Explanipedia

Deformation theory for finite cluster complexes Open

Nathan Ilten, Alfredo Nájera Chávez, Hipolito Treffinger · 2025

A characterisation of higher torsion classes Open

Jenny August, Johanne Haugland, Karin M. Jacobsen, Sondre Kvamme, Yann Palu , et al. · 2025

Let $\mathcal {A}$ be an abelian length category containing a d -cluster tilting subcategory $\mathcal {M}$ . We prove that a subcategory of $\mathcal {M}$ is a d -torsion class if and only if it is closed under d -extensions and d -quotie…

An algebraic approach to Harder-Narasimhan filtrations Open

Hipolito Treffinger · 2024

On tau-tilting subcategories Open

Javad Asadollahi, Somayeh Sadeghi, Hipolito Treffinger · 2024

The main theme of this paper is to study $\tau $ -tilting subcategories in an abelian category $\mathscr {A}$ with enough projective objects. We introduce the notion of $\tau $ -cotorsion torsion triples and investigate a bijection between…

Stability spaces of string and band modules Open

Sibylle Schroll, Aran Tattar, Hipolito Treffinger, Yadira Valdivieso, Nicholas J. Williams · 2023

The stability space of a module is the cone of vectors which make the module semistable. These cones are defined in terms of inequalities; in this paper we draw insights from considering the dual description in terms of non-negative linear…

Weak stability conditions and the space of chains of torsion classes Open

Aran Tattar, Hipolito Treffinger · 2023

In this paper we show an explicit relation between chains of torsion classes and weak stability conditions over an abelian category. In particular, up to a natural equivalence, they coincide. We investigate topological properties of the sp…

Wall-and-chamber structures for finite-dimensional algebras and $τ$-tilting theory Open

Maximilian Kaipel, Hipolito Treffinger · 2023

The wall-and-chamber structure is a geometric invariant that can be associated to any algebra. In this notes we give the definition of this object and we explain its relationship with torsion classes and $τ$-tilting theory.

A geometric perspective on the $τ$-cluster morphism category Open

Sibylle Schroll, Aran Tattar, Hipolito Treffinger, Nicholas J. Williams · 2023

We show how the $τ$-cluster morphism category may be defined in terms of the wall-and-chamber structure of an algebra. This geometric perspective leads to a simplified proof that the category is well-defined.

A characterisation of higher torsion classes Open

Jenny August, Johanne Haugland, Karin M. Jacobsen, Sondre Kvamme, Yann Palu , et al. · 2023

Let $\mathcal{A}$ be an abelian length category containing a $d$-cluster tilting subcategory $\mathcal{M}$. We prove that a subcategory of $\mathcal{M}$ is a $d$-torsion class if and only if it is closed under $d$-extensions and $d$-quotie…

The size of a stratifying system can be arbitrarily large Open

Hipolito Treffinger · 2023

In this short note we construct two families of examples of large stratifying systems in module categories of algebras. The first examples consist on stratifying systems of infinite size in the module category of an algebra . In the second…

Extending (τ-)tilting subcategories and (co)silting modules Open

Javad Asadollahi, Fazad Padashnik, Somayeh Sadeghi, Hipolito Treffinger · 2023

Let B be a finite dimensional algebra and A=B[P 0 ] be the one-point extension algebra of B with respect to the finitely generated projective B-module P 0. The categories of B-modules and A-modules are related by two adjoint functors R and…

Stability spaces of string and band modules Open

Sibylle Schroll, Aran Tattar, Hipolito Treffinger, Yadira Valdivieso, Nicholas J. Williams · 2022

The stability space of a module is the cone of vectors which make the module semistable. These cones are defined in terms of inequalities; in this paper we draw insights from considering the dual description in terms of non-negative linear…

Stratifying systems and g-vectors Open

Octavio Mendoza, Corina Sáenz, Hipolito Treffinger · 2022

Extending ($τ$-)tilting subcategories and (co)silting modules Open

J. Asadollahi, F. Padashnik, S. Sadeghi, Hipolito Treffinger · 2022

Let $B$ be a finite dimensional algebra and $A=B[P_0]$ be the one-point extension algebra of $B$ with respect to the finitely generated projective $B$-module $P_0$. The categories of $B$-modules and $A$-modules are related by two adjoint f…

On $τ$-tilting subcategories Open

Javad Asadollahi, Somayeh Sadeghi, Hipolito Treffinger · 2022

The main theme of this paper is to study $τ$-tilting subcategories in an abelian category $\mathscr{A}$ with enough projective objects. We introduce the notion of $τ$-cotorsion torsion triples and show a bijection between the collection of…

