# Some algebras that are not silting connected

## TL;DR

This paper provides examples of finite-dimensional algebras where silting objects are disconnected via silting mutations, highlighting limitations in silting mutation connectivity.

## Contribution

It introduces specific algebras with silting objects that cannot be connected through silting mutations, expanding understanding of silting theory limitations.

## Key findings

- Silting objects in certain algebras are not connected by silting mutations.
- Invariance under algebra automorphisms affects silting mutation connectivity.
- Existence of spherical modules not invariant under twists impacts silting structures.

## Abstract

We give examples of finite-dimensional algebras $A$ for which the silting objects in $K^b(\mbox{proj-}A)$ are not connected by any sequence of (possibly reducible) silting mutations. The argument is based on the fact that silting mutation preserves invariance under twisting by a fixed algebra automorphism, combined with the existence of spherical modules that are not invariant under such a twist.