We show that the category of representations of the Euclidean group of orientation-preserving isometries of two-dimensional Euclidean space is equivalent to the category of representations of the preprojective algebra of type A∞. We also consider the moduli space of representations of the Euclidean group along with a set of generators. We show that these moduli spaces are quiver varieties of th...

We present a geometric construction of highest weight crystals B(λ) for quantum generalized Kac-Moody algebras. It is given in terms of the irreducible components of certain Lagrangian subvarieties of Nakajima’s quiver varieties associated to quivers with edge loops. Introduction The 1990’s saw a great deal of interesting interplay between the geometry of quiver varieties and the representation...

We study the generalization of the idea of a local quiver of a representation of a formally smooth algebra, to broader classes of finitely generated algebras. In this new setting we can construct for every semisimple representation M a local model and a noncommutative tangent cone. The representation schemes of these new algebras model the local structure and the tangent cone of the representat...

Any moduli space of representations of a quiver (possibly with oriented cycles) has an embedding as a dense open subvariety into a moduli space of representations of a bipartite quiver having the same type of singularities. A connected quiver is Dynkin or extended Dynkin if and only if all moduli spaces of its representations are smooth.

Let Q be a tame quiver of type Ãn and Rep (Q) the category of finite dimensional representations over an algebraically closed field. A representation is simply called a module. We study the number of the GR submodules. It will be shown that only finitely many (central) Gabriel-Roiter measures have no direct predecessors. The quivers Q, whose central part contains no preinjective modules, will a...

We define a graph-based visual query language for object databases, called QUIVER. Most graph-based query languages proposed in the literature view queries either as constraint (pattern) graphs or as data-flow graphs. QUIVER, however, allows constraints and data-flow to be combined in queries. This combination of constraints and data-flow is motivated by the need to support comprehensive object...

For a perfect field k, we study actions of the absolute Galois group of k on the k-valued points of moduli spaces of quiver representations over k; the fixed locus is the set of k-rational points and we obtain a decomposition of this fixed locus indexed by elements in the Brauer group of k. We provide a modular interpretation of this decomposition using quiver representations over division alge...

We provide a common generalization of the Springer map and quiver-graded Springer map due to Lusztig, called generalized quiver-graded Springer map associated to generalized quiver representations introduced by Derksen and Weyman. Following Chriss and Ginzburg for any equivariant projective map π : E → V , there is an algebra structure on the equivariant Borel-Moore homology of Z = E ×V E, we c...