On rooted cluster morphisms and cluster structures in $2$-Calabi-Yau triangulated categories

TL;DR

This paper explores the structure of rooted cluster morphisms in 2-Calabi-Yau triangulated categories, clarifying their properties, introducing new concepts like frozenization, and establishing correspondences with cotorsion pairs.

Contribution

It introduces the notion of frozenization, characterizes when rooted cluster morphisms are ideal, and links cluster subalgebras with cotorsion pairs in 2-Calabi-Yau categories.

Findings

An example of a non-ideal rooted cluster morphism clarifies previous doubts.

Injective rooted cluster morphisms always come from frozenization and subseeds.

A correspondence between rooted cluster subalgebras and cotorsion pairs is established.

Abstract

We study rooted cluster algebras and rooted cluster morphisms which were introduced in \cite{ADS13} recently and cluster structures in $2$-Calabi-Yau triangulated categories. An example of rooted cluster morphism which is not ideal is given, this clarifies a doubt in \cite{ADS13}. We introduce the notion of frozenization of a seed and show that an injective rooted cluster morphism always arises from a frozenization and a subseed. Moreover, it is a section if and only if it arises from a subseed. This answers the Problem 7.7 in \cite{ADS13}. We prove that an inducible rooted cluster morphism is ideal if and only if it can be decomposed as a surjective rooted cluster morphism and an injective rooted cluster morphism. We also introduce the tensor decompositions of a rooted cluster algebra and of a rooted cluster morphism. For rooted cluster algebras arising from a $2$-Calabi-Yau triangulated category $\mathcal{C}$ with cluster tilting objects, we give an one-to-one correspondence between certain pairs of their rooted cluster subalgebras which we call complete pairs (see Definition \ref{def of complete pairs} for precise meaning) and cotorsion pairs in $\mathcal{C}$.