arXiv subjects

Bregje Pauwels

Publications and source records attributed to Bregje Pauwels.

5 recordsLinked to original sources

arXiv2025

Semisimplifying categorical Heisenberg actions and periodic equivalences

We systematically apply semisimplification functors in modular representation theory. Motivated by the Duflo--Serganova functor in Lie superalgebras, we construct various functors of interest. In the setting of finite groups, we refine the cyclic group Brauer construction and categorify the Glauberman correspondence. In the setting of degenerate categorical Heisenberg actions, we obtain a rich collection of functors which commute with the categorical action. Applied to well-known categorifications of the basic representation and Fock space, our functors give explicit realizations of periodic equivalences for polynomial functors and symmetric groups first studied by Henke-Koenig. This allows us to globalize the equivalences of Henke-Koenig by symmetric monoidal functors. We apply these results to deduce branching properties of certain modular representations of $S_n$.

math.RT↗

arXiv2023

Gluing Approximable Triangulated Categories

Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is approximable if this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that the derived category of a quasi compact, separated scheme is approximable has led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier. In this article we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable. Furthermore, the lemmas and techniques developed in this article form a powerful toolbox which has interesting applications in existing and forthcoming work by the authors.

math.CT↗

arXiv2022

Monoidal Abelian Envelopes with a quotient property

We study abelian envelopes for pseudo-tensor categories with the property that every object in the envelope is a quotient of an object in the pseudo-tensor category. We establish an intrinsic criterion on pseudo-tensor categories for the existence of an abelian envelope satisfying this quotient property. This allows us to interpret the extension of scalars and Deligne tensor product of tensor categories as abelian envelopes, and to enlarge the class of tensor categories for which all extensions of scalars and tensor products are known to remain tensor categories. For an affine group scheme G, we show that pseudo-tensor subcategories of RepG have abelian envelopes with the quotient property, and we study many other such examples. This leads us to conjecture that all abelian envelopes satisfy the quotient property.

math.CT↗

arXiv2018

The Gerstenhaber structure on the Hochschild cohomology of a class of special biserial algebras

We determine the Gerstenhaber structure on the Hochschild cohomology ring of a class of self-injective special biserial algebras. Each of these algebras is presented as a quotient of the path algebra of a certain quiver. In degree one, we show that the cohomology is isomorphic, as a Lie algebra, to a direct sum of copies of a subquotient of the Virasoro algebra. These copies share Virasoro degree 0 and commute otherwise. Finally, we describe the cohomology in degree $n$ as a module over this Lie algebra by providing its decomposition as a direct sum of indecomposable modules.

math.RA↗

arXiv2016

Quasi-Galois theory in symmetric-monoidal categories

Given a ring object $A$ in a symmetric monoidal category, we investigate what it means for the extension $\mathbb{1}\rightarrow A$ to be (quasi-)Galois. In particular, we define splitting ring extensions and examine how they occur. Specializing to tensor-triangulated categories, we study how extension-of-scalars along a quasi-Galois ring object affects the Balmer spectrum. We define what it means for a separable ring to have constant degree, which is a necessary and sufficient condition for the existence of a quasi-Galois closure. Finally, we illustrate the above for separable rings occurring in modular representation theory.

math.CT↗