  Anomalous finite-temperature transport has recently been observed in
numerical studies of various integrable models in one dimension; these models
share the feature of being invariant under a continuous non-abelian global
symmetry. This work offers a comprehensive group-theoretic account of this
elusive phenomenon. For an integrable quantum model invariant under a global
non-abelian simple Lie group $G$, we find that finite-temperature transport of
Noether charges associated with symmetry $G$ in thermal states that are
invariant under $G$ is universally superdiffusive and characterized by
dynamical exponent $z = 3/2$. This conclusion holds regardless of the Lie
algebra symmetry, local degrees of freedom (on-site representations), Lorentz
invariance, or particular realization of microscopic interactions: we
accordingly dub it as superuniversal. The anomalous transport behavior is
attributed to long-lived giant quasiparticles dressed by thermal fluctuations.
We provide an algebraic viewpoint on the corresponding dressing transformation
and elucidate formal connections to fusion identities amongst the quantum-group
characters. We identify giant quasiparticles with nonlinear soliton modes of
classical field theories that describe low-energy excitations above
ferromagnetic vacua. Our analysis of these field theories also provides a
complete classification of the low-energy (i.e., Goldstone-mode) spectra of
quantum isotropic ferromagnetic chains.
