67 research outputs found
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 , 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 be a Dynkin quiver and the corresponding set of positive roots.
For the preprojective algebra associated to we produce a rigid
-module with pairwise non-isomorphic indecomposable
direct summands by pushing the injective modules of the Auslander algebra of
to .
If is a maximal unipotent subgroup of a complex simply connected simple
Lie group of type , then the coordinate ring is an upper cluster
algebra. We show that the elements of the dual semicanonical basis which
correspond to the indecomposable direct summands of coincide with certain
generalized minors which form an initial cluster for , and that the
corresponding exchange matrix of this cluster can be read from the Gabriel
quiver of .
Finally, we exploit the fact that the categories of injective modules over
and over its covering 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 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 -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
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