  We explore the Kardar-Parisi-Zhang (KPZ) scaling in the one-dimensional
Hubbard model, which exhibits global $SU_c(2)\otimes SU_s(2)$ symmetry at
half-filling, for the pseudo-charge and the total spin. We analyze dynamical
scaling properties of high temperature charge and spin correlations and
transport. At half-filling, we observe a clear KPZ scaling in both charge and
spin sectors. Away from half-filling, the $SU_c(2)$ charge symmetry is reduced
to $U_c(1)$, while the $SU_s(2)$ symmetry for the total spin is retained.
Consequently, transport in the charge sector becomes ballistic, while KPZ
scaling is preserved in the spin sector. These findings confirm the link
between non-abelian symmetries and KPZ scaling in the presence of
integrability. We study two settings of the model: one involving a quench from
a bi-partitioned state asymptotically close to the $T=\infty$ equilibrium state
of the system, and another where the system is coupled to two markovian
reservoirs at the two edges of the chain.
