The quiver of an affine monoid

Abstract

If R is a finite commutative ring, then the affine monoid of R is the monoid of all affine mappings $x\mapsto ax+b$ on R. Alternatively, it is the semidirect product of the multiplicative monoid of R with the additive group of R. In this article, we compute the Gabriel quiver of the complex algebra of the affine monoid of any finite commutative ring.

Keywords:

• Affine monoids
• monoid representation theory
• quivers of algebras
• simple modules

2020 Mathematics Subject Classification:

• 20M30
• 20M25

Acknowledgments

This work was performed while the first author was visiting the City College of New York. He thanks the College for its warm hospitality. We are grateful to the anonymous referee for their careful reading of the paper.

Additional information

Funding

The first author was partially supported by CMUP, which is financed by national funds through FCT – Fundação para a Ciência e a Tecnologia, I.P., under the project UIDB/00144/2020. The first author also acknowledges FCT support through a contract based on the “Lei do Emprego Científico” (DL 57/2016). The second author was supported in part by a PSC-CUNY grant.