Abstract:
For any valued quiver, by using BGP-reflection functors, an injection from the set of preprojective objects in the cluster category to the set of cluster variables of the corresponding cluster algebra is given, the images are called preprojective cluster variables. It is proved that all preprojective cluster variables have denominators $u^{\underline{dim}M}$ in their irreducible fractions of integral polynomials, where $M$ is the corresponding preprojective module or preinjective module. If the quiver is of Dynkin type, we generalize the denominator theorem in [FZ2] to any seed, and also generalize the corresponding results in [CCS1] [CCS2] [CK1] to non-simply-laced case. Given a finite quiver (with trivial valuations) without oriented cycles, fixed a tilting seed $(V, B_V)$, it is proved that the existence and uniqueness of a bijection (abstractly, not in explicit form, compare [CK2]) from the set of exceptional indecomposable objects in the cluster categories to the set of cluster variables associated to $B_V$ which sends $ V_i[1]$ to $u_i$ and sends cluster tilting objects to clusters.

Abstract:
We construct reflection functors on categories of modules over deformed wreath products of the preprojective algebra of a quiver. These functors give equivalences of categories associated to generic parameters which are in the same orbit under the Weyl group action. We give applications to the representation theory of symplectic reflection algebras of wreath product groups.

Abstract:
The purpose of this paper is to apply the theory of MV polytopes to the study of components of Lusztig's nilpotent varieties. Along the way, we introduce reflection functors for modules over the non-deformed preprojective algebra of a quiver.

Abstract:
The main result of the paper is a natural construction of the spherical subalgebra in a symplectic reflection algebra associated with a wreath-product in terms of quantum hamiltonian reduction of an algebra of differential operators on a representation space of an extended Dynkin quiver. The existence of such a construction has been conjectured in [EG]. We also present a new approach to reflection functors and shift functors for generalized preprojective algebras and symplectic reflection algebras associated with wreath-products.

Abstract:
Quivers play an important role in the representation theory of algebras, with a key ingredient being the path algebra and the preprojective algebra. Quiver grassmannians are varieties of submodules of a fixed module of the path or preprojective algebra. In the current paper, we study these objects in detail. We show that the quiver grassmannians corresponding to submodules of certain injective modules are homeomorphic to the lagrangian quiver varieties of Nakajima which have been well studied in the context of geometric representation theory. We then refine this result by finding quiver grassmannians which are homeomorphic to the Demazure quiver varieties introduced by the first author, and others which are homeomorphic to the graded/cyclic quiver varieties defined by Nakajima. The Demazure quiver grassmannians allow us to describe injective objects in the category of locally nilpotent modules of the preprojective algebra. We conclude by relating our construction to a similar one of Lusztig using projectives in place of injectives.

Abstract:
We determine the PBW deformations of the wreath product of a symmetric group with a deformed preprojective algebra of an affine Dynkin quiver. In particular, we show that there is precisely one parameter which does not come from deformation of the preprojective algebra. We prove that the PBW deformation is Morita equivalent to a corresponding symplectic reflection algebra for wreath product.

Abstract:
This paper studies connections between the preprojective representations of a valued quiver, the (+)-admissible sequences of vertices, and the Weyl group by associating to each preprojective representation a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words. The latter strengthens known results of Howlett, Fomin-Zelevinsky, and the authors.

Abstract:
We introduce "continuous deformed preprojective algebras" attached to infinite affine Dynkin quivers of type A_{\infty}, A_{+\infty}, D_{\infty}. We define a one-parameter family of deformations of the wreath product of a symmetric group with these algebras and, using the generalized McKay correspondence for infinite reductive subgroups of SL(2,C), we establish a Morita equivalence to a corresponding continuous symplectic reflection algebra of wreath product type. We give applications of this Morita equivalence to the representation theory of continuous wreath product symplectic reflection algebras. In particular, we use Crawley-Boevey and Holland's results about the representation theory of deformed preprojective algebras to get a complete classification of the finite dimensional irreducible representations for the rank one continuous wreath product symplectic reflection algebras. In higher rank, we extend Gan's definition of the reflection functors to the continuous case and we use such functors to find an interesting class of finite dimensional representations.

Abstract:
These are the notes for a course on representations of quivers for second year students in Paderborn in summer 2007. My aim was to provide a basic introduction without using any advanced methods. It turns out that a good knowledge of linear algebra is sufficient for proving Gabriel's theorem. Thus we classify the quivers of finite representation type and study their representations using reflection functors. Further material has been added after giving this course in Bielefeld in summer 2010. This includes a discussion of regular representations and wild phenomena. In particular, two classical examples are covered: representations of the Kronecker quiver and representations of the Klein four group.

Abstract:
We define Bernstein-Gelfand-Ponomarev reflection functors in the cluster categories of hereditary algebras. They are triangle equivalences which provide a natural quiver realization of the "truncated simple reflections" on the set of almost positive roots $\Phi_{\ge -1}$ associated to a finite dimensional semisimple Lie algebra. Combining with the tilting theory in cluster categories developed in [4], we give a unified interpretation via quiver representations for the generalized associahedra associated to the root systems of all Dynkin types (a simply-laced or non-simply-laced). This confirms the conjecture 9.1 in [4] in all Dynkin types.