The finite-dimensional representations over $\mathbb C(q)$ of $U_q(\mathfrak{sl}(2))$ are all highest weight. There are two irreducible modules of each dimension. In one, the highest weight vector $v$ transforms under the Cartan element $K$ as $K v = + q^{\dim V - 1}v$. In the other, $Kv = -q^{\dim V - 1}v$. In particular, $U_q(sl(2))$ admits a one-dimensional nontrivial module; tensoring with it interchanges the two $\dim V$-dimensional modules. The subcategory of $\mathrm{mod}_{\mathrm{fd}}(U_q\mathfrak{sl}(2))$ generated by the highest weight modules with positive sign is a monoidal subcategory $\mathrm{mod}^+_{\mathrm{fd}}(U_q\mathfrak{sl}(2))$.

Standard Tannakian reconstruction theory guarantees the existences of a coquasitriangular Hopf algebra $H$ such that $\mathrm{mod}^+_{\mathrm{fd}}(U_q\mathfrak{sl}(2)) \simeq \mathrm{comod}_{\mathrm{fd}}(H)$ as braided categories. (Use the fiber functor that is the restriction to $\mathrm{mod}^+_{\mathrm{fd}}$ of the forgetful functor on $\mathrm{mod}$.) Such $H$ is some version of "quantum SL(2)", but I do not know which version. $H$ is a sub Hopf algebra of the restricted dual to $U_q(\mathfrak{sl}(2))$. The latter is $H[x]/(x^2-1)$.

Does this Hopf algebra $H$ have a standard name?