Silting and Tilting for Weakly Symmetric Algebras

TL;DR

This paper explores the relationship between silting and tilting complexes in weakly symmetric algebras, showing that certain classes behave similarly to symmetric algebras while others do not.

Contribution

It demonstrates that all tilting-discrete weakly symmetric algebras are also silting-discrete and provides a counterexample where silting complexes are not tilting.

Findings

Tilting-discrete weakly symmetric algebras are also silting-discrete

Counterexample of a weakly symmetric algebra with non-tilting silting complexes

Abstract

If A is a finite-dimensional symmetric algebra, then it is well-known that the only silting complexes in $\mathrm{K^b}(\mathrm{proj}A)$ are the tilting complexes. In this note we investigate to what extent the same can be said for weakly symmetric algebras. On one hand, we show that this holds for all tilting-discrete weakly symmetric algebras. In particular, a tilting-discrete weakly symmetric algebra is also silting-discrete. On the other hand, we also construct an example of a weakly symmetric algebra with silting complexes that are not tilting.