Related papers: A categorical framework for glider representations

Let ${\mathcal X}$ be a category fibered in groupoids over a finite field $\mathbb{F}_q$, and let $k$ be an algebraically closed field containing $\mathbb{F}_q$. Denote by $\phi_k\colon {\mathcal X}_k\to {\mathcal X}_k$ the arithmetic…

From the viewpoint of higher homological algebra, we introduce pure semisimple $n$-abelian category, which is analogs of pure semisimple abelian category. Let $\Lambda$ be an Artin algebra and $\mathcal{M}$ be an $n$-cluster tilting…

The main result describes the Brauer-Nesbitt reduction of unipotent representations of a finite group of Lie type, expressing it as an explicit linear combination of the restriction of Weyl modules from the algebraic group to the group of…

Let $\Lambda$ be a finite dimensional algebra over an algebraically closed field. Criteria are given which characterize existence of a fine or coarse moduli space classifying, up to isomorphism, the representations of $\Lambda$ with fixed…

Let C be the category of finite-dimensional representations of a quantum affine algebra of simply-laced type. We introduce certain monoidal subcategories C_l (l integer) of C and we study their Grothendieck rings using cluster algebras.

We study the category $\mathcal{A}$ of smooth semilinear representations of the infinite symmetric group over the field of rational functions in infinitely many variables. We establish a number of results about the structure of…

We consider the abelian group $PT$ generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semi-orthogonal decompositions of corresponding triangulated categories. We introduce an operation of…

We explore several facets of tropical subrepresentations of a linear representation of a group over the tropical semifield $\mathbb{T}$. A key role in the study of tropical subrepresentations is played by two types of modules over a…

We consider algebras defined over a complete, local and noetherian ground ring. They are gentle algebras in case the ground ring is a field. The unbounded homotopy category of complexes of projective modules is considered. Complexes with…

For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion…

The geometric Satake correspondence gives an equivalence of categories between the representations of a semisimple group $ G $ and the spherical perverse sheaves on the affine Grassmannian $Gr$ of its Langlands dual group.…

We develop a unified representation theory for the categories of finite subsets and relation-preserving maps of highly homogeneous relational structures classified by Cameron. For any commutative coefficient ring $k$, we extend the…

Let $k$ be a field of characteristic $0$, let $\mathsf{C}$ be a finite split category, let $\alpha$ be a 2-cocycle of $\mathsf{C}$ with values in the multiplicative group of $k$, and consider the resulting twisted category algebra…

We observe that on the level of derived categories, representations of the Lie algebra of a semisimple algebraic group over a field of characteristic $p> h$ (where $h$ is the Coxeter number), with a given (generalized) central character are…

Let $K$ be a field and $G$ be a group of its automorphisms. If $G$ is precompact then $K$ is a generator of the category of smooth (i.e. with open stabilizers) $K$-semilinear representations of $G$. There are non-semisimple smooth…

This paper surveys bocses, quasi-hereditary algebras and their relationship which was established in a recent result by Koenig, Ovsienko, and the author. Particular emphasis is placed on applications of this result to the representation…

We recall the notions of a graded cocategory, conilpotent cocategory, morphisms of such (cofunctors), coderivations and define their analogs in $\mathbb L$-filtered setting. The difference with the existing approaches: we do not impose any…

We construct a categorification of the braid groups associated with Coxeter groups inside the homotopy category of Soergel's bimodules. Classical actions of braid groups on triangulated categories should come from an action of this monoidal…

Complete filtered $A_\infty$-algebras model certain deformation problems in the noncommutative setting. The formal deformation theory of a group representation is a classical example. With such applications in mind, we provide the…

For an abelian category C and a filtrant preordered set Lambda, we prove that the derived category of the quasi-abelian category of filtered objects in C indexed by Lambda is equivalent to the derived category of the abelian category of…