This first part of a two-volume set offers a modern account of the representation theory of finite dimensional associative algebras over an algebraically closed field. The authors present this topic from the perspective of linear representations of finite-oriented graphs (quivers) and homological algebra. The self-contained treatment constitutes an elementary, up-to-date introduction to the subject using, on the one hand, quiver-theoretical techniques and, on the other, tilting theory and integral quadratic forms. Key features include many illustrative examples, plus a large number of end-of-chapter exercises. The detailed proofs make this work suitable both for courses and seminars, and for self-study. The volume will be of great interest to graduate students beginning research in the representation theory of algebras and to mathematicians from other fields.

We associate an algebra A( ) to a triangulation of a surface S with a set of boundary marking points. This algebra A( ) is gentle and Gorenstein of dimension one. We also prove that A( ) is cluster-tilted if and only if it is cluster-tilted of type ‫ށ‬ or‫,ށ‬ or if and only if the surface S is a disc or an annulus. Moreover all cluster-tilted algebras of type ‫ށ‬ or‫ށ‬ are obtained in this way.

Given a finite dimensional algebra C (over an algebraically closed field) of global dimension at most two, we define its relation-extension algebra to be the trivial extension C ⋉ Ext 2 C (DC, C) of C by the C-C-bimodule Ext 2 C (DC, C). We give a construction for the quiver of the relationextension algebra in case the quiver of C has no oriented cycles. Our main result says that an algebraC is cluster-tilted if and only if there exists a tilted algebra C such thatC is isomorphic to the relation-extension of C.

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In this article, we introduce the notion of cluster automorphism of a given cluster algebra as a Z-automorphism of the cluster algebra that sends a cluster to another and commutes with mutations. We study the group of cluster automorphisms in detail for acyclic cluster algebras and cluster algebras from surfaces, and we compute this group explicitly for the Dynkin types and the euclidean types.