http://www.maths.leeds.ac.uk/~marsh/research_articles/pp18.pdf

Tilting theory and cluster combinatorics

By Vyjayanthi Chari and Andrew Pressley: 651 pp., £22.95 (US$34.95), isbn 0 521 55884 0 (Cambridge University Press, 1994).

A guide to quantum groups

We show that the category of orbits of the bounded derived category of a hereditary category under a well-behaved autoequivalence is canonically triangulated. This answers a question by Aslak Buan, Robert Marsh and Idun Reiten which appeared in their study [8] with M. Reineke and G. Todorov of the link between tilting theory and cluster algebras (cf. also [16]) and a question by Hideto Asashiba about orbit categories. We observe that the resulting triangulated orbit categories provide many examples of triangulated categories with the Calabi-Yau property. These include the category of projective modules over a preprojective algebra of generalized Dynkin type in the sense of Happel-Preiser-Ringel [29], whose triangulated structure goes back to Auslander-Reiten’s work [6], [44], [7].

/pdf/on-triangulated-orbit-categories-4k4m3u3a5b.pdf

On triangulated orbit categories.

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.

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings.

Mutation in triangulated categories and rigid Cohen–Macaulay modules

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U be a cluster algebra of type A_n. We associate to each cluster C of U an abelian category Cat_C such that the indecomposable objects of Cat_C are in natural correspondence with the cluster variables of U which are not in C. We give an algebraic realization and a geometric realization of Cat_C. Then, we generalize the ``denominator Theorem'' of Fomin and Zelevinsky to any cluster.

/pdf/quivers-with-relations-arising-from-clusters-a-n-case-2cfr4gx9i8.pdf

Quivers with relations arising from clusters $(A_n$ case)

/pdf/quivers-with-relations-arising-from-clusters-a-n-case-35qi7nhgti.pdf

Quivers with relations arising from clusters (A_n case)

Positivity for cluster algebras from surfaces

We prove the positivity conjecture for all skew-symmetric cluster algebras.

https://annals.math.princeton.edu/wp-content/uploads/Lee_Schiffler.pdf

Positivity for cluster algebras

We construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, such as principal coefficients.

/pdf/bases-for-cluster-algebras-from-surfaces-28p3dan94a.pdf

Ralf Schiffler

Papers

Quivers with relations arising from clusters $(A_n$ case)

Quivers with relations arising from clusters (A_n case)

Positivity for cluster algebras from surfaces

Positivity for cluster algebras

Bases for cluster algebras from surfaces