One quiver to rule them all

群环上矩阵的突变

Mutation Of Matrices Over Group Rings

本文研究了群环上斜对称可化矩阵的突变定义及其与具有对称性的有向无环图的突变关系，并提出了新的突变规则。

带符号估值有向图的突变与简单复李代数的呈现

Mutation of signed valued quivers and presentations of simple complex Lie algebras

本文提出带符号估值有向图的突变规则，并通过构造李代数的方式呈现了简单复李代数的结构。

量子箭图的 Leavitt 路代数

Leavitt Path Algebras of Quantum Quivers

本文借鉴图 $C^*$-代数扩展至量子图的工作，引入“量子箭图”概念，发展其相关理论，构造其上的 Leavitt 路代数，并计算了该类代数上有限生成投射模的幺半群。

The Auslander-Reiten quivers of string algebras of affine type $\widetilde{C}$ and a conjecture by Geiss-Leclerc-Schröer

In this paper, we study representations of certain string algebras, which are referred to as of affine type $\widetilde{C}$. We introduce minimal string modules and apply them to explicitly describe components of the Auslander-Reiten quivers of the string algebras and $τ$-locally free modules defined by Geiss-Lerclerc-Schröer. As an application, we prove Geiss-Leclerc-Schröer's conjecture on the correspondence between positive roots of type $\widetilde{C}$ and $τ$-locally free modules of the corresponding string algebras.

Quiver Schur algebras for linear quivers

We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke algebras $\mathcal{R}^Λ_n$ of type $A$ when $e=0$ (the linear quiver) or $e\ge n$. We show that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When $e=0$ we show that the KLR grading on the quiver Hecke algebras is compatible with the gradings on parabolic category $\mathcal{O}$ previously introduced in the works of Beilinson, Ginzburg and Soergel and Backelin. As a consequence, we show that when $e=0$ our graded Schur algebras are Koszul over field of chara…

Path algebras of quivers and representations of locally finite Lie algebras

We explore the (noncommutative) geometry of locally simple representations of the diagonal locally finite Lie algebras $\mathfrak{sl}(n^\infty)$, $\mathfrak o(n^\infty)$, and $\mathfrak{sp}(n^\infty)$. Let $\mathfrak g_\infty$ be one of these Lie algebras, and let $I \subseteq U(\mathfrak g_\infty)$ be the nonzero annihilator of a locally simple $\mathfrak g_\infty$-module. We show that for each such $I$, there is a quiver $Q$ so that locally simple $\mathfrak g_\infty$-modules with annihilator $I$ are parameterised by "points" in the "noncommutative space" corresponding to the path algebra of…