Comodule Algebras and 2-Cocycles over the (Braided) Drinfeld Double

We show that for dually paired bialgebras, every comodule algebra over one of the paired bialgebras gives a comodule algebra over their Drinfeld double via a crossed product construction. These constructions generalize to working with bialgebra objects in a braided monoidal category of modules over a quasitriangular Hopf algebra. Hence two ways to provide comodule algebras over the braided Drinfeld double (the double bosonization) are provided. Furthermore, a map of second Hopf algebra cohomology spaces is constructed. It takes a pair of 2-cocycles over dually paired Hopf algebras and produces a 2-cocycle over their Drinfeld double. This construction also has an analogue for braided Drinfeld doubles.Comment: 34 pages, several figures created with Inkscape. Version 2: more references added, small typos corrected. Version 3: Small corrections and further references added. To appear in Comm. Contemp. Mat

Algebras of Quasi-Pl\"ucker Coordinates are Koszul

Motivated by the theory of quasi-determinants, we study non-commutative algebras of quasi-Pl\"ucker coordinates. We prove that these algebras provide new examples of non-homogeneous quadratic Koszul algebras by showing that their quadratic duals have quadratic Gr\"obner bases.Comment: 15 pages, AMS tex, tikz figures. V2: small corrections based on referee report. V3: Further minor changes. To appear in J. Pure Appl. Algebr

A categorification of cyclotomic rings

For any natural number $n \geq 2$, we construct a triangulated monoidal category whose Grothendieck ring is isomorphic to the ring of cyclotomic integers $\mathbb{O}_n$.Comment: 28 pages. Comments welcome! v2, v3: minor correction

The right kind of explanation: Validity in automated hate speech detection

To quickly identify hate speech online, communication research offers a useful tool in the form of automatic content analysis. However, the combined methods of standardized manual content analysis and supervised text classification demand different quality criteria. This chapter shows that a more substantial examination of validity is necessary since models often learn on spurious correlations or biases, and researchers run the risk of drawing wrong inferences. To investigate the overlap of theoretical concepts with technological operationalization, explainability methods are evaluated to explain what a model has learned. These methods proved to be of limited use in testing the validity of a model when the generated explanations aim at sense-making rather than faithfulness to the model. The chapter ends with recommendations for further interdisciplinary development of automatic content analysis

Braided Drinfeld and Heisenberg Doubles

In this paper, the Drinfeld center of a monoidal category is generalized to a class of mixed Drinfeld centers. This gives a unified picture for the Drinfeld center and a natural Heisenberg analogue. Further, there is an action of the former on the latter. This picture is translated to a description in terms of Yetter-Drinfeld and Hopf modules over quasi-bialgebras in a braided monoidal category. Via braided reconstruction theory, intrinsic definitions of braided Drinfeld and Heisenberg doubles are obtained, together with a generalization of the result in [Lu94] that the Heisenberg double is a 2-cocycle twist of the Drinfeld double for general braided Hopf algebras.Comment: 76 pages, minor corrections to V