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?



What is the name of the Hopf algebra whose comodules are the "positive" highest weight modules of $U_{q}(sl(2))$?
It usually goes by "quantized function algebra" of $SL\_2$, or "regular function algebra" of $SL\_q(2)$, of course with various minor variations. In Kassel's book it's denoted by $SL\_q(2)$ (see Section VII.5). In Chari-Pressley book it's $\mathcal{F}\_q(SL\_2(\mathbb{C}))$ (Chapter 13; though they restrict to positive $q$). Other common notation would be $\mathcal{O}\_q(SL\_2)$, $\mathcal{O}(SL\_q(2))$, $\mathbb{C}\_q[SL\_2]$, $\mathbb{C}[SL\_q(2)]$,...


By the way, usually you either want to specialize $q$ to some (not-root-of-unity) complex number already and work over $\mathbb{C}$, or take $\mathbb{Q}(q)$, etc, instead of $\mathbb{C}(q)$, although I don't think it's a serious issue. If you work over formal power series, it's quite common to take $\mathbb{C}[[h]]$ though.


