# Finitary 2-categories associated with dual projection functors

@article{Grensing2014Finitary2A, title={Finitary 2-categories associated with dual projection functors}, author={Anna-Louise Grensing and Volodymyr Mazorchuk}, journal={arXiv: Representation Theory}, year={2014} }

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 quivers of type A) we show that the monoid generated by dual projection functors is the Hecke-Kiselman monoid of the underlying quiver and also obtain a presentation for the monoid of indecomposable subbimodules of the identity bimodule.

## Figures from this paper

## 14 Citations

Bimodules over uniformly oriented A n quivers with radical square zero

- MathematicsKyoto Journal of Mathematics
- 2020

We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly…

Sub-bimodules of the identity bimodule for cyclic quivers

- Mathematics
- 2017

We describe the combinatorics of the multisemigroup with multiplicities for the tensor category of subbimodules of the identity bimodule, foran arbitrary non-uniform orientation of a finite cyclic ...

Diagrams and discrete extensions for finitary 2-representations

- Mathematics, Computer ScienceMathematical Proceedings of the Cambridge Philosophical Society
- 2017

This paper introduces and investigates the notions of diagrams and discrete extensions in the study of finitary 2-representations of finitarian 2-categories.

Cell structure of bimodules over radical square zero Nakayama algebras

- MathematicsCommunications in Algebra
- 2019

Abstract In this paper, we describe the combinatorics of the cell structure of the tensor category of bimodules over a radical square zero Nakayama algebra. This accounts to an explicit description…

Simple transitive $2$-representations and Drinfeld center for some finitary $2$-categories

- Mathematics
- 2015

We classify all simple transitive $2$-representations for two classes of finitary $2$-categories associated with tree path algebras and also for one class of fiat $2$-categories associated with…

Simple transitive $2$-representations of Soergel bimodules in type $B_2$

- Mathematics
- 2015

We prove that every simple transitive $2$-representation of the fiat $2$-category of Soergel bimodules (over the coinvariant algebra) in type $B_2$ is equivalent to a cell $2$-representation. We also…

Multisemigroups with multiplicities and complete ordered semi-rings

- Mathematics
- 2015

Motivated by the appearance of multisemigroups in the study of additive 2-categories, we define and investigate the notion of a multisemigroup with multiplicities. This notion seems to be well suited…

Simple Transitive 2-Representations for Two Nonfiat 2-Categories of Projective Functors

- MathematicsUkrainian Mathematical Journal
- 2019

We show that any simple transitive $2$-representation of the $2$-ca\-te\-go\-ry of projective endofunctors for the quiver algebra of $\Bbbk(\xymatrix{\bullet\ar[r]&\bullet})$ and for the quiver…

ANALOGUES OF CENTRALIZER SUBALGEBRAS FOR FIAT 2-CATEGORIES AND THEIR 2-REPRESENTATIONS

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2018

The main result of this paper establishes a bijection between the set of equivalence classes of simple transitive 2-representations with a fixed apex ${\mathcal{J}}$ of a fiat 2-category…

Semigroups, multisemigroups and representations

- Mathematics
- 2017

This thesis consists of four papers about the intersection between semigroup theory, category theory and representation theory. We say that a representation of a semigroup by a matrix semigroup is ...

## References

SHOWING 1-10 OF 19 REFERENCES

Monoid algebras of projection functors

- Mathematics
- 2012

We study the monoid of so called projection functors $\p{S}$ attached to simple modules $S$ of a finite dimensional algebra, which appear naturally in the study of torsion pairs. We determine…

CATEGORIFICATION OF THE CATALAN MONOID

- 2013

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.…

Categorification of the Catalan monoid

- Mathematics
- 2012

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.

Cell 2-representations of finitary 2-categories

- MathematicsCompositio Mathematica
- 2011

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…

Additive versus abelian 2-representations of fiat 2-categories

- Mathematics
- 2011

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…

Derived equivalences for symmetric groups and sl2-categorification

- Mathematics
- 2004

We define and study sl2-categorifications on abelian categories. We show in particular that there is a self-derived (even homotopy) equivalence categorifying the adjoint action of the simple…

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

- Mathematics
- 2014

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…

On Kiselman quotients of 0-Hecke monoids

- Mathematics
- 2010

Combining the definition of 0-Hecke monoids with that of Kiselman semigroups, we define what we call Kiselman quotients of 0-Hecke monoids associated with simply laced Dynkin diagrams. We classify…

On Arkhipov’s and Enright’s functors

- Mathematics
- 2005

Abstract.We give a description of Arkhipov’s and (Joseph’s and Deodhar-Mathieu’s versions of) Enright’s endofunctors on the category associated with a fixed triangular decomposition of a complex…

A categorification of quantum sl(n)

- Mathematics
- 2010

To an arbitrary root datum we associate a2-category. For root datum corresponding to sl.n/ we show that this 2-category categorifies the idempotented form of the quantum enveloping algebra.