Relations among $\mathbb{P}$-Twists

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 hyperkähler 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.

21 pages. Closed gap in proof of Proposition 3.23 (see also Remark 3.24). Same content as published version