Quivers with relations for symmetrizable Cartan matrices I : Foundations

Abstract

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.

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 Math

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