  We study electroweakly interacting particles in rotating matter. The
existence of the electric current along the axis of the matter rotation is
predicted in this system. This new galvano-rotational effect is caused by the
parity violating interaction between massless charged particles in the rotating
matter. We start with the exact solution of the Dirac equation for a fermion
involved in the electroweak interaction in the rotating frame. This equation
includes the noninertial effects. Then, using the obtained solution, we derive
the induced electric current which turns out to flow along the rotation axis.
We study the possibility of the appearance of the galvano-rotational effect in
dense matter of compact astrophysical objects. The particular example of
neutron and hypothetical quark stars is discussed. It is shown that, using this
effect, one can expect the generation of toroidal magnetic fields comparable
with poloidal ones in old millisecond pulsars. We also briefly discuss the
generation of the magnetic helicity in these stars. Finally we analyze the
possibility to apply the galvano-rotational effect for the description of the
asymmetric neutrino emission from a neutron star to explain pulsars kicks.
