Suppose that $Q$ is a finite quiver and $G\subseteq \Aut(Q)$ is a finite group, $k$ is an algebraic closed field whose characteristic does not divide the order of $G$. For any algebra $Λ=kQ/{\mathcal {I}}$, $\mathcal {I}$ is an arbitrary ideal of path algebra $kQ$, we give all the indecomposable $ΛG$-modules from indecomposable $Λ$-modules when $G$ is abelian. In particular, we apply this result to the deformed preprojective algebra $Π_{Q}^λ$, and get a reflection functor for the module category of $Π_{Q}^λG$. Furthermore, we construct a new quiver $Q_{G}$ and prove that $Π_{Q}^λG$ is Morita equivalent to $Π_{Q_{G}}^η$ for some $η$.
Skew group algebras of path algebras and preprojective algebras
在Morita等价的条件下，我们明确地计算了有限群作用在一个子空间的路代数上的斜群代数和有限群作用在一个预投射代数上的斜群代数.这些结果推广了Reiten和Riedtmann关于循环群作用在环的路代数上的结果，以及Reiten和Van den Bergh关于$\SL(\C X \oplus \C Y)$的有限子群作用在$\C[X, Y]$上的结果.
We construct a numeric algorithm for completing the proof of a conjecture asserting that all deformed preprojective algebras of generalized Dynkin type are periodic. In particular, we obtain an algorithmic procedure showing that non-trivial deformed preprojective algebras of Dynkin types $\mathbb{E}_7$ and $\mathbb{E}_8$ exist only in characteristic 2. As a consequence, we show that deformed preprojective algebras of Dynkin types $\mathbb{E}_6$, $\mathbb{E}_7$ and $\mathbb{E}_8$ are periodic and we obtain an algorithm for a classification of such algebras, up to algebra isomorphism. We do it b…
Deformed preprojective algebras are generalizations of the usual preprojective algebras introduced by Crawley-Boevey and Holland, which have applications to Kleinian singularities, the Deligne-Simpson problem, integrable systems and noncommutative geometry. In this paper we offer three contributions to the study of such algebras: (1) the 2-Calabi-Yau property; (2) the unification of the reflection functors of Crawley-Boevey and Holland with reflection functors for the usual preprojective algebras; and (3) the classification of tilting ideals in 2-Calabi-Yau algebras, and especially in deformed…
We give an interpretation of the $(q,t)$-deformed Cartan matrices of finite type and their inverses in terms of bigraded modules over the generalized preprojective algebras of Langlands dual type in the sense of Geiß-Leclerc-Schröer [Invent. math. 209 (2017)]. As an application, we compute the first extension groups between the generic kernels introduced by Hernandez-Leclerc [J. Eur. Math. Soc. 18 (2016)], and propose a conjecture that their dimensions coincide with the pole orders of the normalized $R$-matrices between the corresponding Kirillov-Reshetikhin modules.
We introduce and study a fermionization procedure for the cohomological Hall algebra $\mathcal{H}_{Π_Q}$ of representations of a preprojective algebra, that selectively switches the cohomological parity of the BPS Lie algebra from even to odd. We do so by determining the cohomological Donaldson--Thomas invariants of central extensions of preprojective algebras studied in the work of Etingof and Rains, via deformed dimensional reduction. Via the same techniques, we determine the Borel-Moore homology of the stack of representations of the $μ$-deformed preprojective algebra introduced by Crawley-…
Motivated by the goal of studying cluster algebras in infinite type, we study the stability domains of modules for the preprojective algebra in the corresponding infinite types. Specifically, we study real bricks: those modules whose endomorphism algebra is a division ring and which have no self-extensions. We define "shard modules" to be those real bricks whose stability domain is as large as possible (meaning, of dimension one less than the rank of the preprojective algebra). We show that all real bricks are obtained by applying the Baumann-Kamnitzer reflection functors to simple modules, an…
定义并研究了同调光滑微分分次（=dg）范畴的变形n-Calabi-Yau完备化.重要的例子有：连通非Dynkin箭图的变形预投射代数，与具有势的箭图相关联的Ginzburg dg代数，与光滑簇的典范丛上的凝聚层范畴相关联的dg范畴。我们表明，变形的Calabi-Yau完成有Calabi-Yau属性，他们的建设是兼容的导出的等价和本地化。特别地，Ginzburg dg代数具有Calabi-Yau性质。我们表明，变形的3-Calabi-Yau完成的代数的全球维度最多为2是准同构的Ginzburg dg代数，并将其应用到研究集群倾斜代数和建设派生的等价突变的箭与潜在的。在附录中，Michel Van den Bergh利用非交换微分几何给出了Ginzburg dg代数具有Calabi-Yau性质的另一种证明。
We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a gr…
We construct infinite-dimensional analogues of finite-dimensional simple modules of the nonstandard $q$-deformed enveloping algebra $U_q'(\mathfrak{so}_n)$ defined by Gavrilik and Klimyk, and we do the same for the classical universal enveloping algebra $U(\mathfrak{so}_n)$. In this paper we only consider the case when $q$ is not a root of unity, and $q\to 1$ for the classical case. Extending work by Mazorchuk on $\mathfrak{so}_n$, we provide rational matrix coefficients for these infinite-dimensional modules of both $U_q'(\mathfrak{so}_n)$ and $U(\mathfrak{so}_n)$. We use these modules with r…
