The commutator algebra of a nilpotent matrix and an application to the theory of commutative Artinian algebras

TL;DR

This paper investigates the structure of the commutator algebra of a nilpotent matrix, characterizes its simple modules, and applies these findings to demonstrate the strong Lefschetz property in certain Artinian complete intersections.

Contribution

It provides a detailed analysis of the commutator algebra of nilpotent matrices and applies these insights to algebraic geometry, specifically to Artinian complete intersections.

Findings

Characterization of simple modules of the commutator algebra

Properties of the commutator algebra of nilpotent matrices

Proof of the strong Lefschetz property for certain Artinian complete intersections

Abstract

We show a number of properties of the commutator algebra of a nilpotent matrix over a field. In particular we determine the simple modules of the commutator algebra. Then the results are applied to prove that certain Artinian complete intersections have the strong Lefscehtz property.