The Grothendieck groups of m-cluster categories of type D_n

Abstract

In this paper, we study the Grothendieck groups of m-cluster categories of type D_n, denoted by ${\mathcal{C}}_{D_n}^m$, when m is odd, via the m-cluster tilting subcategories of ${\mathcal{C}}_{D_n}^m$. When m = 1, we generalize the work of Barot, Kussin and Lenzing for studying the Grothendieck groups of cluster categories of type D_n.

Keywords:

• Grothendieck group
• m-cluster category of type D_n
• m-cluster tilting subcategory

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 18E99
• 18D99
• 18E30

Acknowledgments

We would like to thank the referees for their helpful comments and suggestions.

Additional information

Funding

The first author was supported by the NSF of China (Grant No. 12301047). The second author was supported by the NSF of China (Grant No. 12201045) and R&D Program of Beijing Municipal Education Commission (Grant No. KM202211417002).