Representations of power-associative train algebras of rank 4

Abstract

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.

Keywords:

• Irreducible representation
• power-associative algebra
• rank
• train algebra

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 17A30
• Secondary: 17A05
• 15A04
• 15A78
• 17A60

Acknowledgments

The authors would like to give special thanks to Professor Alexandre Grishkov and to the referee for suggesting several improvements.

Additional information

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) – Finance Code 001.