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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?

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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.

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  • $\begingroup$ Eek, does one really write $\mathrm{SL}_2$ but $\mathrm{SL}_q(2)$? $\endgroup$ – LSpice Jul 9 '17 at 16:27

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