2015/12/11 by Shen Li, Li, Shen, Shunhua Zhang +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum many-body systems #Representation Theory (math.RT) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1512.03613
openalex publication_date 2015/12/11 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
Let A be a finite dimensional algebra over an algebraically closed field k, and M be a partial tilting A-module. We prove that the Bongartz τ-tilting complement of M coincides with its Bongartz complement, and then we give a new proof of that every almost complete tilting A-module has at most two complements. Let A=kQ be a path algebra. We prove that the support τ-tilting quiver \overrightarrowQ(\rm sτ-\rm tilt A) of A is connected. As an application, we investigate the conjecture of Happel and Unger in [9] which claims that each connected component of the tilting quiver \overrightarrowQ(\rm tilt A) contains only finitely many non-saturated vertices. We prove that this conjecture is true for Q being all Dynkin and Euclidean quivers and wild quivers with two or three vertices, and we also give an example to indicates that this conjecture is not true if Q is a wild quiver with four vertices.