Related papers: Introduction to categorification
We introduce the notion of weakly associative algebra and its relations with the notion of nonassociative Poisson algebras.
In this paper we investigate weak polynomial identities for the Weyl algebra $\mathsf{A}_1$ over an infinite field of arbitrary characteristic. Namely, we describe weak polynomial identities of the minimal degree, which is three, and of…
Poisson algebras have become an essential topic in mathematics with a rich structure and wide applicability. Despite numerous resources available on Poisson structures, the algebraic side of the story remains relatively less explored. This…
In a paper by the authors, the associative and the Lie algebras of Weyl type $A[D]=A\otimes F[D]$ were introduced, where $A$ is a commutative associative algebra with an identity element over a field $F$ of any characteristic, and $F[D]$ is…
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…
Recently, we have constructed a non{linear (polynomial) extension of the 1-mode Heisenberg group and the corresponding Fock and Weyl representations. The transition from the 1-mode case to the current algebra level, in which the operators…
The notions of weakly bipartitive and bipartitive families were introduced by Montgolfier (2003) as a general tool for studying some decomposition of graphs and other combinatorial structures. In this paper, we give a matrix description of…
There could be thousands of Introductions/Surveys of representation theory, given that it is an enormous field. This is just one of them, quite personal and informal. It has an increasing level of difficulty; the first part is intended for…
We systematically apply semisimplification functors in modular representation theory. Motivated by the Duflo--Serganova functor in Lie superalgebras, we construct various functors of interest. In the setting of finite groups, we refine the…
Lie algebras are an important class of algebras which arise throughout mathematics and physics. We report on the formalisation of Lie algebras in Lean's Mathlib library. Although basic knowledge of Lie theory will benefit the reader, none…
We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories…
These are notes from an informal mini-course on factorization homology, infinity-categories, and topological field theories. The target audience was imagined to be graduate students who are not homotopy theorists.
`Categorification' is the process of replacing equations by isomorphisms. We describe some of the ways a thoroughgoing emphasis on categorification can simplify and unify mathematics. We begin with elementary arithmetic, where the category…
These are the notes of some lectures given by the author for a workshop held at TIFR, Mumbai in December, 2011, giving an exposition of the Deligne-Lusztig theory.
This is a short introduction to categories with some emphasis on coalgebras. We start from introducing basic notions (categories, functors, natural transformations), move to Kleisli tripels and monads, with a short discussion of monads in…
The rings of symmetric polynomials form an inverse system whose limit, the ring of symmetric functions, is the model for the bosonic Fock space representation of the affine Lie algebra. We categorify this construction by considering an…
We develop a theory of descent and forms of tensor categories over arbitrary fields. We describe the general scheme of classification of such forms using algebraic and homotopical language, and give examples of explicit classification of…
This document contains the notes of a lecture I gave at the "Journ\'ees Nationales du Calcul Formel" (JNCF) on January 2017. The aim of the lecture was to discuss low-level algorithmics for p-adic numbers. It is divided into two main parts:…
In many applications, finding adequate labeled data to train predictive models is a major challenge. In this work, we propose methods to use group-level binary labels as weak supervision to train instance-level binary classification models.…
In these lecture notes, we give a brief introduction to some elements of category theory. The choice of topics is guided by applications to functional programming. Firstly, we study initial algebras, which provide a mathematical…