| Summary: | We perform a series of two-dimensional magnetohydrodynamic simulations of supernova cores. Since the distributions of the angular momentum and the magnetic fields of strongly magnetized stars are quite uncertain, we systematically change the combinations of the strength of the angular momentum, the rotations law, the degree of differential rotation, and the profiles of the magnetic fields to construct the initial conditions. By so doing, we estimate how the rotation-induced anisotropic neutrino heating are affected by the strong magnetic fields through parity-violating effects and first investigate how the north-south asymmetry of the neutrino heating in a strongly magnetized supernova core could be. As for the microphysics, we employ a realistic equation of state based on the relativistic mean field theory and take into account electron captures and the neutrino transport via the neutrino leakage scheme. With these computations, we find that the parity-violating corrections reduce $ \lesssim 0.5 %$ of the neutrino heating rate than that without the magnetic fields in the vicinity of the north pole of a star, on the other hand, enhance about $ \lesssim 0.5 %$ in the vicinity of the south pole. If the global asymmetry of the neutrino heating in the both of the poles develops in the later phases, the newly born neutron star might be kicked toward the north pole in the subsequent time.
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