  The thermoelectric transport through a ring threaded by an Aharonov-Bohm
flux, with a molecular bridge on one of its arms, is analyzed. The transport
electrons also interact with the vibrational excitations of that molecule. This
nano-system is connected to three terminals: two are electronic reservoirs,
which supply the transport electrons, and the third is the phonon bath which
thermalizes the molecular vibrations. Expressions for the transport
coefficients, relating all charge and heat currents to the temperature and
chemical potential differences between the terminals, are derived to second
order in the electron-vibration coupling. At linear response, all these
coefficients obey the full Onsager-Casimir relations. When the phonon bath is
held at a temperature different from those of the electronic reservoirs, a heat
current exchanged between the molecular vibrations and the transport electrons
can be converted into electric and/or heat electronic currents. The related
transport coefficients, which exist only due to the electron-vibration
coupling, change sign under the interchange between the electronic terminals
and the sign change of the magnetic flux. It is also demonstrated that the
Aharonov-Bohm flux can enhance this type of conversion.
