Algebra Seminar: Uncoiled Temperley-Lieb algebras, Jones-Wenzl projectors and their trace decategorification

Abstract: The affine and periodic Temperley-Lieb algebras are families of infinite-dimensional algebras with a diagrammatic presentation. They have been studied in the last 30 years, mostly for their physical applications in statistical mechanics, where the diagrammatic presentation encodes the connectivity property of the models. Most of the relevant representations for physics are finite-dimensional. In the first part of the talk, we will present the diagrammatic calculus related to these algebras and define finite-dimensional quotients of these algebras, which we name uncoiled algebras in reference to the diagrammatic interpretation. Afterwards, we construct a family of Jones-Wenzl idempotents, each of which projects onto one of the one-dimensional modules these algebras admit. The second part of the talk will go in depth on the construction of the Jones-Wenzl idempotents. We will then discuss what do the related uncoiled categories categorify by investigating the trace decatorification of the idempotents we defined.

This is joint work with Alexi Morin-Duchesne

Welcome to join on site or via Zoom link (Meeting ID: 645 5572 6999, please contact the organizer for passcode)

This is a seminar in our algebra seminar series.