2021/06/22 by Peigen Cao, Cao, Peigen, Yasuaki Gyoda +3 · 2 citations
Mathematics · #13F60 #16G20 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2106.11668
openalex publication_date 2021/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In the present paper, we first give a characterization for Bongartz completion in τ-tilting theory via c-vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using c-vectors. Then we prove the existence and uniqueness of Bongartz completion in cluster algebras. We also prove that Bongartz completion admits certain commutativity. We give two applications for Bongartz completion in cluster algebras. As the first application, we prove the full subquiver of the exchange quiver (or known as oriented exchange graph) of a cluster algebra \mathcal A whose vertices consist of seeds of \mathcal A containing particular cluster variables is isomorphic to the exchange quiver of another cluster algebra. As the second application, we prove that in a cluster Poisson algebra \mathcal X_\bullet, each cluster Poisson seed (up to seed equivalence) of \mathcal X_\bullet is uniquely determined by its negative cluster Poisson variables.