Related papers: Tilting theory of preprojective algebras and $c$-s…

We study the stable category of the factor algebra of the preprojective algebra associated with an element $w$ of the Coxeter group of a quiver. We show that there exists a silting object $M(\bf{w})$ of this category associated with each…

Let $Q$ be an acyclic quiver and $\Lambda$ be the complete preprojective algebra of $Q$ over an algebraically closed field $k$. To any element $w$ in the Coxeter group of $Q$, Buan, Iyama, Reiten and Scott have introduced and studied in…

In this paper we investigate the endomorphism algebras of standard cluster tilting objects in the stably 2-Calabi-Yau categories $\Sub{\Lambda_w}$ with elements $w$ in Coxeter groups in \cite{BIRSc}. They are examples of the 2-Auslander…

In representation theory of graded Iwanaga-Gorenstein algebras, tilting theory of the stable category $\underline{\mathsf{CM}}^{\mathbb{Z}} A$ of graded Cohen-Macaulay modules plays a prominent role. In this paper we study the following two…

We give an interpretation of the map $\pi^c$ defined by Reading, which is a map from the elements of a Coxeter group to the $c$-sortable elements, in terms of the representation theory of preprojective algebras. Moreover, we study a close…

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\leq 2$. We construct a triangulated category $\Cc_A$ associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$.…

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 $\Lambda=kQ/{\mathcal {I}}$, $\mathcal {I}$ is an…

We give a generalization of the classical tilting theorem. We show that for a 2-term silting complex $\mathbf{P}$ in the bounded homotopy category $K^b(\mathop{\rm proj}\nolimits A)$ of finitely generated projective modules of a finite…

We study the quiver of the descent algebra of a finite Coxeter group W. The results include a derivation of the quiver of the descent algebra of types A and B. Our approach is to study the descent algebra as an algebra constructed from the…

Let Q be a finite quiver without oriented cycles, and let k be an algebraically closed field. The main result in this paper is that there is a natural bijection between the elements in the associated Coxeter group W_Q and the cofinite…

Let $Q$ be an acyclic quiver. Associated with any element $w$ of the Coxeter group of $Q$, triangulated categories $\underline{\Sub}\Lambda_w$ were introduced in \cite{Bua2}. There are shown to be triangle equivalent to generalized cluster…

The Hecke algebra $\mathbb{C}_q[W]$ of a Coxter group $W$, associated to parameter $q$, can be completed to a von Neumann algebra $\mathcal{N}_q(W)$. We study such algebras in case where $W$ is right-angled. We determine the range of $q$…

The descent algebra $\Sigma(W)$ is a subalgebra of the group algebra $\Q W$ of a finite Coxeter group $W$, which supports a homomorphism with nilpotent kernel and commutative image in the character ring of $W$. Thus $\Sigma(W)$ is a basic…

Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be its Weyl group. With w in W is associated a…

For any given symmetrizable Cartan matrix $C$ with a symmetrizer $D$, Gei\ss~ et al. (2016) introduced a generalized preprojective algebra $\Pi(C, D)$. We study tilting modules and support $\tau$-tilting modules for the generalized…

Buan, Iyama, Reiten and Smith proved that cluster-tilting objects in triangulated 2-Calabi--Yau categories are closely connected with mutation of quivers with potentials over an algebraically closed field. We prove a more general statement…

A (semi)brick over an algebra $A$ is a module $S$ such that the endomorphism ring $\operatorname{\mathsf{End}}_A(S)$ is a (product of) division algebra. For each Dynkin diagram $\Delta$, there is a bijection from the Coxeter group $W$ of…

The descent algebra of a finite Coxeter group $W$ is a basic algebra, and as such it has a presentation as quiver with relations. In recent work, we have developed a combinatorial framework which allows us to systematically compute such a…

We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster tilting objects in 2-Calabi-Yau triangulated categories, hence all their non-projective indecomposable modules are…

In this paper we introduce a special kind of relative (co)resolutions associated to a pair of classes of objects in an abelian category $\mathcal{C}.$ We will see that, by studying these relative (co)resolutions, we get a possible…