Transitive 2-representations of finitary 2-categories

@article{Mazorchuk2014Transitive2O,
  title={Transitive 2-representations of finitary 2-categories},
  author={Volodymyr Mazorchuk and Vanessa Miemietz},
  journal={Transactions of the American Mathematical Society},
  year={2014},
  volume={368},
  pages={7623-7644}
}
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 composition subquotients are given by simple transitive $2$-representations. For a large class of finitary $2$-categories we prove that simple transitive $2$-representations are exhausted by cell $2$-representations. Finally, we show that this large class contains finitary quotients of $2$-Kac-Moody… 

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References

SHOWING 1-10 OF 33 REFERENCES

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

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

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

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.

Quiver Hecke Algebras and 2-Lie Algebras

We provide an introduction to the 2-representation theory of Kac-Moody algebras, starting with basic properties of nil Hecke algebras and quiver Hecke algebras, and continuing with the resulting

Categorification of highest weight modules via Khovanov-Lauda-Rouquier algebras

In this paper, we prove Khovanov-Lauda’s cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let $U_{q}(\mathfrak{g})$ be the quantum group associated with a

A categorification of the Temperley-Lieb algebra and Schur quotients of $ U({\frak sl}_2) $ via projective and Zuckerman functors

Author(s): Bernstein, Joseph; Frenkel, Igor; Khovanov, Mikhail | Abstract: We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with

Implicit structure in 2-representations of quantum groups

Given a strong 2-representation of a Kac–Moody Lie algebra (in the sense of Rouquier), we show how to extend it to a 2-representation of categorified quantum groups (in the sense of Khovanov–Lauda).

Classical Finite Transformation Semigroups: An Introduction

Ordinary and Partial Transformations.- The Semigroups T n, PT n, and IS n.- Generating Systems.- Ideals and Green ' s Relations.- Subgroups and Subsemigroups.- Other Relations on Semigroups.-