Many of the original quantum invariants are based on representations of the quantum group U_q(sl_2) or, more or less equivalently, corepresentations or its Hopf dual O_q(SL_2). Whereas U_q(sl_2) is a deformation of the enveloping algebra of the Lie algebra sl_2, its dual O_q(SL_2) is a deformation of the algebra of functions on the Lie group SL_2. It turns out that, when the quantum parameter q is a root of unity, O_q(SL_2) also has interesting representations. I will discuss several features of the corresponding representation category, and in particular how it is closely connected to the geometry of the Lie group SL_2. This is joint work with Matt Harper.