We explain the physical origin of a curious property of algebras
$\mathcal{A}_\mathfrak{q}$ which encode the rotation-equivariant fusion ring of
half-BPS line defects in four-dimensional $\mathcal{N}=2$ supersymmetric
quantum field theories. These algebras are a quantization of the algebras of
holomorphic functions on the three-dimensional Coulomb branch of the SQFTs,
with deformation parameter $\log \mathfrak{q}$. They are known to acquire a
large center, canonically isomorphic to the undeformed algebra, whenever
$\mathfrak{q}$ is a root of unity. We give a physical explanation of this fact.
We also generalize the construction to characterize the action of this center
in the $\mathcal{A}_\mathfrak{q}$-modules associated to three-dimensional
$\mathcal{N}=2$ boundary conditions. Finally, we use dualities to relate this
construction to a construction in the Kapustin-Witten twist of four-dimensional
$\mathcal{N}=4$ gauge theory. These considerations give simple physical
explanations of certain properties of quantized skein algebras and cluster
varieties, and quantum groups, when the deformation parameter is a root of
unity.