Vanessa Miemietz’s research while affiliated with University of East Anglia and other places

In this article, we define and investigate Hochschild cohomology for finitary 2-representations of quasi-fiat 2-categories.

We prove that applying a projective functor to a holonomic simple module over a semisimple finite‐dimensional complex Lie algebra produces a module that has an essential semisimple submodule of finite length. This implies that holonomic simple supermodules over certain Lie superalgebras are quotients of modules that are induced from simple modules over the even part. We also provide some further insight into the structure of Lie algebra modules that are obtained by applying projective functors to simple modules.

A well-known theorem of Buchweitz provides equivalences between three categories: the stable category of Gorenstein projective modules over a Gorenstein algebra, the homotopy category of acyclic complexes of projectives, and the singularity category. To adapt this result to N-complexes, one must find an appropriate candidate for the N-analogue of the stable category. We identify this “N-stable category” via the monomorphism category and prove Buchweitz’s theorem for N-complexes over a Grothendieck abelian category. We also compute the Serre functor on the N-stable category over a self-injective algebra and study the resultant fractional Calabi–Yau properties.

We give a complete combinatorial answer to Kostant’s problem for simple highest weight modules indexed by fully commutative permutations.We also propose a reformulation of Kostant’s problem in the context of fiab bicategories and classify annihilators of simple objects in the principal birepresentations of such bicategories generalizing the Barbasch–Vogan theorem for Lie algebras.

In this paper, we show that Soergel bimodules for finite Coxeter types have only finitely many equivalence classes of simple transitive 2‐representations and we complete their classification in all types but H3$H_{3}$ and H4$H_{4}$.

Motivated by the so-called H-cell reduction theorems, we investigate certain classes of bicategories which have only one H-cell apart from possibly the identity. We show that H0-simple quasi fiab bicategories with unique H-cell H0 are fusion categories. We further study two classes of non-semisimple quasi-fiab bicategories with a single H-cell apart from the identity. The first is , indexed by a finite-dimensional radically graded basic Hopf algebra A, and the second is , consisting of symmetric projective A-A-bimodules. We show that can be viewed as a 1-full subbicategory of and classify simple transitive birepresentations for . We point out that the number of equivalence classes of the latter is finite, while that for is generally not.

This article develops a theory of cell combinatorics and cell 2-representations for differential graded 2-categories. We introduce two types of partial preorders, called the strong and weak preorder. We then analyse and compare them. The weak preorder is more easily tractable, while the strong preorder is more closely related to the combinatorics of the associated homotopy 2-representations. To each left cell, we associate a maximal ideal spectrum, and each maximal ideal gives rise to a differential graded cell 2-representation. We prove that any strong cell is contained in a weak cell and that there is a bijection between the corresponding maximal ideal spectra. Finally, we classify weak and strong cell 2-representations for dg 2-categories of projective bimodules over finite-dimensional differential graded algebras.

Motivated by the so-called H-cell reduction theorems, we investigate certain classes of bicategories which have only one H-cell apart from possibly the identity. We show that H_0-simple quasi fiab bicategories with unique H-cell H_0 are fusion categories. We further study two classes of non-semisimple quasi-fiab bicategories with a single H-cell apart from the identity. The first is \cH_A, indexed by a finite-dimensional radically graded basic Hopf algebra A, and the second is \cG_A, consisting of symmetric projective A-A-bimodules. We show that \cH_A can be viewed as a 1-full subbicategory of \cG_A and classify simple transitive birepresentations for \cG_A. We point out that the number of equivalence classes of the latter is finite, while that for \cH_A is generally not.

In this paper, we use Soergel calculus to define a monoidal functor, called the evaluation functor, from extended affine type A Soergel bimodules to the homotopy category of bounded complexes in finite type A Soergel bimodules. This functor categorifies the well-known evaluation homomorphism from the extended affine type A Hecke algebra to the finite type A Hecke algebra. Through it, one can pull back the triangulated birepresentation induced by any finitary birepresentation of finite type A Soergel bimodules to obtain a triangulated birepresentation of extended affine type A Soergel bimodules. We show that if the initial finitary birepresentation in finite type A is a cell birepresentation, the evaluation birepresentation in extended affine type A has a finitary cover, which we illustrate by working out the case of cell birepresentations with subregular apex in detail.