  We investigate properties of strange quark matter in the framework of SU(3)
Nambu--Jona-Lasinio(NJL) model with vector interaction under strong magnetic
fields. The effects of vector-isoscalar and vector-isovector interaction on the
equation of state of strange quark matter are investigated, and it is found
that the equation of state is not sensitive to the vector-isovector
interaction, however, a repulsive interaction in the vector-isoscalar channel
gives a stiffer equation of state for cold dense quark matter. In the presence
of magnetic field, gluons will be magnetized via quark loops, and the
contribution from magnetized gluons to the equation of state is also estimated.
The sound velocity square is a quantity to measure the hardness or softness of
dense quark matter, and in the NJL model without vector interaction at zero
magnetic field the sound velocity square is always less than 1/3. It is found
that a repulsive vector-isoscalar interaction and a positive pressure
contribution from magnetized gluons can enhance the sound velocity square,
which can even reach 1. To construct quark magnetars under strong magnetic
fields, we consider anisotropic pressures and use a density-dependent magnetic
field profile to mimic the magnetic field distribution in a quark star. We also
analyze the parameter region for the magnitude of vector-isoscalar interaction
and the contribution from magnetized gluons in order to produce 2 solar mass
quark magnetars.
