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Yu-Zhe Liu

Publications and source records attributed to Yu-Zhe Liu.

At least 19 recordsLinked to original sources
arXiv2026

Kleisli convolution representations and a Margolis--Sakurai version of the modular isomorphism problem

We introduce Kleisli convolution representations for groups, rings, and algebras. We show that their representation categories are equivalent to the usual ones for groups and finite-dimensional algebras, but for rings only recover modules whose underlying Abelian groups are free of finite rank. We also apply the Kleisli convolution representation to provide a partial answer to the Margolis--Sakurai version of the modular isomorphism problem.

math.GR↗
arXiv2026

Integral coefficient rings and homological dimensions of algebras

We define the integral profiles of all modules and introduce integral coefficient rings ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)$ for all finite-dimensional complex algebras $A$. The integral profile of a module is a matrix with parameters. We provide a classification theorem for modules, to be precise, (1) two modules $M\cong N$ are isomorphic if and only if their integral profiles are similar, i.e., $M\cong N$ if and only if $\displaystyle \int M \sim \int N$. That is, the integral profile is a complete invariant of finite-dimensional modules. Furthermore, we show the following results in this paper: (2) we introduce the central integrals of algebras and show that it is isomorphic to the center of algebras; (3) we provide a descriptions for some special modules; (4) integral coefficient ring of $A$ (with a compatible orthogonal fixed embedding system) has Morita invariance; (5) the global dimension of $A$ is finite if and only if the embedded integral profile of $\mathrm{top}(A)$ lies in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$; (6) the finitistic dimension of $A$ is finite if and only if each embedded integral profile of $M$ lying in ${^{\mathscr{P}}\mkern-6.5mu\text{\&}\mkern-5.5mu{_\mathscr{I}}}(A)[x]$ implies that its degree is less than or equal to a fixing integer $d\in\mathbb{N}^+$.

math.RA↗
arXiv2026

Auslander-Reiten (n+2)-angles and local finiteness

Let $\mathcal C$ be an $(n+2)$-angulated category. Zhou proved that, when $n$ is odd, if the Auslander-Reiten $(n+2)$-angles generate the relations for the Grothendieck group of $\mathcal C$, then $\mathcal C$ is locally finite. Whether the corresponding statement remains valid for even $n$ is still open. In this paper, we give a partial affirmative answer to this problem by establishing a sufficient condition under which the same implication holds for even $n$. We further show that our sufficient condition is satisfied by a broad class of examples, thereby demonstrating that the result extends well beyond isolated cases.

math.RT↗
arXiv2026

Kleisli convolution representations of power monoids

We show that power semigroups of groups, and more generally reduced finitary power monoids, arise naturally as convolution monoids in Kleisli categories of powerset monads: (1) for the non-empty powerset monad, the Kleisli Hom-space $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_+)}(1,G)$ is isomorphic to the power monoid $\mathcal P_+(G)$; (2) for the reduced finite powerset monad on pointed sets, the Kleisli Hom-space $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_{\mathrm{fin}})}$ $(\mathbb Z/2\mathbb Z,H)$ is isomorphic to the reduced finitary power monoid $\mathcal P_{\mathrm{fin},1}(H)$. This unifies several constructions in power semigroup theory: Kleisli convolution representations of semigroups, base change along surjective group homomorphisms, and rigidity of automorphism groups. As an application, we prove that for every proper numerical monoid $S$, the Kleisli Hom-monoid $\mathrm{Hom}_{\mathbf{Kl}(\mathscr P_{\mathrm{fin}})}(\mathbb Z/2\mathbb Z,S)$ is rigid. This gives a proof of the Tringali--Yan conjecture in the language of Kleisli categories. It should be mentioned that this conjecture was already proved by Bhowmik and Tringali in a preprint posted on arXiv on July 25, 2026.

math.GR↗
arXiv2026

Semi-orthogonal and derived decompositions for gentle algebras

We study semi-orthogonal decompositions of perfect derived categories of gentle algebras via marked ribbon surfaces. We characterize such decompositions in terms of suitable disjoint union decompositions of full formal arc systems, and relate this description to good cuts of the corresponding surfaces. For gentle algebras, rotations of curves induce fully faithful functors from extension-closed subcategories of module categories to the components of the associated semi-orthogonal decompositions. Under additional Abelian and extension-comparison conditions, these constructions give derived decompositions of the module categories.

math.CT↗
arXiv2026

Support $\tau$-tilting posets and Hochschild reconstruction for matrix centralizer algebras

Let $R$ be a field and $A$ the endomorphism algebra of a finite direct sum of cyclic modules over a finite-dimensional commutative local principal ideal $R$-algebra. We construct a central quotient showing that the support $\tau$-tilting poset of $A$ is isomorphic to the poset of the symmetric group with the weak order. We show that the center of $A$ and degree-zero Hochschild homology, viewed as a module over the center, determine the truncated local algebra and the multiset of successive length gaps. For centralizer matrix algebras, the support $\tau$-tilting poset determines the multiset of distinct-exponent counts of the primary blocks. The corresponding algebra--module pair also recovers their local algebras and gap multisets. Combining this reconstruction with the known derived equivalence classification, we characterize derived equivalence by isomorphism of these algebra-module pairs. We apply the results to Morita reconstruction in the string and gentle classes.

