Julian Külshammer

Pro-Species of Algebras I: Basic Properties

In this paper, we generalise part of the theory of hereditary algebras to the context of pro-species of algebras. Here, a pro-species is a generalisation of Gabriel’s concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Geiß et al. (2016). In particular, we construct a corresponding preprojective algebra, and establish a theory of a separated pro-species yielding a stable equivalence between certain functorially finite subcategories.

Auslander–Reiten theory of Frobenius–Lusztig kernels

In this paper we show that the tree class of a component of the stable Auslander-Reiten quiver of a Frobenius-Lusztig kernel is one of the three infin te Dynkin diagrams. For the special case of the small quantum group we show that the periodic comp onents are homogeneous tubes and that the non-periodic components have shape Z[A∞] if the component contains a module for the infinite-dimensional quantum group.

Auslander-Reiten theory of small half quantum groups

For the small half quantum groups and we show that the components of the stable Auslander-Reiten quiver containing gradable modules are of the form Z[A_\infty]

Biserial algebras via subalgebras and the path algebra of D4

We give two new criteria for a basic algebra to be biserial. T he first one states that an algebra is biserial i ff all subalgebras of the formeAewheree is supported by at most 4 vertices are biserial. The second one gives some condition on modules that must not exist for a biserial algebra. These modules have properties similar to the modul e with dimension vector (1 , 1, 1, 1) for the path algebra of the quiver D4. Both criteria generalize criteria for an algebra to be Nakay ma. They rely on the description of a basic biserial algebra in terms of quiver and relations giv en by R. Vila-Freyer and W. CrawleyBoevey [CBVF98].