In a k-linear triangulated category (where k is a field) we show that the existence of Auslander-Reiten triangles implies that objects are determined, up to shift, by knowing dimensions of homomorphisms between them. In most cases the objects themselves are distinguished by this information, a conclusion which was also reached under slightly different hypotheses in a theorem of Jensen, Su and Z...

This is a survey paper of our recent results [6]. We present a novel method to detect robust and common topological structures of two geometric objects. The idea is to extend the notion of persistent homology [5, 12] to representations on a commutative triple ladder quiver. Our contributions of this paper are given as follows: (i) We prove that the commutative triple ladder quiver is representa...

In [3], Bautista and Salmeron explained how, given a finite quiver Z without double arrows or oriented cycles, certain sets of subsets of ZZ (called (P)-generating systems) may, under appropriate conditions, generate algebras whose preprojective components can be embedded in ZZ in a nice way, and which are (P)-algebras [2] whenever they are representation-finite. Here, we define numerical funct...

Following Braverman–Finkelberg–Feigin–Rybnikov (arXiv: 1008.3655), we study the convolution algebra of a handsaw quiver variety, a.k.a. a parabolic Laumon space, and a finite W -algebra of type A. This is a finite analog of the AGT conjecture on 4-dimensional supersymmetric Yang–Mills theory with surface operators. Our new observation is that the C-fixed point set of a handsaw quiver variety is...

the k-vectorspace of outer k-derivations, that is, the quotient of all k-derivations of Λ modulo the inner ones. It has been suspected for some time, and [4] seems to be the earliest reference, that vanishing of the first Hochschild cohomology precludes the existence of oriented cycles in the ordinary quiver. There are various supporting partial results, among them [8, (2.3), (3.2)], [2, (2.2)]...

We study linear free divisors, that is, free divisors arising as discriminants in prehomogeneous vector spaces, and in particular in quiver representation spaces. We give a characterization of the prehomogeneous vector spaces containing such linear free divisors. For reductive linear free divisors, we prove that the numbers of geometric and representation theoretic irreducible components coinci...

the k-vectorspace of outer k-derivations , that is, the quotient of all k-derivations of Λ modulo the inner ones. It has been suspected for some time, and [4] seems to be the earliest reference, that vanishing of the first Hochschild cohomology precludes the existence of oriented cycles in the ordinary quiver. There are various supporting partial results, among them [8, (2.3), (3.2)], [2, (2.2)...

Starting from the N = 2 SYM4 quiver theory living on wrapped NiD5 branes around S 2 i spheres of deformed ADE fibered Calabi-Yau threefolds (CY3) and considering deformations using massive vector multiplets, we explicitly build a new class of N = 1 quiver gauge theories. In these models, the quiver gauge group ∏ i U (Ni) is spontaneously broken down to ∏ i SU (Ni) and Kahler deformations are sh...

(1) Study the class of hyper-Kähler manifolds which are hyper-Kähler reductions of finite dimensional quaternion vector spaces by products of unitary groups. (Probably it is better to assume that the action is linear.) Besides quiver varieties, hyper-Kähler toric varieties in the sense of Bielawski and Dancer [2] are such examples. When the quotients are nonsingular ? How much of geometric prop...

Let Q be a quiver. M. Reineke and A. Hubery investigated the connection between the composition monoid CM(Q), as introduced by M. Reineke, and the generic composition algebra Cq(Q), as introduced by C. M. Ringel, specialised at q = 0. In this thesis we continue their work. We show that if Q is a Dynkin quiver or an oriented cycle, then C0(Q) is isomorphic to the monoid algebra QCM(Q). Moreover,...