math.RA↗
arXiv2026

Quasi-abelian quotients in extriangulated categories

Let $(\mathcal{E}, \mathbb{E}, \mathfrak{s})$ be an extriangulated category. Motivated by the theory of hereditary algebras, we introduce the notion of a hereditary-type subcategory $\mathcal{W}\subseteq \mathcal{E}$. We prove that the quotient $\mathcal{E}/\mathcal{W}$ is a quasi-abelian category, that is, an additive category with kernels and cokernels in which kernels are stable under pushouts and cokernels are stable under pullbacks. Moreover, we show that $\mathcal{E}/\mathcal{W}$ is abelian if and only if $\mathcal{W}$ is a cluster tilting subcategory in a suitable relative extriangulated structure. Several examples are provided to illustrate the main results, showing that our approach both recovers known abelian hearts and yields new abelian or quasi-abelian quotients beyond classical settings.

math.RT↗
arXiv2026

Module-valued ordinary differential equations and structure of solution spaces

We define and study ordinary differential equations (ODEs) for functions valued in a Banach module $V$ over a finite-dimensional $\Bbbk$-algebra $\mathit{\Lambda}$ by using the tensor of Banach modules. Furthermore, we show that the solution space of a homogeneous linear ODE as above is shown to be a finitely generated $\mathit{\Lambda}$-submodule.

math.FA↗
arXiv2026

Recollements of Cohen-Macaulay Auslander algebras for gentle algebras

We construct two recollements of module categories for the Cohen--Macaulay Auslander algebra $A^{\mathrm{CMA}}$ of a gentle algebra $A$. In this paper, we establish three equivalent characterizations for the quotient algebra $A^{\mathrm{CMA}}/A^{\mathrm{CMA}}(1-\epsilon_{\star}) A^{\mathrm{CMA}}$ of the CM--Auslander algebra of $A$ to be quasi-tilted, precisely, the following statements are equivalent: (1) $A^{\mathrm{CMA}}/A^{\mathrm{CMA}}(1-\epsilon_{\star}) A^{\mathrm{CMA}}$ is quasi-tilted; (2) $\mathrm{findim} A\leqslant 2$, and for each forbidden $A$-module $M$, $\mathrm{proj.dim}M+\mathrm{inj.dim}M\leqslant 2$; (3) for any homotopy string/band $\mathsf{h}$ none of whose arrows lie on any forbidden cycle, the cohomological width of the indecomposable object in $\mathsf{D}^b(A)$ corresponding to $\mathsf{h}$ is $\leqslant 2$. Moreover, we prove that the Krull--Gabriel dimension of $A$ is bounded by 2 if and only if the Krull--Gabriel dimension of $A^{\mathrm{CMA}}$ is bounded by 2 in the case where $A$ is gentle one-cycle.

math.RT↗
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.

math.RT↗
arXiv2026

Igusa-Todorov properties of recollements of abelian categories

In this paper, we investigate the behavior of Igusa-Todorov properties under recollements of abelian categories. In particular, we study how the Igusa-Todorov distances of the categories involved in a recollement are related. Applications are given to Artin algebras, especially to Morita context rings.

math.RT↗
arXiv2025

Homological Bounds of Gentle algebras

This paper studies the homological bounds of gentle algebras, i.e., the upper bounds for the sum of the projective and injective dimensions of indecomposable modules over gentle algebras. We provide conditions under which this sum is strictly less than twice the global dimension, and as an application, we give a characterization of quasi-tilted gentle algebras.

math.RT↗
arXiv2025

Normed representations of weight quivers

Let $A$ and $B$ be two tensor rings given by weight quivers. We introduce norms for tensor rings and $(A,B)$-bimodules, and define an important category $\mathscr{A}^p_{\varsigma}$ in this paper whose object is a triple $(N,v,\delta)$ given by an $(A,B)$-bimodule $N$, a special element $v\in V$ satisfying some special conditions, and a special $(A,B)$-homomorphism $\delta: N^{\oplus_p 2^{\dim A}} \to N$ and each morphism $(N,v,\delta) \to (N',v',\delta')$ is given by an $(A,B)$-homomorphism $\theta: N\to N'$ such that $\theta(v)=v'$ and $\delta' \theta^{\oplus 2^{\dim A}} = \theta\delta$ hold. We show that $\mathscr{A}^p_{\varsigma}$ has an initial object such that Daniell integration, Bochner integration, Lebesgue integration, Stone--Weierstrass Approximation Theorem, power series expansion, and Fourier series expansion are morphisms in $\mathscr{A}^p_{\varsigma}$ starting with this initial object.

math.RT↗
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.

math.RT↗
arXiv2025

On the Gorensteiness of string algebras

In this paper, we give a description of the self-injective dimension of string algebras and obtain a necessary and sufficient condition for a string algebra to be Gorenstein.

math.RT↗
arXiv2025

Homological dimensions over almost gentle algebras

We provide a method for computing the global dimension and self-injective dimension of almost gentle algebras,and prove that an almost gentle algebra is Gorenstein if it satisfies the Auslander condition.

math.RT↗
arXiv2024

Normed modules, integral sequences, and integrals with variable upper limits

This paper provides a new categorification of the Lebesgue integral with variable upper limits by using normed modules over finite-dimensional $\Bbbk$-algebras $\mathit{\Lambda}$ and the category $\mathscr{A}^p_{\mathit{\Lambda}}$ associated with $\mathit{\Lambda}$. The integration process is redefined through the introduction of an integral partially ordered set and an abstract integral with variable upper limits. Finally, we present two important applications: (1) the categorification of basic elementary functions, including (anti-)trigonometric and logarithmic functions, and (2) a new approach for characterizing the global dimensions of gentle algebras.

math.CT↗