Diagrams and discrete extensions for finitary 2-representations

@article{Chan2017DiagramsAD,
  title={Diagrams and discrete extensions for finitary 2-representations},
  author={Aaron Chan and Volodymyr Mazorchuk},
  journal={Mathematical Proceedings of the Cambridge Philosophical Society},
  year={2017},
  volume={166},
  pages={325 - 352}
}
  • Aaron ChanV. Mazorchuk
  • Published 1 January 2016
  • Mathematics
  • Mathematical Proceedings of the Cambridge Philosophical Society
Abstract In this paper we introduce and investigate the notions of diagrams and discrete extensions in the study of finitary 2-representations of finitary 2-categories. 

Simple transitive 2-representations of 2-categories associated to self-injective cores

Abstract Given a finite dimensional algebra A, we consider certain sets of idempotents of A, called self-injective cores, to which we associate 2-subcategories of the 2-category of projective

Simple transitive 2-representations via (co)algebra 1-morphisms

For any fiat 2-category C, we show how its simple transitive 2-representations can be constructed using coalgebra 1-morphisms in the injective abelianization of C. Dually, we show that these can also

Weighted colimits of 2-representations and star algebras

We apply the theory of weighted bicategorical colimits to study the problem of existence and computation of such colimits of birepresentations of finitary bicategories. The main application of our

COIDEMPOTENT SUBCOALGEBRAS AND SHORT EXACT SEQUENCES OF FINITARY 2-REPRESENTATIONS

In this article, we study short exact sequences of finitary 2-representations of a weakly fiat 2-category. We provide a correspondence between such short exact sequences with fixed middle term and

Pyramids and 2-representations

We describe a diagrammatic procedure which lifts strict monoidal actions from additive categories to categories of complexes avoiding any use of direct sums. As an application, we prove that every

Classification problems in 2-representation theory

This article surveys recent advances and future challenges in the 2-representation theory of finitary 2-categories with a particular emphasis on problems related to classification of various classes

Simple transitive 2-representations of left cell 2-subcategories of projective functors for star algebras

Abstract In this article, we study simple transitive 2-representations of certain 2-subcategories of the 2-category of projective functors over a star algebra. We show that in the simplest case,

Special modules over positively based algebras

We use the Perron-Frobenius Theorem to define, study and, in some sense, classify special simple modules over arbitrary finite dimensional positively based algebras. For group algebras of finite Weyl

On Simple Transitive 2-representations of Bimodules over the Dual Numbers

We study the problem of classification of simple transitive 2-representations for the (non-finitary) 2-category of bimodules over the dual numbers. We show that simple transitive 2-representations

2-categories of symmetric bimodules and their 2-representations

In this article we analyze the structure of $2$-categories of symmetric projective bimodules over a finite dimensional algebra with respect to the action of a finite abelian group. We determine under

References

SHOWING 1-10 OF 37 REFERENCES

Morita theory for finitary 2-categories

We develop Morita theory for finitary additive 2-representations of finitary 2-categories. As an application we describe Morita equivalence classes for 2-categories of projective functors associated

CATEGORIFICATION OF THE CATALAN MONOID

We construct a finitary additive 2-category whose Grothendieck ring is isomorphic to the semigroup algebra of the monoid of order-decreasing and order-preserving transformations of a finite chain.

Finitary 2-categories associated with dual projection functors

We study finitary 2-categories associated to dual projection functors for finite dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin

Transitive 2-representations of finitary 2-categories

In this article, we define and study the class of simple transitive $2$-representations of finitary $2$-categories. We prove a weak version of the classical Jordan-H{\"o}lder Theorem where the weak

Isotypic faithful 2-representations of J-simple fiat 2-categories

We introduce the class of isotypic 2-representations for finitary 2-categories and the notion of inflation of 2-representations. Under some natural assumptions we show that isotypic 2-representations

Isotypic faithful 2-representations of $${\mathcal {J}}$$J-simple fiat 2-categories

We introduce the class of isotypic 2-representations for finitary 2-categories and the notion of inflation of 2-representations. Under some natural assumptions we show that isotypic 2-representations

Additive versus abelian 2-representations of fiat 2-categories

We study connections between additive and abelian 2-rep- resentations of fiat 2-categories, describe combinatorics of 2-categories in terms of multisemigroups and determine the annihilator of a cell

Cell 2-representations of finitary 2-categories

Abstract We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite-dimensional algebras. In particular, we define, construct and

Gabriel 2‐quivers for finitary 2‐categories

The theory of 2-quivers and quiver 2-categories is developed to run in parallel with the classical theory of quiver algebras and every finitary 2-category is always finitary.