Relations among $\mathbb{P}$-Twists

TL;DR

This paper explores the relations among $bP$-twists associated with $bP$-objects in algebraic triangulated categories, establishing conditions under which these twists commute or have no relations, especially in hyperk"ahler varieties.

Contribution

It characterizes when $bP$-twists commute or are unrelated, linking their behavior to orthogonality of $bP$-objects and relating $bP$-twists to spherical twists.

Findings

$bP$-twists commute iff $bP$-objects are orthogonal.

Most known pairs of $bP$-objects on hyperk"ahler varieties are orthogonal.

No relations exist among $bP$-twists unless they commute.

Abstract

Given two $\mathbb{P}$-objects in some algebraic triangulated category, we investigate the possible relations among the associated $\mathbb{P}$-twists. The main result is that, under certain technical assumptions, the $\mathbb{P}$-twists commute if and only if the $\mathbb{P}$-objects are orthogonal. Otherwise, there are no relations at all. In particular, this applies to most of the known pairs of $\mathbb{P}$-objects on hyperk\"ahler varieties. In order to show this, we relate $\mathbb{P}$-twists to spherical twists and apply known results about the absence of relations between pairs of spherical twists.