Noncommutative Tensor Triangular Geometry

Abstract: In this talk, I will show how to develop a general noncommutative version of Balmer's tensor triangular geometry that is applicable to arbitrary monoidal triangulated categories (M$\Delta$C). Insights from noncommutative ring theory is used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of an M$\Delta$C, $\mathbf K $, and then to associate to $\mathbf K$ a topological space --the Balmer spectrum $\text{Spc }{\mathbf K}$.

We develop a general framework for (noncommutative) support data, coming in three different flavors, and show that $\text{Spc }{\mathbf K}$ is a universal terminal object for the first two notions (support and weak support). The first two types of support data are then used in a theorem that gives a method for the explicit classification of the thick (two-sided) ideals and the Balmer spectrum of an M$\Delta$C. The third type (quasi support) is used in another theorem that provides a method for the explicit classification of the thick right ideals of $\mathbf K$, which in turn can be applied to classify the thick two-sided ideals and $\text{Spc }{\mathbf K}$.

If time permits applications will be given for quantum groups and non-cocommutative finite-dimensional Hopf algebras studied by Benson and Witherspoon.

This is joint and ongoing work with Milen Yakimov and Kent Vashaw

category theoryquantum algebra

Audience: researchers in the topic

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Quantum Groups Seminar [QGS]

Series comments: This online seminar aims to bring together experts in the area of quantum groups. The seminar topics will cover the theory of quantum groups and related structures in a large sense: Hopf algebras, operator algebras, q-deformations, higher categories and related branches of noncommutative mathematics.

The zoom links will be distributed by mail, so please join the mailing list if you are interested in attending the seminar.

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