The central simple modules of Artinian Gorenstein algebras

Let AA be a standard graded Artinian KK-algebra, with char K=0K=0. We prove the following. 1.AA has the Weak Lefschetz Property (resp. Strong Lefschetz Property) if and only if Gr(z)(A) has the Weak Lefschetz Property (resp. Strong Lefschetz Property) for some linear form zz of AA.2.If AA is Gorenstein, then AA has the Strong Lefschetz Property if and only if there exists a linear form zz of AA such that all central simple modules of (A,z)(A,z) have the Strong Lefschetz Property. As an application of these theorems, we give some new classes of Artinian complete intersections with the Strong Lefschetz Property.