Representation dimension of generalized path algebras of cyclic quivers

ABSTRACT

Let Λ be a non semisimple symmetric artin algebra over a commutative artinian ring R and let repdim(Λ) be its representation dimension. Assume also that 𝒬 is a finite quiver whose connected components are amalgamations of single cycles possibly with multiple arrows, and that ℐ is an admissible ideal in the generalized path algebra Λ𝒬. We prove that $repdim\left(\frac{\Lambda \mathcal{𝒬}}{\mathcal{ℐ}}\right)\le$ repdim(Λ)+1.

KEYWORDS:

• Quiver representations
• representation dimension
• representation type of an algebra

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16G20
• 16G60
• 16G10
• 16G99

Acknowledgments

The author would like to thank the anonymous referee for his/her useful comments on the paper that improved the exposition.