Baby Pizza ZMP Seminar

Introduction to Quantum Groups (Christopher Raymond)

I have the pleasure of introducing quantum groups, which (in)famously are not quantum and not groups. My goal for the talk will be to describe a little bit of the physics that motivates the definition of what we now refer to as a "quantum group", then give some version of a definition, and finish off with some motivation for the talks that are to come. I'll be sacrificing some precision and detail in order to make things as understandable as possible for those with either physics or math backgrounds.

Notes from the talk.

Modular Tensor Categories associated to Quantum Groups (Felipe Ruiz)

Modular Tensor Categories are relevant objects in mathematical physics partly because they encode algebraic data of  3d TQFT's and 2d RCFT´s. The goal of this talk is to show how certain quantum groups produce modular tensor categories and state the Kazhdan-Lusztig correspondence relating them to categories of representations of affine vertex operator algebras.

Notes from the talk.

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Braided Reconstruction (Tyler Franke)

Some motivation coming from physics: A solution to the Yang--Baxter equation describes factorisable scattering in the bulk of a system. Factorisable scattering at the boundary of a system is described by the so-called reflection equation. Since quantum groups are a nice source of solutions to Yang--Baxter, it is natural to ask what algebraic structure is a good source for solutions to the reflection equation. Given a quantum group U, the reflection equation algebra for U can be identified with the coend of a certain functor to U-mod. I can try to convince people why that sentence makes sense, and explain in some detail what this looks like for U_q(sl_2). I could then give a gist of how it should work for quantum groups in general. For the history I can refer to [Lyubashenko, Majid; 1994], Majid's book (where he called this ``braided reconstruction") and his many other papers about this. [Donin, Kulish, Mudrov; 2002] and [Donin, Mudrov; 2002] make a refinement upon the ``universality" of solutions to the reflection equation coming from this procedure. [Cooke, Laugwitz; 2025] is a nice resource for some technical details (one doesn't really need to know what a coend is to be able to follow what they do). [Cherednik; 1984] is where the reflection equation first appeared in physics. I would also want to mention how this is related to braided module categories over U-mod [Brochier; 2012], and time-permitting factorisation homology of surfaces with coefficients in U-mod [Ben-Zvi, Brochier, Jordan; 2015], and thus skein theory on stratified manifolds.