  We study the lowest order conservation laws in one-dimensional (1D)
integrable quantum many-body models (IQM) as the Heisenberg spin 1/2 chain, the
Hubbard and t-J model. We show that the energy current is closely related to
the first conservation law in these models and therefore the thermal transport
coefficients are anomalous. Using an inequality on the time decay of current
correlations we show how the existence of conserved quantities implies a finite
charge stiffness (weight of the zero frequency component of the conductivity)
and so ideal conductivity at finite temperatures.
