An elementary introduction to representations of quivers. Covered material:
1. The objects we study. Definition of quiver, representation of a quiver and a morphism between two representations. The Lemma of Fitting and the Theorem of Krull Remak Schmidt.
2. The classification problem. The problem is explained and studied in three examples: the linear quiver, the loop and the three Kronecker. The phenomenon of wildness.
3. Morphisms. Morphisms as the structure for the list of indecomposables. Radical and irreducible morphisms, the Auslander-Reiten quiver. The theorem of Auslander-Reiten (without proof) and the technique of knitting.
4. Independence on orientation. The affine space of representations with a fixed dimension vector, the group action on it. Gabriels result. As a generalization: Kac's Theorem (without proof).
5. Connection to algebras. The path algebra and its modules. Morita equivalence. Quotients of path algebras and its modules.

This article studies cluster-tilted algebras whose quiver is cyclically oriented. In this case an explicit description of the defining relations is given. For this kind of algebras, it is also shown that there exists an admissible cut and moreover that each arrow of the quiver is contained in an admissible cut. Furthermore, we show that if the endomorphism ring of an algebra of global dimension two over its cluster category, in the sense of Amiot, is cluster-tilted and has a cyclically oriented quiver, then the original algebra is a quotient by an admissible cut. In the case of cluster tilted algebras of Dynkin or extended Dynkin type, the connection is stronger and also the converse statement holds. Even more, in that case the original algebra is derived equivalent to the hereditary algebra.

We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.

Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the automorphism group of the corresponding cluster category and second by giving explicit sequences of mutations.

Canonical algebras, introduced by C. M. Ringel in 1984, play an important role in the representation theory of nite-dimensional algebras. They also feature in many other mathematical areas like function theory, 3-manifolds, singularity theory, commutative algebra, algebraic geometry and mathematical physics. We show that canonical algebras are characterized by a number of interesting extremal properties (among concealedcanonical algebras, that is, the endomorphism rings of tilting bundles on a weighted projective line). We also investigate the corresponding class of algebras antipodal to canonical ones. Our study yields new insights into the nature of concealed-canonical algebras, and sheds a new light on an old question: Why are the canonical algebras canonical?

We present a categorification of four mutation finite cluster algebras by the cluster category of the category of coherent sheaves over a weighted projective line of tubular weight type. Each of these cluster algebras which we call tubular is associated to an elliptic root system. We show that via a cluster character the cluster variables are in bijection with the positive real Schur roots associated to the weighted projective line. In one of the four cases this is achieved by the approach to cluster algebras of Fomin–Shapiro–Thurston using a 2-sphere with 4 marked points whereas in the remaining cases it is done by the approach of Geiss–Leclerc–Schröer using preprojective algebras.

We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan-Iyama-Reiten-Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently, if A is of tame (domestic or tubular) representation type.

In this paper the relationship between iterated tilted algebras and cluster-tilted algebras and relation-extensions is studied. In the Dynkin case, it is shown that the relationship is very strong and combinatorial.

For the cluster category of a hereditary or a canonical algebra, equivalently for the hereditary category of coherent sheaves on a weighted projective line, we study the Grothendieck group with respect to an admissible triangulated structure.

Every semisimple Lie algebra defines a root system on the dual space of a Cartan subalgebra and a Cartan matrix, which expresses the dual of the Killing form on a root base. Serre's Theorem gives then a representation of the given Lie algebra by generators and relations in terms of the Cartan matrix.
In this work, we generalize Serre's Theorem to give an explicit representation by generators and relations for any simply laced semisimple Lie algebra in terms of a positive quasi-Cartan matrix. Such a quasi-Cartan matrix expresses the dual of the Killing form for a Z-base of roots. Here, by a Z-base of roots, we mean a set of lineary independent roots which generate all roots as linear combinations with integral coefficients.

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.

An explicit construction for calculating the functor from the bounded derived category of an algebra to the stable category of the repetitive algebra is given.

To a unit form a Lie algebra is associated and it is shown that for a non-negative, connected unit form of corank 0,1 or 2 respectively, the resulting Lie algebras are simply-laced finite-dimensional simple, simply-laced affine Kac-Moody or simply-laced elliptic.

Given an integral quadratic unit form q: Zⁿ→Z and a finite tuple of q-roots r=(r^j)_{j∈ J} the induced q-root form q_r is considered as in [Gabriel, Roiter: Algebra VIII, Chapter 6]. We show that two non-negative unit forms are of the same Dynkin type precisely when they are root-induced one from the other. Moreover, there are only finitely many unit forms without double edges of a given Dynkin type. Root-induction yields an interesting partial order on the Dynkin types, which is studied in the paper.

A sufficient criteria is given for respective one-point extensions of two derived equivalent algebras to be derived equivalent again.

Module varieties over canonical algebras are studied using the stratification of Richmond. For that sake, the subfinite canonical algebras are classified. Easy combinatorial criteria for irreducibility, Cohen-Macaulauy and normailty are given for many cases.

Explicit probabilities, expectancy and variance are calculated for the number of black balls in the following urn problem: from an urn, filled initially with white balls, you take out n balls, color them black and put them back and repeat this for a finite number of times.

Positive unit forms of Dynkin type D_n are characterized by a combinatorial description of their associated bigraphs, the possible bigraphs for type E₆ are listed.

It is shown, that if an algebra contains a convex subcategory, which is derived equivalent to a hereditary algebra of type E or a tubular algebra, then the algebra is derived tame precisely when its Euler form is non-negative.

Given an algebra A, a set of necessary and sufficient conditions on the Euler form and the Tits form is given for A to be representation finite and derived tubular.

Positive unit forms of Dynkin type A_n are characterized by a combinatorial description of their associated bigraphs.

Given an algebra A, which is a tree algebra or a strongly simply connected poset algebra, it is shown that, if the Euler form of A is non-negative of corank 2, then the algebra is derived equivalent to a tubular algebra or a poset algebra of very explicit type (certain pg-critical algebra).

A necessary and sufficient condition is given for a form to be non-negative.
It is shown, how to associate a Dynkin diagram to a connected non-negative unit form and that this Dynkin diagram, called the Dynkin type, together with the corank of the form define its equivalence class.

An algorithm is presented for deciding whether a given strongly simply connected algebra is derived tubular or not.

A necessary and sufficient condition expressed by the bilinear homological form is given for a strongly simply connected algebra to be derived tubular.

The question when a one-point extension of a finite-dimensional algebra A by a A-module M is derived canonical, i.e. derived equivalent to a canonical algebra, are answered by giving necessary conditions on the algebra A and the module M. If the canonical algebra associated with A is tame the conditions are even sufficient.

A concrete recipe to construct all derived tubular algebras which are reflection equivalent to given tubular algebra is given. Furthermore the repetitive algebra of a tubular algebra is divided into slices, each of it containg the vertices to which the corresponding projectives are contained in the same tubular family.