On the composition subalgebras of Bridgeland’s Hall algebras

ABSTRACT

Let A be a finite dimensional hereditary algebra over a finite field k and 𝒫 the category consisting of finite dimensional projective (left) A-modules. In this paper, we consider the composition subalgebra of Bridgeland’s Hall algebra of the category 𝒞_m(𝒫) of m-cyclic complexes for any positive integer m≥2 and determine its generating relations.

KEYWORDS:

• Bridgeland’s Hall algebra
• composition subalgebra
• cyclic complexes
• generating relations

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 16G10
• 16T10
• 17B37

Acknowledgments

The author would like to thank the anonymous referee for many corrections and suggestions. He is also grateful to Professor Bangming Deng for many comments, suggestions and discussions.

Notes

¹For each fixed A-module M, we fix a minimal projective resolution (2.7) of M using notations P_M and Ω_M throughout the paper.