Aaron Chan

Bio: Aaron Chan is an academic researcher from Nagoya University. The author has contributed to research in topics: Injective function & Symmetric group. The author has an hindex of 7, co-authored 24 publications receiving 175 citations. Previous affiliations of Aaron Chan include Uppsala University & University of Aberdeen.

Classification of two-term tilting complexes over Brauer graph algebras

Abstract: Using only the combinatorics of its defining ribbon graph, we classify the two-term tilting complexes, as well as their indecomposable summands, of a Brauer graph algebra. As an application, we determine precisely the class of Brauer graph algebras which are tilting-discrete.

Classification of two-term tilting complexes over Brauer graph algebras

Abstract: Using only the combinatorics of its defining ribbon graph, we classify the two-term tilting complexes, as well as their indecomposable summands, of a Brauer graph algebra. As an application, we determine precisely the class of Brauer graph algebras which are tilting-discrete.

Diagrams and discrete extensions for finitary 2-representations

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.

Diagrams and discrete extensions for finitary 2-representations

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.

I and i

Abstract: There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality. Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

$\boldsymbol{\tau}$-Tilting Finite Algebras, Bricks, and $\boldsymbol{g}$-Vectors

Abstract: The class of support $\\tau$-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study $\\tau$-tilting finite algebras, i.e. finite dimensional algebras $A$ with finitely many isomorphism classes of indecomposable $\\tau$-rigid modules. We show that $A$ is $\\tau$-tilting finite if and only if very torsion class in $\\mod A$ is functorially finite. We observe that cones generated by $g$-vectors of indecomposable direct summands of each support $\\tau$-tilting module form a simplicial complex $\\Delta(A)$. We show that if $A$ is $\\tau$-tilting finite, then $\\Delta(A)$ is homeomorphic to an $(n-1)$-dimensional sphere, and moreover the partial order on support $\\tau$-tilting modules can be recovered from the geometry of $\\Delta(A)$. Finally we give a bijection between indecomposable $\\tau$-rigid $A$-modules and bricks of $A$ satisfying a certain finiteness condition, which is automatic for $\\tau$-tilting finite algebras.

$\tau$-tilting finite algebras, bricks and $g$-vectors

Abstract: The class of support $\tau$-tilting modules was introduced to provide a completion of the class of tilting modules from the point of view of mutations. In this article we study $\tau$-tilting finite algebras, i.e. finite dimensional algebras $A$ with finitely many isomorphism classes of indecomposable $\tau$-rigid modules. We show that $A$ is $\tau$-tilting finite if and only if very torsion class in $\mod A$ is functorially finite. We observe that cones generated by $g$-vectors of indecomposable direct summands of each support $\tau$-tilting module form a simplicial complex $\Delta(A)$. We show that if $A$ is $\tau$-tilting finite, then $\Delta(A)$ is homeomorphic to an $(n-1)$-dimensional sphere, and moreover the partial order on support $\tau$-tilting modules can be recovered from the geometry of $\Delta(A)$. Finally we give a bijection between indecomposable $\tau$-rigid $A$-modules and bricks of $A$ satisfying a certain finiteness condition, which is automatic for $\tau$-tilting finite algebras.

Classifying tilting complexes over preprojective algebras of Dynkin type

Abstract: We study support τ-tilting modules over preprojective algebras of Dynkin type. We classify basic support τ-tilting modules by giving a bijection with elements in the corre- sponding Weyl groups. Moreover we show that they are in bijection with the set of torsion classes, the set of torsion-free classes and many other important objects in representation theory. We also study g-matrices of support τ-tilting modules, which show terms of minimal projective presentations of indecomposable direct summands. We give an explicit descrip- tion of g-matrices and prove that cones given by g-matrices coincide with chambers of the associated root systems.

Lattice theory of torsion classes

Abstract: The aim of this paper is to establish a lattice theoretical framework to study the partially ordered set $\operatorname{\mathsf{tors}} A$ of torsion classes over a finite-dimensional algebra $A$. We show that $\operatorname{\mathsf{tors}} A$ is a complete lattice which enjoys very strong properties, as bialgebraicity and complete semidistributivity. Thus its Hasse quiver carries the important part of its structure, and we introduce the brick labelling of its Hasse quiver and use it to study lattice congruences of $\operatorname{\mathsf{tors}} A$. In particular, we give a representation-theoretical interpretation of the so-called forcing order, and we prove that $\operatorname{\mathsf{tors}} A$ is completely congruence uniform. When $I$ is a two-sided ideal of $A$, $\operatorname{\mathsf{tors}} (A/I)$ is a lattice quotient of $\operatorname{\mathsf{tors}} A$ which is called an algebraic quotient, and the corresponding lattice congruence is called an algebraic congruence. The second part of this paper consists in studying algebraic congruences. We characterize the arrows of the Hasse quiver of $\operatorname{\mathsf{tors}} A$ that are contracted by an algebraic congruence in terms of the brick labelling. In the third part, we study in detail the case of preprojective algebras $\Pi$, for which $\operatorname{\mathsf{tors}} \Pi$ is the Weyl group endowed with the weak order. In particular, we give a new, more representation theoretical proof of the isomorphism between $\operatorname{\mathsf{tors}} k Q$ and the Cambrian lattice when $Q$ is a Dynkin quiver. We also prove that, in type $A$, the algebraic quotients of $\operatorname{\mathsf{tors}} \Pi$ are exactly its Hasse-regular lattice quotients.