Cores of graded algebras with triangular decomposition

TL;DR

This paper investigates the core of self-injective graded algebras with triangular decomposition, showing it is cellular and can be used to construct a quasi-hereditary algebra that captures essential representation-theoretic information.

Contribution

It introduces a canonical method to derive a quasi-hereditary algebra from the core of such graded algebras, linking cellularity and highest weight structures.

Findings

The core is cellular and encodes key representation-theoretic data.

A canonical construction of a highest weight cover from the core is provided.

The approach applies to various important algebraic examples.

Abstract

We consider self-injective finite-dimensional graded algebras admitting a triangular decomposition. In a preceding paper, we have shown that the graded module category of such an algebra is a highest weight category and has tilting objects in the sense of Ringel. In this paper we focus on the degree zero part of the algebra, the core of the algebra. We show that the core captures essentially all relevant information about the graded representation theory. Using tilting theory, we show that the core is cellular. We then describe a canonical construction of a highest weight cover, in the sense of Rouquier, of this cellular algebra using a finite subquotient of the highest weight category. Thus, beginning with a self-injective graded algebra admitting a triangular decomposition, we canonically construct a quasi-hereditary algebra which encodes key information, such as graded multiplicities, of the original algebra. Our results are general and apply to a wide variety of examples, including restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras. We expect that the cell modules and quasi-hereditary algebras introduced here will provide a new way of understanding these important examples.