Tilting theory and cluster combinatorics

TL;DR

This paper introduces the cluster category as a new mathematical framework linking tilting theory with cluster algebra combinatorics, revealing regularities and connections to algebra classification.

Contribution

It constructs the cluster category as a quotient of derived categories, connecting tilting modules with cluster algebra clusters, and explores its regularity and classification implications.

Findings

Cluster category models Fomin-Zelevinsky cluster algebra combinatorics.

Tilting theory in the cluster category is more regular than in module categories.

Links between tilting theory and classification of self-injective algebras are established.

Abstract

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting modules correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.