Cluster algebras and triangulated orbifolds

Quantum cluster algebras from unpunctured triangulated surfaces: arbitrary coefficients and quantization

We study quantum cluster algebras from unpunctured surfaces with arbitrary coefficients and quantization. We first give a new proof of the Laurent expansion formulas for commutative cluster algebras from unpunctured surfaces, we then give the quantum Laurent expansion formulas for the quantum cluster algebras. Particularly, this gives a combinatorial proof of the positivity for such class of quantum cluster algebras.

Bases for cluster algebras from orbifolds with one marked point

We generalize the construction of the bangle, band and the bracelet bases for cluster algebras from orbifolds to the case where there is only one marked point on the boundary.

Cluster algebras, quiver representations and triangulated categories

This is an introduction to some aspects of Fomin-Zelevinsky's cluster algebras and their links with the representation theory of quivers and with Calabi-Yau triangulated categories. It is based on lectures given by the author at summer schools held in 2006 (Bavaria) and 2008 (Jerusalem). In addition to by now classical material, we present the outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams (details will appear elsewhere) and recent results on the interpretation of mutations as derived equivalences.

来自三角化曲面的分母向量与维数向量

Denominator vectors and dimension vectors from triangulated surfaces

本文研究了来自三角化曲面的簇代数中簇变量的分母向量与对应模维数向量的一致性问题，通过比较带标记弧的两种相交数，推广了Buan、Marsh和Reiten在无环簇代数中的结果。

来自三角剖分曲面的丛代数的 $F$-矩阵

$F$-matrices of cluster algebras from triangulated surfaces

本文证明标记曲面的带标记三角剖分可由弧间相交数唯一确定，并由此推出丛代数中每个丛可由其 $F$-矩阵唯一刻画。

Positivity for quantum cluster algebras from unpunctured orbifolds

We give the quantum Laurent expansion formula for the quantum cluster algebras from unpunctured orbifolds with arbitrary coefficients and quantization. As an application, positivity for such class of quantum cluster algebras is given. For technical reasons, it will always be assumed that the weights of the orbifold points are 2.

Bases for cluster algebras from orbifolds

We generalize the construction of the bracelet and bangle bases defined by Musiker, Schiffler and Williams, and the band basis defined by D. Thurston to cluster algebras arising from orbifolds. We prove that the bracelet bases are positive, and the bracelet basis for the affine cluster algebra of type $C_n^{(1)}$ is atomic. We also show that cluster monomial bases of all skew-symmetrizable cluster algebras of finite type are atomic.

局部有限三角范畴中的高阶簇倾斜对象

Higher cluster tilting objects in locally finite triangulated categories

研究通过遗传代数导出范畴的覆盖函子探讨高阶簇倾斜对象，覆盖形式可将存在性问题转化，对有限不可分解对象三角范畴给出存在准则与公式，证明有限Frobenius模型中相关自同态代数导出等价，还应用于多个领域。

来自轨道流形的量子簇代数的正性

Positivity for quantum cluster algebras from orbifolds

针对源自标记轨道流形的具任意系数与量子化的量子簇代数，提出了其量子簇变量关于任意量子种子的量子Laurent展开组合公式，并由此证明了该代数的正性。

簇代数与闭轨道面关联的极大绿色序列

Maximal Green Sequences for Cluster Algebras Associated to Closed Orbifolds

本文研究了闭轨道面关联的簇代数中极大绿色序列的存在条件，通过构造三角剖分来确定其存在性。

The Large $N$ Limit of Orbifold Vertex Operator Algebras

We investigate the large $N$ limit of permutation orbifolds of vertex operator algebras. To this end, we introduce the notion of nested oligomorphic permutation orbifolds and discuss under which conditions their fixed point VOAs converge. We show that if this limit exists, then it has the structure of a vertex algebra. Finally, we give an example based on $\mathrm{GL}(N,q)$ for which the fixed point VOA limit is also the limit of the full permutation orbifold VOA.

三角剖分曲面上的τ-刚性模的维数向量与簇单项式的f-向量

Dimension vectors of $τ$-rigid modules and $f$-vectors of cluster monomials from triangulated surfaces

本文针对三角剖分曲面相关的簇代数与Jacobian代数，利用Qiu-Zhou相交数工具，刻画了不同非初始簇单项式具有不同f-向量以及不同τ-刚性模具有不同维数向量的条件，并由此证明了特定曲面情形下的分母猜想。

Gentle algebras arising from surfaces with orbifold points of order 3, Part I: scattering diagrams

To each triangulation of any surface with marked points on the boundary and orbifold points of order three, we associate a quiver (with loops) with potential whose Jacobian algebra is finite dimensional and gentle. We study the stability scattering diagrams of such gentle algebras and use them to prove that the Caldero--Chapoton map defines a bijection between reachable $τ$-rigid pairs and cluster monomials of the generalized cluster algebra associated to the surface by Chekhov and Shapiro.

格顶点算子代数的循环轨道折叠的额外自同构

Extra automorphisms of cyclic orbifolds of lattice vertex operator algebras

本文研究了无根偶格顶点算子代数在奇素数阶无不动点等距提升下的循环轨道折叠自同构群，证明了额外自同构存在的充要条件是该格由 Construction B 构造或同构于 Leech 格的余不变格。

Orbifold diagrams

We study alternating strand diagrams on the disk with an orbifold point. These are quotients by rotation of Postnikov diagrams on the disk, and we call them orbifold diagrams. We associate a quiver with potential to each orbifold diagram, in such a way that its Jacobian algebra and the one associated to the covering Postnikov diagram are related by a skew-group algebra construction. We moreover realise this Jacobian algebra as the endomorphism algebra of a certain explicit cluster-tilting object. This is similar to (and relies on) a result by Baur-King-Marsh for Postnikov diagrams on the disk.