Title: The Structure of the Inverse System of Level $K$-Algebras,
Abstract: Macaulay's inverse system is an effective method to construct Artinian
K-algebras with additional properties like, Gorenstein, level, more generally
with any socle type. Recently, Elias and Rossi gave the structure of the
inverse system of $d$-dimensional Gorenstein K-algebras for any $d>0$. In this
paper we extend their result by establishing a one-to-one correspondence
between $d$-dimensional level K-algebras and certain submodules of the divided
power ring. We give several examples to illustrate our result.