Asian Journal of Mathematics

Volume 22 (2018)

Number 1

Representation and derived representation rings of Nakayama truncated algebras and a viewpoint under monoidal categories

Pages: 41 – 74

DOI: http://dx.doi.org/10.4310/AJM.2018.v22.n1.a2

Authors

Min Huang (Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China)

Fang Li (Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang, China)

Yichao Yang (Département de Mathématiques, Université de Sherbrooke, Québec, Canada)

Abstract

The main aim of this study is to characterize representation rings and derived representation rings of a class of finite dimensional Hopf algebras constructed from the Nakayama truncated algebras $KZ_n / J^d$ with certain constraints. For the representation ring $r (KZ_n / J^d)$, we completely determine its generators and the relations of generators via the method of the Pascal triangle. For the derived representation ring $dr (KZ_n / J^2)$ (i.e., $d = 2$), we determine its generators and give the relations of generators. For these two aspects, the polynomial characterizations of the representation ring and the derived representation rings are both given.

Representation rings are well-known as Green rings from module categories over Hopf algebras. We have studied Green rings in the context of monoidal categories such that representations of Hopf algebras can be investigated through Green rings of various levels from module categories to derived categories from a unified viewpoint. Firstly, as analogues of representation rings of Hopf algebras, we set up so-called Green rings of monoidal categories, and then we list some such categories including module, complex, homotopy, derived and (derived) shift categories, and the relationship among their corresponding Green rings.

Keywords

representation ring, derived representation ring, shift ring, Nakayama truncated algebra, Pascal triangle, monoidal category

2010 Mathematics Subject Classification

16T05, 18D10, 19A22

Full Text (PDF format)

This project was supported by the National Natural Science Foundation of China (No. 11671350 and No.11571173).

Received 30 June 2015

Accepted 12 October 2016

Published 10 May 2018