In [2][3], the mathematicians Fomin and Zelevinsky described the mathematical object known as a quiver, and connected it with the theory of cluster algebras. In particular, each quiver can be represented by a seed of a cluster algebra, which couples a set of n variables with the adjacency matrix of the quiver. By performing a transformation on the quiver, we change accordingly the values of the...

Coverings in the representation theory of algebras were introduced for the Auslander-Reiten quiver of a representation finite algebra in [15] and later for finite dimensional algebras in [2, 7, 11]. The best understood class of covering functors is that of Galois covering functors F : A → B determined by the action of a group of automorphisms of A. In this work we introduce the balanced coverin...

We show that the dimension of the mesh algebra of a finite Auslander-Reiten quiver over a field is a purely combinatorial invariant that does not depend on the ground field. Moreover, a combinatorial algorithm for computing this dimension is given along our proof of this result. Translation quivers appear naturally in the representation theory of finite dimensional algebras; see, for example, [...

These are notes from introductory survey lectures given at the Institute for Studies in Theoretical Physics and Mathematics (IPM), Teheran, in 2008 and 2010. We present the definition and the fundamental properties of Fomin-Zelevinsky’s cluster algebras. Then, we introduce quiver representations and show how they can be used to construct cluster variables, which are the canonical generator...

There are three types of monopole in gauge theories with fundamental matter and N = 2 supersymmetry broken by a superpotential. There are unconfined 0-monopoles and also 1 and 2-monopoles confined respectively by one or two vortices transforming under distinct components of the unbroken gauge group. If a Fayet-Iliopoulos term is added then there are only 2-monopoles. Monopoles transform in the ...

Articles-Karin BAUR: Cluster categories, m-cluster categories and diagonals in polygons-Ada BORALEVI: On simplicity and stability of tangent bundles of rational homogeneous varieties-Laurent EVAIN: Intersection theory on punctual Hilbert schemes-Daniel JUTEAU, Carl MAUTNER and Geordie WILLIAMSON: Perverse sheaves and modular representation theory-Manfred LEHN and Christoph SORGER: A symplectic ...

We prove a structure theorem for triangulated Calabi-Yau categories: An algebraic 2-Calabi-Yau triangulated category over an algebraically closed field is a cluster category iff it contains a cluster tilting subcategory whose quiver has no oriented cycles. We prove a similar characterization for higher cluster categories. As an application to commutative algebra, we show that the stable categor...

We consider SU(3)-equivariant dimensional reduction of Yang-Mills theory on Kähler manifolds of the form M × SU(3)/H , with H = SU(2) × U(1) or H = U(1) × U(1). The induced rank two quiver gauge theories on M are worked out in detail for representations of H which descend from a generic irreducible SU(3)-representation. The reduction of the Donaldson-UhlenbeckYau equations on these spaces induc...

We introduce and explore the relation between knot invariants and quiver representation theory, which follows from the identification of quiver quantum mechanics in D-brane systems representing knots. We identify various structural properties of quivers associated to knots, and identify such quivers explicitly in many examples, including some infinite families of knots, all knots up to 6 crossi...

We introduce the tropical $F$-polynomial $f_M$ of a quiver representation $M$. study its interplay with general presentation for any finite-dimensional basic algebra. give an interpretation evaluating at weight vector. As consequence, we Newton polytope ${\sf N}(M)$ dual fan and 1-skeleton N}(M)$. propose algorithm to determine generic polytopes, show it works path algebras. application, repres...