Representations Of Power-Associative Train Algebras Of Rank 4

This paper is devoted to the study of four-dimensional commutative power-associative non-Jordan train algebras. The existence of a unique orthogonal idempotent element in such kind of algebras enables us to define a non-zero algebra homomorphism over their ground field, which gives rise to train algebras of rank 4. We describe the irreducible representations over an algebraically closed field of characteristic prime to 30 by means of multilinear identities which define it. Furthermore, we provide sufficient conditions to guarantee whenever a representation is decomposable.