# Fortnightly links (109)

Dmitrii Pirozhkov: Admissible subcategories of del Pezzo surfaces is a very interesting preprint. It's the first non-trivial case where we know that the derived category of a smooth projective variety which is expected not to contain phantom subcategories, actually doesn't contain any. Great!

Federico Barbacovi: On the composition of two spherical twists is an extension of Segal's result that every autoequivalence is a spherical twist, by showing explicitly how the composition of spherical twists in this setting works, given the non-uniqueness of the source category for the spherical functor (spoiler: the answer is given by spherical functor from the gluing).

Alexander Polishchuk: Geometrization of trigonometric solutions of the associative and classical Yang-Baxter equations discusses how sheaves of orders (and sheaves of Lie algebras) can be used to construct solutions of the Yang–Baxter equation. This equation is studied in many different guises, and appears in many different areas of mathematics. The connection between the Yang–Baxter equation and 1-Calabi-Yau $\mathrm{A}_\infty$-categories was realised by Polishchuk almost 20 years ago, and in the current paper he discusses various explicit examples coming from noncommutative algebraic geometry which fit into this framework.

If your brain is wired somewhat similar to how mine is, this paper (and the references therein) might be a good introduction to the subject. I saw Yang–Baxter-y things before in seminar and conference talks, but the subject never clicked. With this preprint it did.