter group $GL(2, \mathbb{C}) \times GL (1,\mathbb{C})$.
This solution doesn't come from any known quantum group, though it is related
to both solutions coming from $U_{\pm i}(\widehat{\mathfrak{sl}_2})$ and
$U_q(\widehat{\mathfrak{gl}(1|1)})$. We then study certain irreducible
comodules of this newly built object.
