Related papers: Commutative algebra in tensor categories

A semisimple algebraic tensor category over an algebraically closed field k of characteristic zero is the representation category of all finite dimensional twisted super representations of an affine reductive supergroup G over k. Such a…

The Kuperberg Program asks to find presentations of planar algebras and use these presentations to prove results about their corresponding categories purely diagrammatically. This program has been completed for index less than 4 and is…

Using face algebras (i.e. algebras of L-operators of IRF models), we construct modular tensor categories with positive definite inner product, whose fusion rules and S-matrices are the same as (or slightly different from) those obtained by…

For a large class of geometric objects, the passage to categories of quasi-coherent sheaves provides an embedding in the 2-category of abelian tensor categories. The notion of weakly Tannakian categories introduced by the author gives a…

We construct certain tensor categories that are dominated by finitely many simple objects. Objects in these categories are modules over rings of algebra integers. We show how to obtain TQFTs defined over algebra integers from these…

We introduce a notion of complexity of diagrams (and in particular of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several…

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,…

We consider the 2-generated free metabelian associative and Lie algebras over the complex field and the invariants of the dihedral groups of finite order acting on these algebras. In the associative case we find a finite set of generators…

Let $G$ be a finite group. There is a standard theorem on the classification of $G$-equivariant finite dimensional simple commutative, associative, and Lie algebras (i.e., simple algebras of these types in the category of representations of…

Given a finite modular tensor category, we associate with each compact surface with boundary a cochain complex in such a way that the mapping class group of the surface acts projectively on its cohomology groups. In degree zero, this action…

This article presents a natural extension of the tensor algebra. In addition to "left multiplications" by vectors, we can consider "derivations" by covectors as basic operators on this extended algebra. These two types of operators satisfy…

We introduce and analyse a general notion of fundamental group for noncommutative spaces, described by differential graded algebras. For this we consider connections on finitely generated projective bimodules over differential graded…

In the authors book, Associative Algebraic Geometry, 2023, and the following article Shemes of Associative Algebras,\\ https://doi.org/10.48550/arXiv.2410.17703,2024, we use an algebraization of the semi-local formal moduli of simple…

We give a new construction of the algebraic $K$-theory of small permutative categories that preserves multiplicative structure, and therefore allows us to give a unified treatment of rings, modules, and algebras in both the input and…

We prove a constructive existence theorem for abelian envelopes of non-abelian monoidal categories. This establishes a new tool for the construction of tensor categories. As an example we obtain new proofs for the existence of several…

We classify affine varieties with an action of a connected, reductive algebraic group such that the group is isomorphic to an open orbit in the variety. This is accomplished by associating a set of one-parameter subgroups of the group to…

A modular tensor category provides the appropriate data for the construction of a three-dimensional topological field theory. We describe the following analogue for two-dimensional conformal field theories: a 2-category whose objects are…

Given a positive definite even lattice and a commutative ring, there is a standard construction of a lattice vertex algebra over the commutative ring, and it admits a natural grading by non-negative integers. We describe the groups of…

As part of the study of correspondence functors, the present paper investigates their tensor product and proves some of its main properties. In particular, the correspondence functor associated to a finite lattice has the structure of a…

This is Part I of a series of papers constructing intertwining operator superalgebras and vertex tensor categories associated to the superconformal minimal models and other related models. In this paper, we construct the intertwining…