Related papers: Tensor categories: A selective guided tour
These lecture notes cover 13 sessions and are presented as an e-print, intended to evolve over time. Quantum invariants do more than distinguish topological objects; they build bridges between topology, algebra, number theory and quantum…
These notes attempt to give a short survey of the approach to support theory and the study of lattices of triangulated subcategories through the machinery of tensor triangular geometry. One main aim is to introduce the material necessary to…
We develop general foundations of topological algebra over a linearly topologized ring k in a format applicable to both formal schemes and analytic adic spaces. We are especially interested in determining exact closed tensor categories of…
We study exact module categories over the representation categories of finite-dimensional quasi-Hopf algebras. As a consequence we classify exact module categories over some families of pointed tensor categories with cyclic group of…
For any essentially small triangulated category the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for…
This article is meant as a gentle introduction to the "topological terms" that often play a decisive role in effective theories describing topological quantum effects in condensed matter systems. We first take up several prominent examples,…
We develop some techniques to the study of exact module categories over some families of pointed finite-dimensional Hopf algebras. As an application we classify exact module categories over the tensor category of representations of the…
The present paper are the notes of a mini-course addressed mainly to non-experts. It purpose it to provide a first approach to the theory of mapping class groups of non-orientable surfaces.
We start the general structure theory of not necessarily semisimple finite tensor categories, generalizing the results in the semisimple case (i.e. for fusion categories), obtained recently in our joint work with D.Nikshych. In particular,…
Tensor network diagram (graphical notation) is a useful tool that graphically represents multiplications between multiple tensors using nodes and edges. Using the graphical notation, complex multiplications between tensors can be described…
In this lecture note, we discuss a fundamental concept, referred to as the {\it characteristic rank}, which suggests a general framework for characterizing the basic properties of various low-dimensional models used in signal processing.…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
This is an expanded version of the notes by the second author of the lectures on symmetric tensor categories given by the first author at Ohio State University in March 2019 and later at ICRA-2020 in November 2020. We review some aspects of…
Modelling compositionality has been a longstanding area of research in the field of vector space semantics. The categorical approach to compositionality maps grammar onto vector spaces in a principled way, but comes under fire for requiring…
We introduce the notion of restricted dynamical quantum groups through their category of representations, which are monoidal categories with a forgetful functor to the category of $\pi$-graded vector spaces for a groupoid $\pi$.
These are lecture notes compiled for a short lecture series at the 2023 Condensed Matter Summer School at the University of Minnesota. They are designed to be conversational and fun, and not to take the place of review articles that do a…
A subunit in a monoidal category is a subobject of the monoidal unit for which a canonical morphism is invertible. They correspond to open subsets of a base topological space in categories such as those of sheaves or Hilbert modules. We…
We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of finite-dimensional pointed Hopf algebras over abelian groups only in terms of the (cohomology class of the) associator of the pointed part.…
We develop a notion of iterated monoidal category and show that this notion corresponds in a precise way to the notion of iterated loop space. Specifically the group completion of the nerve of such a category is an iterated loop space and…
Tensor networks are a popular and computationally efficient approach to simulate general quantum systems on classical computers and, in a broader sense, a framework for dealing with high-dimensional numerical problems. This paper presents a…