Matrix quantum groups as matrix product operator representations of Lie groups

TL;DR

This paper shows how matrix quantum groups can generate matrix product operator representations of Lie groups, revealing new insights into symmetries in quantum spin chains and their algebraic structures.

Contribution

It introduces explicit matrix product operator representations of the quantum group $SL_q(2)$ for the Lie group $SL(2)$, connecting quantum groups with tensor network methods.

Findings

Matrix product operators form a closed, non-injective set under multiplication.

Fusion tensors and recoupling coefficients satisfy pentagon relations.

The framework aligns with bimodule category descriptions of quantum groups.

Abstract

We demonstrate that the matrix quantum group $SL_q(2)$ gives rise to nontrivial matrix product operator representations of the Lie group $SL(2)$, providing an explicit characterization of the nontrivial global $SU(2)$ symmetry of the XXZ model with periodic boundary conditions. The matrix product operators are non-injective and their set is closed under multiplication. This allows to calculate the fusion tensors acting on the virtual or quantum degrees of freedom and to obtain the recoupling coefficients, which satisfy a type of pentagon relation. We argue that the combination of this data with the well known $q$-deformed Clebsch-Gordan coefficients and 6j-symbols is consistent with a description of this quantum group in terms of bimodule categories.