We introduce higher dimensional analogues of the Nakayama algebras from the viewpoint of Iyama's higher Auslander–Reiten theory. More precisely, for each Nakayama algebra A and each positive integer d, we construct a finite dimensional algebra A(^{d}) having a distinguished d-cluster-tilting A(^{d})-module whose endomorphism algebra is a higher dimensional analogue of the Auslander algebra of A. We also construct higher dimensional analogues of the mesh category of type ZΑ∞ and the tubes.

The full text will be freely available from 2021-06-05 10:25

Highest weight categories arising in Lie theory are known to be associated with finite dimensional quasi-hereditary algebras such as Schur algebras or blocks of category O. An analogue of the PBW theorem will be shown to hold for quasi-hereditary algebras: Up to Morita equivalence each such algebra has an exact Borel subalgebra. The category F(\Delta) of modules with standard (Verma, Weyl, …) filtration, which is exact, but rarely abelian, will be shown to be equivalent to the category of representations of a directed box. This box is constructed as a quotient of a dg algebra associated with the A-infinity-structure on Ext(\Delta,\Delta). Its underlying algebra is an exact Borel subalgebra.

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 infinite Dynkin diagrams. For the special case of the small quantum group, we show that the periodic components 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.

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

We give two new criteria for a basic algebra to be biserial. The first one states that an algebra is biserial iff all subalgebras of the form eAe where e 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 module with dimension vector (1,1,1,1) for the path algebra of the quiver D4.

Both criteria generalize criteria for an algebra to be Nakayama. They rely on the description of a basic biserial algebra in terms of quiver and relations given by R. Vila-Freyer and W. Crawley-Boevey.

In the bocs seat: quasi-hereditary algebras and representation type2017In: Representation theory—current trends and perspectives / [ed] Henning Krause, Peter Littelmann, Gunter Malle, Karl-Hermann Neeb, Christoph Schweigert, European Mathematical Society Publishing House, 2017, p. 375-426Conference paper (Refereed)

Abstract [en]

This paper surveys bocses, quasi-hereditary algebras and their relationship which was established in a recent result by Koenig, Ovsienko, and the author. Particular emphasis is placed on applications of this result to the representation type of the category filtered by standard modules for a quasi-hereditary algebra. In this direction, joint work with Thiel is presented showing that the subcategory of modules filtered by Weyl modules for tame Schur algebras is of finite representation type. The paper also includes a new proof for the classication of quasi-hereditary algebras with two simple modules, a result originally obtained by Membrillo–Hernández.

Pro-species of algebras I: Basic properties2017In: Algebras and Representation Theory, ISSN 1386-923X, E-ISSN 1572-9079, Vol. 20, no 5, p. 1215-1238Article in journal (Refereed)

Abstract [en]

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.

In this article, we show that almost all blocks of all Frobenius–Lusztig kernels are of wild representation type extending results of Feldvoss and Witherspoon, who proved this result for the principal block of the zeroth Frobenius–Lusztig kernel. Furthermore, we verify the conjecture that there are infinitely many Auslander–Reiten components for a finite-dimensional algebra of infinite representation type for selfinjective algebras whose cohomology satisfies certain finiteness conditions.

S. Koenig, S. Ovsienko and the second author showed that every quasi-hereditary algebra is Morita equivalent to the right algebra, i.e. the opposite algebra of the left dual, of a coring. Let A be an associative algebra and V an A-coring whose right algebra R is quasi-hereditary. In this paper, we give a combinatorial description of an associative algebra B and a B-coring W whose right algebra is the Ringel dual of R. We apply our results in small examples to obtain restrictions on the A-infinity-structure of the Ext-algebra of standard modules over a class of quasi-hereditary algebras related to birational morphisms of smooth surfaces.

Nakayama-type phenomena in higher Auslander-Reiten theory2018In: Representations of algebras / [ed] Graham J. Leuschke, Frauke Bleher, Ralf Schiffler, Dan Zacharia, American Mathematical Society (AMS), 2018, Vol. 705Conference paper (Refereed)

Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Algebra and Geometry.

Psaroudakis, Chrysostomos

Skartsæterhagen, Øystein

Derived invariance of support varieties2019In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 147, no 1, p. 1-14Article in journal (Refereed)

Abstract [en]

The (Fg) condition on Hochschild cohomology as well as the sup-port variety theory are shown to be invariant under derived equivalence.