Quivers with relations for symmetrizable Cartan matrices I : Foundations

We introduce and study a class of Iwanaga-Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices. These algebras generalize the path algebras of quivers associated with symmetric Cartan matrices. We also define a corresponding class of generalized preprojective algebras. Without any assumption on the ground field, we obtain new representation-theoretic realizations of all finite root systems.Comment: 72 pages. v2 : We restructured some of the sections, improved the exposition, fixed a few typos, and added new references. Title has slightly changed v3 : A few typos fixed. Section 9.3 on APR-tilting has been rewritten. v4 : an acknowledgement added. v5 : Exposition improved after comments from a referee, references updated and a few typos corrected. Final version, to appear in Inventiones Mat

Quivers with relations for symmetrizable Cartan matrices V. Caldero-Chapoton formula

We generalize the Caldero-Chapoton formula for cluster algebras of finite type to the skew-symmetrizable case. This is done by replacing representation categories of Dynkin quivers by categories of locally free modules over certain Iwanaga-Gorenstein algebras introduced in Part I. The proof relies on the realization of the positive part of the enveloping algebra of a simple Lie algebra of the same finite type as a convolution algebra of constructible functions on representation varieties of $H$, given in Part III. Along the way, we obtain a new result on the PBW basis of this convolution algebra.Comment: 24 pages. V2: Exposition improved and a few typos fixed after referee report. Final version, to appear in Proc. London Math. So

Auslander algebras and initial seeds for cluster algebras

Let $Q$ be a Dynkin quiver and $\Pi$ the corresponding set of positive roots. For the preprojective algebra $\Lambda$ associated to $Q$ we produce a rigid $\Lambda$-module $I_Q$ with $r=|\Pi|$ pairwise non-isomorphic indecomposable direct summands by pushing the injective modules of the Auslander algebra of $kQ$ to $\Lambda$. If $N$ is a maximal unipotent subgroup of a complex simply connected simple Lie group of type $|Q|$, then the coordinate ring $C[N]$ is an upper cluster algebra. We show that the elements of the dual semicanonical basis which correspond to the indecomposable direct summands of $I_Q$ coincide with certain generalized minors which form an initial cluster for $C[N]$, and that the corresponding exchange matrix of this cluster can be read from the Gabriel quiver of $End_{\Lambda}(I_Q)$. Finally, we exploit the fact that the categories of injective modules over $\Lambda$ and over its covering $\tilde{\Lambda}$ are triangulated in order to show several interesting identities in the respective stable module categories.Comment: 23 pages, Version 2: Reference [7] corrected+update

Generic bases for cluster algebras and the Chamber Ansatz

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 unipotent cell N^w of the Kac-Moody group with Lie algebra g. In previous work we proved that the coordinate ring \C[N^w] of N^w is a cluster algebra in a natural way. A central role is played by generating functions \vphi_X of Euler characteristics of certain varieties of partial composition series of X, where X runs through all modules in a Frobenius subcategory C_w of the category of nilpotent $\Lambda$-modules. We show that for every X in C_w, \vphi_X coincides after appropriate changes of variables with the cluster characters of Fu and Keller associated with any cluster-tilting module T of C_w. As an application, we get a new description of a generic basis of the cluster algebra obtained from \C[N^w] via specialization of coefficients to 1. For the special case of coefficient-free acyclic cluster algebras this proves a conjecture by Dupont.Comment: 48 pages. Version 2: Minor improvements, and a few typos corrected. v3: New section 2 (reminder on cluster algebras), so subsequent sections renumbered; section 6 (was section 5) reorganized and extended; several small corrections; references updated. Final version, now 55 pages, to appear JAM

Verma modules and preprojective algebras

We give a geometric construction of the Verma modules of a symmetric Kac-Moody Lie algebra in terms of constructible functions on the varieties of nilpotent finite-dimensional modules of the corresponding preprojective algebra.Comment: Minor changes. Final version. To appear in Nagoya Math. Journa