On representation rings in the context of monoidal categories

TL;DR

This paper develops a unified framework for Green rings in monoidal categories, characterizes representation and derived rings of specific Hopf algebras, and provides explicit generators and relations using combinatorial methods.

Contribution

It introduces Green rings for monoidal categories and fully characterizes the representation and derived rings of certain Nakayama truncated Hopf algebras.

Findings

Complete generators and relations for the representation ring of $KZ_{n}/J^{d}$.

Generators and relations for the derived ring of $KZ_{n}/J^{2}$.

Polynomial descriptions of the representation and derived rings.

Abstract

In general, representation rings are well-known as Green rings from module categories of Hopf algebras. In this paper, we study 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 in the unified view-point. Firstly, as analogue of representation rings of Hopf algebras, we set up the so-called Green rings of monoidal categories, and then list some such categories including module categories, complex categories, homotopy categories, derived categories and (derived) shift categories, etc. and the relationship among their corresponding Green rings. The main part of this paper is to characterize representation rings and derived rings of a class of inite 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 Pascal triangle. For the derived ring $dr(KZ_{n}/J^{2})$(i.e., $d=2$), we determine its generators and give the relations of generators. In these two aspects, the polynomial characterizations of the representation ring and the derived ring are both given.