Tilting modules of tilted algebras and related topics

Tilting theory has been a central research topic in representation theory for a long time. Let $Λ$ be a hereditary algebra, $T_0$ be a tilting Λ-module, $B := \End_Λ(T_0)$, then $B$ is called a tilted algebra of type $Λ$. In this thesis, we will study some related topics of tilted algebras.

We will give some general ways to construct tilting $B$-modules from tilting $Λ$-modules. The main tools we will use in the constructions are the traces and rejects.

Let $A$ be a finite dimensional $k$ algebra over an algebraically closed field $k$, the study of the equivalence classes of tilting $A$-modules was initiated by Riedtman and Schofield in 1991. Later in 2005, Happel and Unger defined the tilting quiver $\overrightarrow{\cK_A}$ for a given algebra $A$. With the constructions of tilting modules we have obtained, we will show how to get the tilting quiver of $B$ from that of $Λ$, when $B$ is a BB-tilted algebra of type $Λ$.

As a generalization of the classic tilting theory, $τ$-tilting theory was introduced by Adachi, Iyama and Reiten in 2014. For a given algebra $A$, in analogy with the tilting quiver of $A$, the authors in 2014 defined a quiver which is called the support $τ$-tilting quiver $Q(sτ-tiltA)$. We will consider the reconstruction of the support $τ$-tilting quiver $Q(sτ-tiltB)$ from the support $τ$-tilting quiver $Q(sτ-tiltΛ)$.

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