Joseph Grant (University of East Anglia) – Fractionally Calabi-Yau quivers and Temperley-Lieb categories

Representation theory involves studying mathematical objects by interpreting them in linear algebra: for example, we interpret group elements as matrices. It can be useful to abstract our linear algebra problem using a quiver, which is a directed graph where the vertices correspond to vector spaces and the edges to linear transformations. Gabriel showed that a quiver has finitely many representations precisely when its underlying graph is of ADE Dynkin type and noticed a pattern which Kontsevich later formalised as the fractionally Calabi-Yau property, based on categorical properties occurring in geometry. I will explain work with Mathew Pugh where we show how this is a shadow of a property of the Temperley-Lieb category, formed from non-crossing lines between dots in the plane, when the quantum parameter is a complex root of unity. This involves working with new definitions of Frobenius algebra objects and Nakayama morphisms in monoidal categories.