# Highest weight theory for finite-dimensional graded algebras with triangular decomposition

## TL;DR

This paper establishes that graded modules over finite-dimensional graded algebras with a triangular decomposition form a highest weight category, revealing new structures and insights in their representation theory, especially for self-injective cases.

## Contribution

It introduces a highest weight category framework for these algebras and demonstrates the existence of tilting modules when the algebra is self-injective, offering a new perspective on their representation theory.

## Key findings

- Category of graded modules forms a highest weight category
- Existence of tilting modules in self-injective cases
- Applicable to various algebraic structures like quantum groups and Cherednik algebras

## Abstract

We show that the category of graded modules over a finite-dimensional graded algebra admitting a triangular decomposition can be endowed with the structure of a highest weight category. When the algebra is self-injective, we show furthermore that this highest weight category has tilting modules in the sense of Ringel. This provides a new perspective on the representation theory of such algebras, and leads to several new structures attached to them. There are a wide variety of examples in algebraic Lie theory to which this applies: restricted enveloping algebras, Lusztig's small quantum groups, hyperalgebras, finite quantum groups, and restricted rational Cherednik algebras.