Short proof of the classification of representation-finite symmetric algebras up to stable equivalence

Assume K is an algebraically closed field and A a finite dimensional K-algebra. Assume additionally that A is symmetric and representation-finite. Then one has the following classification of such algebras up to stable equivalence( source without proof section 3.14 in: Andrzej Skowroński – Selfinjective algebras: finite and tame type): A is stable equivalent to a … Read more

Auslander-Reiten-Quivers of representation-finite algebras having different 3-dimensional forms

I am looking for references, where I can find (pictures of) connected Auslander-Reiten-Quivers of representation-finite k-algebras (k is a (preferably, but not necessarily finite) field) with one or more of the following properties: 1) the Auslander-Reiten quiver contains two cylinders (so, there are periodic modules), but also many non-periodic modules 2) the Auslander-Reiten quiver contains … Read more

Kac’s theorem for quiver representations over an arbitrary ground field

Let Q be a quiver without loops (cycles of length 1). Kac proved that if K is algebraically closed, the dimension vectors of indecomposable representations of Q over K are exactly the positive real roots and the imaginary roots; further, there is a single isomorphism class of representations if the root is real, and infinitely … Read more

Vertex embeddings of quantum groups via quivers

Let Uq be a quantised enveloping algebra of type affine ADE (untwisted). By the loop presentation of Uq, we see that for each vertex of the finite Dynkin diagram, there is an inclusion Uq(^sl2)→Uq. Now let us restrict to the positive part U+ of Uq. It is well known how to construct U+ from the … Read more

Quiver invariants as polynomials/algebraic curves

I’m interested in algebraic curves one can associate to gauge or string theories. Examples involve Seiberg-Witten curves or family of A-polynomials which define holomorphic Lagrangian submanifolds for SL(2,C) Chern-Simons theory or alternatively mirror Calabi-Yau manifolds to resolved conifold. I’m wondering if something like this has been done for quiver gauge theories or quivers in general, … Read more

Rigid regular objects of path algebras of tame quivers

In the paper On Maximal Green Sequences by Brustle, Dupont and Perotin the authors argued that in a path algebra Λ=kQ of a tame quiver Q with n vertices each tilting module contains at most n−2 regular components. The same applies to silting objects between Λ and Λ[1]. I think this is because for each … Read more

Indecomposable representations of euclidean quivers

The classification of indecomposable representations of a Euclidean quiver is well-known over an algebraically closed field. I am interested in an analogous classification, but over an arbitrary field. It is sometimes said that the classification in the more general case can be done similarly to the algebraically closed, however I see some problems in doing … Read more

When is a given quiver algebra a hopf algebra?

Given a finite dimensional selfinjective quiver algebra A over a finite field (or more generally an arbitrary field). Whats the best way to check if the algebra A has a Hopf algebra structure or not? If we assume the field to be finite it is a finite problem, so there might be some good algorithm? … Read more