Contents Online
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: https://dx.doi.org/10.4310/AJM.2018.v22.n1.a2
Authors
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
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