What is the group of a quantum group? From $S{L}_q(2)$ to MPO representations of $SL(2)$

Abstract

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 $S{L}_q(2)$ 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. On special loci it is non-semisimple, and yields reducible but indecomposable tensors with Jordan blocks. Differentiating $\mathcal{D}(g)$ at the identity produces the $\mathfrak{sl}(2)$ generators as $d$-body operators: the MPO algebra supplies the exponential map for quantum group algebras.

5+23 pages, v2: major revision