arXiv subjects

Hanpeng Gao

Publications and source records attributed to Hanpeng Gao.

13 recordsLinked to original sources

arXiv2026

Projectively Wakamatsu Tilting Modules over One-Point Extensions

Let $\Gamma = \Lambda[M]$ be the one-point extension of an algebra $\Lambda$ by a $\Lambda$-module $M$. We establish a method to lift projectively Wakamatsu tilting (PWT) modules from $\mathrm{mod}\,\Lambda$ to $\mathrm{mod}\,\Gamma$ by adding the new projective module, and prove that this lifting process perfectly preserves mutation relations under certain homological conditions. Furthermore, for source point extensions of representation-finite algebras, we obtain a complete classification of PWT $\Gamma$-modules in terms of those over $\Lambda$. In particular, we establish a bijection \[ \mathrm{PWT}(\Gamma) \longleftrightarrow \mathrm{PWT}(\Lambda) \coprod \mathrm{RPWT}(\Lambda, S_i). \] which yields the counting formula about $|\mathrm{PWT}(\Gamma)|$.

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arXiv2026

A characterization of IE-closed subcategories via $\tau$-tilting theory

Enomoto and Sakai classified functorially finite IE-closed subcategories over hereditary algebras in terms of twin rigid modules. Their approach uses the hereditary assumption essentially and therefore does not extend directly to arbitrary finite-dimensional algebras. In this paper, we introduce canonical twin support $\tau$-tilting modules and prove that, for an arbitrary finite-dimensional algebra, they are in bijection with left-and-right finite IE-closed subcategories, namely those whose generated torsion and torsion-free classes are both functorially finite. We further give a characterization of canonicality via the torsion-pair decompositions associated with $\operatorname{Fac} M$ and $\operatorname{Sub} N$, which yields a canonicalization procedure whenever the associated IE-closed subcategory is left-and-right finite. We also introduce canonical Ext-pairs. If the algebra is hereditary or $\tau$-tilting finite, then functorially finite IE-closed subcategories are in bijection with isomorphism classes of canonical Ext-pairs, where the corresponding pair is given by the basic Ext-progenerator and the basic Ext-injective cogenerator. In the hereditary case, this recovers the twin rigid classification of Enomoto and Sakai.

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arXiv2025

Some functors preserving exceptionality

We constructed some tensor functors that send each exceptional sequence in a module category to another exceptional sequence in another module category by using split extensions and recollements.

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arXiv2024

Normed modules and The Stieltjes integrations of functions defined on finite-dimensional algebras

We define integrals for functions on finite-dimensional algebras, adapting methods from Leinster's research. This paper discusses the relationships between the integrals of functions defined on subsets $\mathbb{I}_1 \subseteq {\mathitΛ}_1$ and $\mathbb{I}_2 \subseteq {\mathitΛ}_2$ of two finite-dimensional algebras, under the influence of a mapping $ω$, which can be an injection or a bijection. We explore four specific cases: $\bullet$ $ω$ as a monotone non-decreasing and right-continuous function; $\bullet$ $ω$ as an injective, absolutely continuous function; $\bullet$ $ω$ as a bijection; $\bullet$ and $ω$ as the identity on $\mathbb{R}$. These scenarios correspond to the frameworks of Lebesgue-Stieltjes integration, Riemann-Stieltjes integration, substitution rules for Lebesgue integrals, and traditional Lebesgue or Riemann integration, respectively.

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arXiv2024

ICE-closed subcategories and epibricks over one-point extensions

Let $B$ be the one-point extension algebra of $A$ by an $A$-module $M$. We proved that every ICE-closed subcategory in$\mod A$ can be extended to be some ICE-closed subcategories in$\mod B$.In the same way, every epibrick in $\mod A$ can be extended to be some epibricks in $\mod B$.The number of ICE-closed subcategories in $\mod B$ and the number of ICE-closed subcategories in $\mod A$ are denoted respectively as $m$, $n$.We can conclude the following inequality:$$m \geq 2n$$ This is the analogical in epibricks.As an application, we can get some wide $τ$-tilting modules of $B$ by wide $τ$-tilting modules of $A$.

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arXiv2023

Homological Dimensions of Gentle Algebras via Geometric Models

Let $A=kQ/I$ be a finite dimensional basic algebra over an algebraically closed field $k$ which is a gentle algebra with the marked ribbon surface $(\mathcal{S}_A,\mathcal{M}_A,Γ_A)$. It is known that $\mathcal{S}_A$ can be divided into some elementary polygons $\{Δ_i\mid 1\le i\le d\}$ by $Γ_A$ which has exactly one side in the boundary of $\mathcal{S}_A$. Let $\mathfrak{C}(Δ_i)$ be the number of sides of $Δ_i$ belonging to $Γ_A$ if the unmarked boundary component of $\mathcal{S}_A$ is not a side of $Δ_i$; otherwise, $\mathfrak{C}(Δ_i)=\infty$, and let $\mathsf{f}\text{-}Δ$ be the set of all non-$\infty$-elementary polygons and $\mathcal{F}_A$ (respectively, ${\mathsf{f}\text{-}\mathcal{F}}_A$) the set of all forbidden threads (respectively, of finite length). Then we have \begin{enumerate} \item[{\rm (1)}] The global dimension of $A=\max\limits_{1\leq i\leq d}{\mathfrak{C}(Δ_i)}-1 =\max\limits_{\mathitΠ\in\mathcal{F}_A} l(\mathitΠ)$, where $l(\mathitΠ)$ is the length of $\mathitΠ$. \item[{\rm (2)}] The left and right self-injective dimensions of $A=$ \begin{center} $\begin{cases} 0,\ \mbox{\text{if {\it Q} is either a point or an oriented cycle with full relations};}\\ \max\limits_{Δ_i\in{\mathsf{f}\text{-}Δ}}\big\{1, {\mathfrak{C}(Δ_i)}-1 \big\}= \max\limits_{\mathitΠ\in{\mathsf{f}\text{-}\mathcal{F}}_A} l(\mathitΠ),\ \mbox{\text{otherwise}.} \end{cases}$ \end{center} \end{enumerate} As a consequence, we get that the finiteness of the global dimension of gentle algebras is invariant under AG-equivalence. In addition, we get that the number of indecomposable non-projective Gorenstein projective modules over gentle algebras is also invariant under AG-equivalence.

