Analytic theory of tensor C*-categories
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MSCA-2020-KDCommer01
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Beschrijving van het project
Tensor C*-categories are categories with a monoidal structure and a compatible operator algebra structure on their endomorphism spaces. One instance of a tensor C*-category is obtained by considering the category of finite dimensional representations of a compact group. Another instance is obtained by considering finite dimensional Hilbert spaces graded over a discrete group. Both these particular cases are subsumed by the example of representation categories of compact quantum groups, and these provide a large but non-exhaustive class of tensor C*-categories with duals.
In recent years, there has been an increasing interest in tensor C*-categories. One line of research, closely following similar developments in the purely algebraic setting of fusion categories, is geared towards classification results for finite tensor C*-categories up to an appropriate form of Morita equivalence. Such tensor C*-categories and their associated module C*-categories are modeled on subfactors of finite index and finite depth. Another line of research considers non-finite tensor C*-categories, and aims to understand better the analytic theory of tensor C*-categories. This approach is heavily inspired by (geometric) group theory, and has provided an important framework in which to consider for example the finer analytic structure of finite index subfactors (of infinite depth). There are however many opportunities and unexplored research questions related to the further analytic structure of tensor C*-categories, for example with regards to developing an associated equivariant KK-theory. This is particularly challenging in light of the rich inherent structure of tensor C*-categories, which calls for a categorification of many of the classical tools of operator algebra analysis.
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TACT: Topological Algebra, Functional Analysis and Category Theory
The research team investigates mathematical structures that are important in several basic areas of mathematics like geometry, representation theory, functional analysis, differential calculus or theory of approximation. The motivation also comes from outside mathematics, from computer science or physics where some of the mathematical structures that are studied are called upon as models. By application of methods from category theory the relation between these mathematical structures is studied. The compatibility of their fundamental constructions is investigated and a general study of their representability, as well as of their function space theory is undertaken.
More specifically the theory of frames or locales uses order-theoretic notions to gain more insight in topological structures and to shed light on the use of choice principles in topology (or sometimes simply avoid them altogether).
The theory of approach spaces provides the tools for obtaining quantified results in topology and in functional analysis, extending the isometric theory of Banach spaces.
The team contributes to the development of the theory of semi-abelian categories and tensor categories.
Semi-abelian categories allow a unified setting for many important homological properties of non-abelian categories. Categories of quantum groups, of rings, of Lie-algebras and of crossed modules are typical non-abelian categories, often with a tensor structure.
Abstract tensor categories lead to interesting non-commutative spaces (operator algebras) whose analytical properties are studied in connection with the properties of the associated category. The main emphasis is on representation categories of quantum deformations of semi-simple Lie groups.
Research Tracks:
•Theory of approach spaces: E. Colebunders, M. Sioen
•Categorical topology: E. Colebunders, M. Sioen
•Pointfree topology: M. Sioen
•Abstract tensor categories: K. De Commer
•Quantum groups: K. De Commer
•Operator algebras: K. De Commer