A geometric model for semilinear locally gentle algebras
Joint work with Esther Banaian, Raphael Bennett-Tennenhaus and Kayla Wright
We consider certain generalizations of gentle algebras that we call semilinear locally gentle algebras. These rings are examples of semilinear clannish algebras as introduced by the second author and Crawley-Boevey. We generalise the notion of a nodal algebra from work of Burban and Drozd and prove that semilinear gentle algebras are nodal by adapting a theorem of Zembyk. We also provide a geometric realization of Zembyk's proof, which involves cutting the surface into simpler pieces in order to endow our locally gentle algebra with a semilinear structure. We then consider this surface glued back together, with the seams in place, and use it to give a geometric model for the finite-dimensional modules over the semilinear locally gentle algebra.
We have published a preprint.
A characterisation of higher torsion classes
Joint work with Jenny August, Johanne Haugland, Sondre Kvamme, Yann Palu and Hipolito Treffinger.
We prove that a subcategory is a d-torsion class if and only if it is closed under d-extensions and d-quotients. This generalises an important result for classical torsion classes. As an application, we prove that the d-torsion classes in M form a complete lattice. Moreover, we use the characterisation to classify the d-torsion classes associated to higher Auslander algebras of type 𝔸, and give an algorithm to compute them explicitly.
We have a preprint out, along with some rather nifty code for calculating torsion classes.
Admissible ideals for k-linear categories
Joint work with J. Daisie Rock.
We generalize the notion of an admissible ideal from path algebras to (small) k-linear categories that satisfy the Krull-Remak-Schmidt-Azumaya assumption.
The role of gentle algebras in higher homological algebra
Joint work with Johanne Haugland and Sibylle Schroll.
We use the geometric model of Opper-Plamondon-Schroll to investigate the existence of d-cluster-tilting subcategories in the module and derived categories of gentle algebras.
Published in Forum Mathematicum.
Infinite friezes and triangulations of annuli
Joint work with Karin Baur, Ilke Canakci, Maitreyee Kulkarni and Gordana Todorov.
We show that each periodic infinite frieze determines a triangulation of an annulus in essentially a unique way. Since each triangulation of an annulus determines a pair of friezes, we study such pairs and show how they determine each other.
Published in Journal of Algebra and its Applications.
Maximal τd-rigid pairs
Joint work with Peter Jørgensen.
We give a higher homological analogue of τ-tilting theory as defined by Adachi, Iyama and Reiten
Published in Journal of Algebra
d-abelian quotients of (d+2)-angulate.d categories
Joint work with Peter Jørgensen
We give neccessary and sufficient conditions for the quotient of a (d+2)-angulated category to be d-abelian.
Published in Journal of Algebra.
Realizing orbit categories as stable module categories: a complete classification
Joint work with Benedikte Grimeland
We classify all triangulated orbit categories of path-algebras of Dynkin diagrams that are triangle equivalent to a stable module category of a representation-finite self-injective standard algebra.
Published in Contributions to Algebra and Geometry.
Abelian quotients of triangulated categories
Joint work with Benedikte Grimeland
We give neccessary and sufficient conditions for the quotient of a triangulated category to be abelian.
Published in Journal of Algebra.