Modules for 2 x 2 matrices over commutative power-associative algebras

The aim of this paper is to describe the irreducible modules for the Jordan algebra of 2 x 2 matrices over an algebraically closed field of characteristic different from 2, 3 and 5 in the class of the commutative power-associative algebras. All irreducible non-unital modules, and irreducible unital modules up to dimension three are classified, namely we find seven non-parameterized and five families of parameterized modules of dimension three. For every k >= 2, an irreducible module of dimension 3k is also constructed.