Yuya Mizuno, Kota Yamaura
We show that $m$-APR tilting preserves $n$-representation infiniteness for $1\leq m\leq n$. Moreover, we show that these tilting modules provide different tilting modules for the corresponding higher preprojective algebras, which is $(n+1)$-CY algebras. We also study the interplay of the two kinds of tilting modules.
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The higher APR tilting modules and higher BB tilting modules were introduced and studied in higher Auslander-Reiten theory. Our objective is to consider these tilting modules by the corresponding simple modules, and show that the tensor product of higher APR (BB) tilting modules is a higher APR (BB) tilting module.
We discuss finiteness/infiniteness of $τ$-tilting modules over tensor products of two symmetric algebras. As an application, we discuss that over block algebras of direct products of finite groups.
We study the quiver with relations of the endomorphism algebra of an APR tilting module. We give an explicit description of the quiver with relations by graded quivers with potential (QPs) and mutations. The result also implies that mutations of QPs with algebraic cuts induce tilting mutations between the associated truncated Jacobian algebras. Consequently, mutations of QPs provide a rich source of derived equivalence classes of algebras. As an application, we give a sufficient condition of QPs such that Derksen-Weyman-Zelevinsky's question has a positive answer.
We introduce the notion of n-representation-finiteness, generalizing representation-finite hereditary algebras. We establish the procedure of n-APR tilting, and show that it preserves n-representation-finiteness. We give some combinatorial description of this procedure, and use this to completely describe a class of n-representation-finite algebras called "type A".
APR tilts for path algebra $kQ$ can be realized as the mutation of the quiver $Q$ in $\mathbb Z Q$ with respect to the translation. In this paper, we show that we have similar results for the quadratic dual of truncations of $n$-translation algebras, that is, under certain condition, the $n$-APR tilts of such algebras are realized as $τ$-mutations.For the dual $τ$-slice algebras with bound quiver $Q^{\perp}$, we show that their iterated $n$-APR tilts are realized by the iterated $τ$-mutations in $\mathbb Z|{n-1}Q^{\perp}$.
A finite directed category is a $k$-linear category with finitely many objects and an underlying poset structure, where $k$ is an algebraically closed field. This concept unifies structures such as $k$-linerizations of posets and finite EI categories, quotient algebras of finite-dimensional hereditary algebras, triangular matrix algebras, etc. In this paper we study representations of finite directed categories, discuss their stratification properties, and show the existence of generalized APR tilting modules for triangular matrix algebras under some assumptions.
We study the relationship between $n$-cluster tilting modules over $n$ representation finite algebras and the Euler forms. We show that the dimension vectors of cluster-indecomposable modules give the roots of the Euler form. Moreover, we show that cluster-indecomposable modules are uniquely determined by their dimension vectors. This is a generalization of Gabriel's theorem by cluster tilting theory. We call the above roots cluster-roots and investigate their properties. Furthermore, we provide the description of quivers with relations of $n$-APR tilts. Using this, we provide a generalization…
We treat the $τ$-tilting finiteness of those minimal representation-infinite (min-rep-infinite) algebras which are non-distributive. Building upon the new results of Bongartz, we fully determine which algebras in this family are $τ$-tilting finite and which ones are not. This complements our previous work in which we carried out a similar analysis for the min-rep-infinite biserial algebras. Consequently, we obtain nontrivial explicit sufficient conditions for $τ$-tilting infiniteness of a large family of algebras. This also produces concrete families of "minimal $τ$-tilting infinite algebras"-…
Jacobian Algebras of Species with Potentials and 2-Representation Finite Algebras
本研究围绕带势物种的雅可比代数展开，证明其截断集合在截断突变下的传递性，明确截断、截断突变分别对应截断雅可比代数与2-APR倾斜，构造了新的2-表示有限$\u211a$-代数实例。
We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting modules correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demon…
$τ$-Tilting finiteness of group algebras over generalized symmetric groups
本研究通过证明弱对称τ-倾斜有限代数的Cartan矩阵正定性，得到自内射代数τ-倾斜无限的矩阵条件，并将其应用于广义对称群的群代数，给出τ-倾斜无限的判定条件及p'-子群情形下的决定因素。
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation theory of hereditary algebras. As an application of this, we prove a generalised version of so-called APR-tilting.
Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object $T$ in a hereditary abelian category $\mathcal{H}$, we verify that the tilting functor Hom$_\mathcal{H}(T,-)$ induces a triangle equivalence from the cluster category $\mathcal{C(H)}$ to the cluster category $\mathcal{C}(A)$, where $A$ is the quasi-tilted algebra End$_{\mathcal{H}}T.$ Under the condition that one of derived categories of hereditary abelian categories $\mathc…
Coxeter-Dynkin algebras of canonical type
本文提出Coxeter-Dynkin代数的标准类型定义，并通过构造倾斜对象证明其终止代数与该代数一致，同时展示了其与Saito分类的联系以及在domestic情况下有限表示类型的结论。
Higher torsion classes, $τ_d$-tilting theory and silting complexes
本文研究了高维 $τ_d$-tilting 理论，通过 $τ_d$-rigid 对和 $(d+1)$-项 silting 复合体揭示了 $d$-torsion 类与 $d$-exact 类别的关系。
The aim of this paper is mainly to build a new representation-theoretic realization of finite root systems through the so-called Frobenius-type triangular matrix algebras by the method of reflection functors over any field. Finally, we give an analog of APR-tilting module for this class of algebras. The major conclusions contains the known results as special cases, e.g. that for path algebras over an algebraically closed field and for path algebras with relations from symmetrizable cartan matrices. Meanwhile, it means the corresponding results for some other important classes of algebras, that…
Higher representation infinite algebras and toric Fano stacks of Picard number one or two
本文证明了Picard数一或二的光滑环面Fano栈上由线丛构成的d-倾斜丛的存在性并进行了分类，其自同态代数给出了d-表示无限代数的自然例子，并建立了与dimer模型的联系。
A classification of $n$-representation infinite algebras of type Ã
本文通过高阶预投射代数与高度函数对Ã型n-表示无限代数进行完整分类，证明同类型代数经迭代n-APR倾斜导出等价，并将其组织为有限分配格。
On compatibility of Koszul- and higher preprojective gradings
本文研究Koszul或几乎Koszul代数的高预投射分次与（几乎）Koszul分次的相容性，证明在$n$-表示有限或无限情形下，可赋予相应分次蕴含原代数Koszul性，且分次同构于截断，并应用于$n$-APR倾斜保持Koszul性质。
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