Examples of tilting-discrete symmetric algebras

TL;DR

This work investigates tilting-discreteness for finite-dimensional algebras, focusing on symmetric cases, Brauer graph algebras, and trivial extensions, and clarifies its relationship with $ au$-tilting finiteness and derived invariants. It provides a negative answer to the conjecture that $ au$-tilting finite symmetric algebras are tilting-discrete by constructing counterexamples from trivial extensions of $ ext{$ extell$-Kronecker}$ algebras, while also identifying conditions under which tilting-discreteness follows from Cartan matrix positivity and other invariants. The paper establishes equivalences for Brauer graph algebras, classifies gentle algebras whose trivial extensions are tilting-discrete, and describes tilting-disconnected Brauer graph algebras, together with implications for cluster-tilted and derived-discrete settings. It develops an upper-bound criterion on Cartan-invariant–type data that guarantees tilting-discreteness in the $ au$-tilting finite regime and discusses lifting and silting-discreteness questions in nonsymmetric contexts, supported by extensive mutation and reduction techniques. Overall, the results map the landscape of tilting/disconnectedness phenomena, highlighting the nuanced interplay between $ au$-tilting finiteness, Cartan data, and derived structure across broad algebra classes.

Abstract

We give several examples of tilting-discrete symmetric algebras; in particular, one explores which algebra has tilting-discrete trivial extension. We provide a counter example of the conjecture stating any τ -tilting finite symmetric algebra is tiltingdiscrete. Also, we discuss the tilting-disconnectedness of symmetric algebras and give new examples of tilting-disconnected symmetric algebras.

Theorems & Definitions (66)

• Conjecture
• Proposition
• proof
• Theorem 1: Example \ref{['Am']} and Theorem \ref{['Kronecker']}
• Remark
• Theorem 2: Theorem \ref{['deriveddiscrete']}
• Theorem 3: Theorem \ref{['clustertilted']}
• Theorem 4: Proposition \ref{['TEofAm2']}
• Theorem 5: Corollary \ref{['upperbound2']}
• Theorem 6: Theorem \ref{['tdcBGA']}