  We consider transport properties of the hypernuclear matter in neutron star
cores. In particular, we calculate the thermal conductivity, the shear
viscosity, and the momentum transfer rates for np$\Sigma^{-}\Lambda e\mu$
composition of dense matter in $\beta$--equilibrium for baryon number densities
in the range $0.1-1$~fm$^{-3}$. The calculations are based on baryon
interactions treated within the framework of the non-relativistic
Brueckner-Hartree-Fock theory. Bare nucleon-nucleon (NN) interactions are
described by the Argonne v18 phenomenological potential supplemented with the
Urbana IX three-nucleon force. Nucleon-hyperon (NY) and hyperon-hyperon (YY)
interactions are based on the \new{NSC97e and NSC97a models} of the Nijmegen
group. We find that the baryon contribution to transport coefficients is
dominated by the neutron one as in the case of neutron star cores containing
only nucleons. In particular, we find that neutrons dominate the total thermal
conductivity over the whole range of densities explored and that, due to the
onset of $\Sigma^-$ which leads to the deleptonization of the neutron star
core, they dominate also the shear viscosity in the high density region, in
contrast with the pure nucleonic case where the lepton contribution is always
the dominant one.
