  In this article, we study the $k$-Lefschetz properties for non-Artinian
algebras, proving that several known results in the Artinian case can be
generalized in this setting. Moreover, we describe how to characterize the
graded algebras having the $k$-Lefschetz properties using sectional matrices.
We then apply the obtained results to the study of the Jacobian algebra of
hyperplane arrangements, with particular attention to the class of free
arrangements.
