  We find that the presence of strong non-Abelian conserved quantities can lead
to highly entangled stationary states even for unital quantum channels. We
derive exact expressions for the bipartite logarithmic negativity, R\'enyi
negativities, and operator space entanglement for stationary states restricted
to one symmetric subspace, with focus on the trivial subspace. We prove that
these apply to open quantum evolutions whose commutants, characterizing all
strongly conserved quantities, correspond to either the universal enveloping
algebra of a Lie algebra or to the Read-Saleur commutants. The latter provides
an example of quantum fragmentation, whose dimension is exponentially large in
system size. We find a general upper bound for all these quantities given by
the logarithm of the dimension of the commutant on the smaller bipartition of
the chain. As Abelian examples, we show that strong U($1$) symmetries and
classical fragmentation lead to separable stationary states in any symmetric
subspace. In contrast, for non-Abelian SU$(N)$ symmetries, both logarithmic and
R\'enyi negativities scale logarithmically with system size. Finally, we prove
that while R\'enyi negativities with $n>2$ scale logarithmically with system
size, the logarithmic negativity (as well as generalized R\'enyi negativities
with $n<2$) exhibits a volume law scaling for the Read-Saleur commutants. Our
derivations rely on the commutant possessing a Hopf algebra structure in the
limit of infinitely large systems, and hence also apply to finite groups and
quantum groups.
