What is the group of a quantum group? From $SL_q(2)$ to MPO representations of $SL(2)$

Understand

Generalized symmetries realized by matrix product operators (MPOs) have so far been tied to discrete data leaving continuous symmetries outside the framework.

• Quantum groups fill this gap: starting from the RTT relations of $SL_q(2)$ at an odd root of unity $q^d=1$, we construct a three-parameter family of periodic-boundary MPOs of bond dimension $d^2$ that multiply according to the group law of $SL(2)$.
• This gives Lusztig's quantum Frobenius map a concrete operator form, and endows the XXZ chain on the ring at $\Delta=(q+q^{-1})/2$ with an intrinsically non-local $SU(2)$ symmetry.
• Fusion of the local tensors is generically semisimple.