Characterisations of trivial extensions Open

Elsa Fernández, Sibylle Schroll, Hipolito Treffinger, Sonia Trepode, Yadira Valdivieso · 2022

In this paper we give a characterisation of trivial extension algebras in terms of quivers with relations. This result is based on a explicit description of the ideal of relations of the trivial extension of an algebra, given by the first …

Stability conditions and maximal green sequences in abelian categories Open

Thomas Brüstle, David S. Smith, Hipolito Treffinger · 2022

We study the stability functions on abelian categories introduced by Rudakov and their relation with torsion classes and maximal green sequences.Moreover, we introduce the concept of red paths, a stability condition in the sense of Rudakov…

ON HIGHER TORSION CLASSES Open

Javad Asadollahi, Peter Jørgensen, Sibylle Schroll, Hipolito Treffinger · 2022

Building on the embedding of an n -abelian category $\mathscr {M}$ into an abelian category $\mathcal {A}$ as an n -cluster-tilting subcategory of $\mathcal {A}$ , in this paper, we relate the n -torsion classes of $\mathscr {M}$ with the …

Towards a categorification of scattering amplitudes Open

Severin Barmeier, Prafulla Oak, Aritra Pal, Koushik Ray, Hipolito Treffinger · 2021

Categorification of scattering amplitudes for planar Feynman diagrams in scalar field theories with a polynomial potential is reported. Amplitudes for cubic theories are directly written down in terms of projectives of hearts of intermedia…

Stratifying systems and $g$-vectors Open

Octavio Mendoza, Corina Sáenz, Hipolito Treffinger · 2021

In this paper we study the Cartan matrix associated to the Ext-projective stratifying system induced by a basic and $τ$-rigid object $M$ in mod$(A)$ by means of the $g$-vectors of the indecomposable direct summands of $M$. In particular we…

Deformation Theory for Finite Cluster Complexes Open

Nathan Ilten, Alfredo Nájera Chávez, Hipolito Treffinger · 2021

We study the deformation theory of the Stanley-Reisner rings associated to cluster complexes for skew-symmetrizable cluster algebras of geometric and finite cluster type. In particular, we show that in the skew-symmetric case, these cluste…

$τ$-tilting theory -- An introduction Open

Hipolito Treffinger · 2021

The notion of $τ$-tilting theory was introduced by Adachi, Iyama and Reiten at the beginning of the last decade and quickly became one of the most active areas of research in the representation theory of finite dimensional algebras. The ai…

$\tau$-tilting theory -- An introduction Open

Hipolito Treffinger · 2021

The notion of $\\tau$-tilting theory was introduced by Adachi, Iyama and\nReiten at the beginning of the last decade and quickly became one of the most\nactive areas of research in the representation theory of finite dimensional\nalgebras.…

On band modules and $$\tau $$-tilting finiteness Open

Sibylle Schroll, Hipolito Treffinger, Yadira Valdivieso · 2021

The size of a stratifying system can be arbitrarily large Open

Hipolito Treffinger · 2021

In this short note we construct two families of examples of large stratifying systems in module categories of algebras. The first examples consists on stratifying systems of infinite size in the module category of an algebra $A$. In the se…

The size of an exceptional sequence can be arbitrarily large Open

Hipolito Treffinger · 2021

In this short note we construct two families of examples of large exceptional sequences in module categories of algebras. The first examples consists on exceptional sequences of infinite size in the module category of an algebra $A$. In th…

A $τ$-tilting approach to the first Brauer-Thrall conjecture Open

Sibylle Schroll, Hipolito Treffinger · 2020

In this paper we study the behaviour of modules over finite dimensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the lengt…

A $\tau$-tilting approach to the first Brauer-Thrall conjecture Open

Sibylle Schroll, Hipolito Treffinger · 2020

In this paper we study the behaviour of modules over finite di-mensional algebras whose endomorphism algebra is a division ring. We show that there are finitely many such modules in the module category of an algebra if and only if the leng…

Stratifying Systems through $\tau$-Tilting Theory Open

Octavio Mendoza, Hipolito Treffinger · 2020

In this paper we first show that every non-zero \tau -rigid A -module induces at least one stratifying system in the module category of A . Moreover, we show that each of these stratifying systems can be seen as a signed \tau -exceptional …

On band modules and $τ$-tilting finiteness Open

Sibylle Schroll, Hipolito Treffinger, Yadira Valdivieso · 2019

In this paper, motivated by a $τ$-tilting version of the Brauer-Thrall Conjectures, we study general properties of band modules and their endomorphisms in the module category of a finite dimensional algebra. As an application we describe p…

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