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arXiv2021

$τ$-Tilting modules over one-point extensions by a simple module at a source point

Let $B$ be an one-point extension of a finite dimensional $k$-algebra $A$ by a simple $A$-module at a source point $i$. In this paper, we classify the $τ$-tilting modules over $B$. Moreover, it is shown that there are equations $$|\tilt B|=|\tilt A|+|\tilt A/\langle e_i\rangle|\quad \text{and}\quad |\stilt B|=2|\stilt A|+|\stilt A/\langle e_i\rangle|.$$ As a consequence, we can calculate the numbers of $τ$-tilting modules and support $τ$-tilting modules over linearly Dynkin type algebras whose square radical are zero.

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arXiv2020

Support $τ$-tilting modules over one-point extensions

Let $B$ be the one-point extension algebra of $A$ by an $A$-module $X$. We proved that every support $τ$-tilting $A$-module can be extended to be a support $τ$-tilting $B$-module by two different ways. As a consequence, it is shown that there is an inequality $$|\stilt B|\geqslant 2|\stilt A|.$$

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arXiv2020

A note on the Hasse quiver of $τ$-tilting modules

Let $Λ$ be an algebra with a indecomposable projective-injective module. Adachi gave a method to construct the Hasse quiver of support $τ$-tilting $Λ$-modules. In this paper, we will show that it can be restricted to $τ$-tilting modules.

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arXiv2020

On the Number of $τ$-Tilting Modules over Nakayama Algebras

Let $Λ^r_n$ be the path algebra of the linearly oriented quiver of type $\mathbb{A}$ with $n$ vertices modulo the $r$-th power of the radical, and let $\widetildeΛ^r_n$ be the path algebra of the cyclically oriented quiver of type $\widetilde{\mathbb{A}}$ with $n$ vertices modulo the $r$-th power of the radical. Adachi gave a recurrence relation for the number of $τ$-tilting modules over $Λ^r_n$. In this paper, we show that the same recurrence relation also holds for the number of $τ$-tilting modules over $\widetildeΛ^r_n$. As an application, we give a new proof for a result by Asai on recurrence formulae for the number of support $τ$-tilting modules over $Λ^r_n$ and $\widetildeΛ^r_n$.

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arXiv2020

Extending silted algebras to cluster-tilted algebras

It is well known that the relation-extensions of tilted algebras are cluster-tilted algebras. In this paper, we extend the result to silted algebras and prove some extension of silted algebras are cluster-tilted algebras.

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arXiv2020

Silting Modules over Triangular Matrix Rings

Let $Λ,Γ$ be rings and $R=\left(\begin{array}{cc}Λ& 0 \\ M & Γ\end{array}\right)$ the triangular matrix ring with $M$ a $(Γ,Λ)$-bimodule. Let $X$ be a right $Λ$-module and $Y$ a right $Γ$-module. We prove that $(X, 0)$$\oplus$$(Y\otimes_ΓM, Y)$ is a silting right $R$-module if and only if both $X_Λ$ and $Y_Γ$ are silting modules and $Y\otimes_ΓM$ is generated by $X$. Furthermore, we prove that if $Λ$ and $Γ$ are finite dimensional algebras over an algebraically closed field and $X_Λ$ and $Y_Γ$ are finitely generated, then $(X, 0)$$\oplus$$(Y\otimes_ΓM, Y)$ is a support $τ$-tilting $R$-module if and only if both $X_Λ$ and $Y_Γ$ are support $τ$-tilting modules, $\Hom_Λ(Y\otimes_ΓM,τX)=0$ and $\Hom_Λ(eΛ, Y\otimes_ΓM)=0$ with $e$ the maximal idempotent such that $\Hom_Λ(eΛ, X)=0$.

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arXiv2020

Support $τ$-Tilting Modules under Split-by-Nilpotent Extensions

Let $Γ$ be a split extension of a finite-dimensional algebra $Λ$ by a nilpotent bimodule $_ΛE_Λ$, and let $(T,P)$ be a pair in $\modΛ$ with $P$ projective. We prove that $(T\otimes_ΛΓ_Γ, P\otimes_ΛΓ_Γ)$ is a support $τ$-tilting pair in $\mod Γ$ if and only if $(T,P)$ is a support $τ$-tilting pair in $\mod Λ$ and $\Hom_Λ(T\otimes_ΛE,τT_Λ)=0=\Hom_Λ(P,T\otimes_ΛE)$. As applications, we obtain a necessary and sufficient condition such that $(T\otimes_ΛΓ_Γ, P\otimes_ΛΓ_Γ)$ is support $τ$-tilting pair for a cluster-tilted algebra $Γ$ corresponding to a tilted algebra $Λ$; and we also get that if $T_1,T_2\in\modΛ$ such that $T_1\otimes_ΛΓ$ and $T_2\otimes_ΛΓ$ are support $τ$-tilting $Γ$-modules, then $T_1\otimes_ΛΓ$ is a left mutation of $T_2\otimes_ΛΓ$ if and only if $T_1$ is a left mutation of $T_2$.

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