Singularity categories of skewed-gentle algebras

偏斜-温和代数表示模空间

Moduli of Representations of Skewed-Gentle Algebras

本文证明了偏斜-温和代数（更一般地，宗族代数）的半稳定表示模空间的不可约分支同构于射影空间的乘积，通过将宗族代数表示簇的不可约分支视为偏斜-温和代数的不可约分支并证明其正规性，推广了Carroll-Chindris-Kinser-Weyman关于特殊双列代数表示模空间的类似结果。

关于斜 gentle 代数的同态与一般 $\tau$-正则不可约分支

On homomorphisms and generically $τ$-regular components for skewed-gentle algebras

本文针对斜 gentle 代数，利用容许词给出同态基，推广 fringe 与 kiss，得到 E-不变量和 g-向量公式，并描述一般 τ-正则不可约分支。

q-Cartan matrices and combinatorial invariants of derived categories for skewed-gentle algebras

Cartan matrices are of fundamental importance in representation theory. For algebras defined by quivers (i.e. directed graphs) with relations the computation of the entries of the Cartan matrix amounts to counting nonzero paths in the quivers, leading naturally to a combinatorial setting. In this paper we study a refined version, so-called q-Cartan matrices, where each nonzero path is weighted by a power of an indeterminant q according to its length. Specializing q=1 gives the classical Cartan matrix. Our main motivation are derived module categories and their invariants: the invariant factors…

Derived tame Nakayama algebras

We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.

Higher gentle algebras

We introduce higher gentle algebras. Our definition allows us to determine the singularity categories and subsequently show that higher gentle algebras are Iwanaga-Gorenstein. Under extra assumptions, we show that cluster-tilted algebras (in the sense of Oppermann-Thomas) of higher Auslander algebras of type $A$ are higher gentle.

Singularity categories of gentle algebras

We determine the singularity category of an arbitrary finite dimensional gentle algebra $Λ$. It is a finite product of $n$-cluster categories of type $\mathbb{A}_{1}$. Equivalently, it may be described as the stable module category of a selfinjective gentle algebra. If $Λ$ is a Jacobian algebra arising from a triangulation $\ct$ of an unpunctured marked Riemann surface, then the number of factors equals the number of inner triangles of $\ct$.