2014/10/21 by Wen Chang, Bin Zhu, Chang, Wen +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Representation Theory (math.RT)
paper · pdf · doi:10.48550/arxiv.1410.5702
openalex publication_date 2014/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study rooted cluster algebras and rooted cluster morphisms which were introduced in \citeADS13 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 \citeADS13. 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 \citeADS13. 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 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 \refdef of complete pairs for precise meaning) and cotorsion pairs in C.