Thomas Brüstle

On the cluster category of a marked surface without punctures

We study in this paper the cluster category C(S,M) of a marked surface (S,M). We explicitly describe the objects in C(S,M) as direct sums of homotopy classes of curves in (S,M) and one-parameter families related to closed curves in (S,M). Moreover, we describe the Auslander-Reiten structure of the category C(S,M) in geometric terms and show that the objects without self-extensions in C(S,M) correspond to curves in (S,M) without self-intersections. As a consequence, we establish that every rigid indecomposable object is reachable from an initial triangulation.

The Δ-Filtered Modules Without Self-Extensions for the Auslander Algebra of k[T]/〈T n 〉

It is well known that the Auslander algebra of any representation finite algebra is quasi-hereditary. We consider the Auslander algebra An of k[T]/〈n〉 (here, k is a field, T a variable and n a natural number). We determine all Δ-filtered An-modules without self-extensions. They can be described purely combinatorially. Given any Δ-filtered module N, we show that there is (up to isomorphism) a unique Δ-filtered module M without self-extensions which has the same dimension vector. In the case where k is an infinite field, N is a degeneration of this module M. In particular, we see that in this case, the set of Δ-filtered modules with a fixed dimension vector is the closure of an open orbit (thus irreducible). As observed by Hille and Röhrle, the problem of describing all Δ-filtered An-modules is the same as that of describing the conjugacy classes of elements in the unipotent radical of a parabolic subgroup P of GL(m, k) under the action of P, thus we recover Richardsons dense orbit theorem in this instance.

Tagged mapping class groups: Auslander–Reiten translation

We give a geometric realization, the tagged rotation, of the AR-translation on the generalized cluster category associated to a surface

Mutation Classes of Skew-Symmetric 3 × 3-Matrices

\mathbf {S}

TAME TWO-POINT ALGEBRAS WITHOUT LOOPS

S with marked points and non-empty boundary, which generalizes Brüstle–Zhang’s result for the puncture free case. As an application, we show that the intersection of the shifts in the 3-Calabi–Yau derived category

On positive roots of pg-critical algebras

\mathrm{\mathcal {D} }(\Gamma _{\mathbf {S}})