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

Abstract

The aim of this paper is to describe the irreducible modules for the Jordan algebra of 2 × 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 $k\ge 2$, an irreducible module of dimension 3k is also constructed.

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 17A05
• Secondary: 17A60
• 17B10

KEYWORDS:

• Irreducible modules
• power-associative modules

Acknowledgments

The authors would like to thank the anonymous referee for his/her helpful comments that improved the quality of the manuscript. E. Quintero Vanegas would like to thank the Universidade Federal da Bahia (UFBA) where part of the results were obtained.

Additional information

Funding

I. Hernández was supported by grant CONAHCYT A1-S-45886. This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001.