Apr. 12, 2016
Speaker: Alejandro Ginory
Title: Generalized Kac-Moody algebras II.
Abstract: I will introduce the concept of generalized Kac-Moody algebras and briefly study some of its representation theory. We will then see some examples of interest, in particular the (non-fake!) Monster Lie algebra introduced by Borcherds in his proof of monstrous moonshine. Finally, we will discuss a `triangular decomposition' of a certain class of generalized Kac-Moody algebras that is analogous to the well-known triangular decomposition of Kac-Moody algebras.
Mar. 9, 2016
Speaker: Alejandro Ginory
Title: Generalized Kac-Moody algebras.
Abstract: I will introduce the concept of generalized Kac-Moody algebras and briefly study some of its representation theory. We will then see some examples of interest, in particular the (non-fake!) Monster Lie algebra introduced by Borcherds in his proof of monstrous moonshine. Finally, we will discuss a `triangular decomposition' of a certain class of generalized Kac-Moody algebras that is analogous to the well-known triangular decomposition of Kac-Moody algebras.
Mar. 2, 2016
Speaker: Johannes Flakes
Title: Quantum Groups II.
Abstract: Continued from last time.
Feb. 24, 2016
Speaker: Johannes Flakes
Title: Quantum Groups.
Abstract: Let's return to our tensor categories. Interesting examples for modular tensor categories are obtained from (modules of) quantum groups, which are deformations of universal enveloping algebras of Lie algebras. We will discuss definitions and basic properties of quantum groups and the mentioned module categories.
Feb. 17, 2016
Speaker: Fei Qi
Title: Meromorphic Open-String Vertex Algebras II.
Abstract: We'll continue with the paper arxiv:1204.1774v1 and hopefully we can finish it.
Feb. 10, 2016
Speaker: Fei Qi
Title: Meromorphic Open-String Vertex Algebras.
Abstract: This week we are going to take a break from the tensor categories. I'll talk about one of my current research topics, namely the meromorphic open-string vertex algebras. I'll basically go over Huang's paper arxiv:1204.1774v1 starting from axioms. Hopefully I'll be able to finish the example in the paper.
Feb. 3, 2016
Speaker: Semeon Artamonov
Title: Graphical Calculus for Modular Tensor Categories.
Abstract: This is the second talk in a series of discussions on Modular Tensor Categories. It will mainly follow Chapter II of V. Turaev book "Quantum Invariants of Knots and 3-manifolds". This time we will go over graphical calculus technique for Ribbon Tensor Categories, in particular we will prove Quantum Yang-Baxter equation and several other ingridients of graphical calculus. This technique drastically simplifies calculations, and as name suggests make them visual. Next, we will pass to the case of semisimple categories and define Modular Tensor Categories. Finally, we will introduce the Fourier Transform matrix and prove that along with a twist it defines an SL(2,Z) projective action on MTC. If time permits we will prove Verlinde formula which expresses the structure constants of the Grothendieck ring of Modular Tensor Category via matrix elements of the Fourier transform.
Jan. 27, 2016
Speaker: Semeon Artamonov
Title: Introduction to Modular Tensor Categories
Abstract: This semester we will launch a series of talks on Modular Tensor Categories, starting by discussing V. Turaev "Quantum invariants of knots and 3-manifolds". In the first talk I will start with definitions of Braided Tensor Categories. Then I will introduce abstract notions of evaluation and coevaluation in monoidal categories followed by the definition of Ribbon categories. I will conclude this introductory talk with a technique of Graphical Calculus for ribbon categories. In my talk I wouldn't assume any knowledge beyond graduate algebra courses, I would even say that it might be easier to learn abstract MTC from scratch.