All Outputs (24)

Presentations for small reflection equation algebras of type A (2025)
Journal Article
Cooke, J., & Laugwitz, R. (2025). Presentations for small reflection equation algebras of type A. Journal of Algebra, 682, 131-187. https://doi.org/10.1016/j.jalgebra.2025.05.037

We give presentations, in terms of the generators and relations, for the reflection equation algebras of type GLn and SLn, i.e., the covariantized algebras of the dual Hopf algebras of the small quantum groups of gln and sln. Our presentations displa... Read More about Presentations for small reflection equation algebras of type A.

Reflective centers of module categories and quantum K-matrices (2025)
Journal Article
Laugwitz, R., Walton, C., & Yakimov, M. (2025). Reflective centers of module categories and quantum K-matrices. Forum of Mathematics, Sigma, 13, Article e95. https://doi.org/10.1017/fms.2025.10055

Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category C and C-module category M, we introduce a version of the Drinfeld center ZpCq of C adapted for M... Read More about Reflective centers of module categories and quantum K-matrices.

Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers (2025)
Journal Article
Flake, J., Laugwitz, R., & Posur, S. (2025). Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers. Advances in Mathematics, 475, Article 110344. https://doi.org/10.1016/j.aim.2025.110344

We identify general conditions, formulated using the projection formula morphisms, for a functor that is simultaneously left and right adjoint to a strong monoidal functor to be a Frobenius monoidal functor. Moreover, we identify stronger conditions... Read More about Frobenius monoidal functors from ambiadjunctions and their lifts to Drinfeld centers.

Pretriangulated 2-representations via dg algebra 1-morphisms (2025)
Journal Article
Laugwitz, R., & Miemietz, V. (in press). Pretriangulated 2-representations via dg algebra 1-morphisms. Documenta Mathematica,

This paper develops a theory of pretriangulated 2-representations of dg 2-categories. We characterize cyclic pretriangulated 2-representations, under certain compactness assumptions, in terms of dg modules over dg algebra 1-morphisms internal to asso... Read More about Pretriangulated 2-representations via dg algebra 1-morphisms.

Infinitesimal 2-braidings from 2-shifted Poisson structures (2025)
Journal Article
Kemp, C., Laugwitz, R., & Schenkel, A. (2025). Infinitesimal 2-braidings from 2-shifted Poisson structures. Journal of Geometry and Physics, 212, Article 105456. https://doi.org/10.1016/j.geomphys.2025.105456

It is shown that every 2-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra A defines a very explicit infinitesimal 2-braiding on the homotopy 2-category of the symmetric monoidal dg-category of finite... Read More about Infinitesimal 2-braidings from 2-shifted Poisson structures.

Differential graded cell 2-representations (2025)
Journal Article
Laugwitz, R., & Miemietz, V. (in press). Differential graded cell 2-representations. Arkiv för Matematik,

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 preorde... Read More about Differential graded cell 2-representations.

The Braids on Your Blanket (2024)
Journal Article
Cheng, M., & Laugwitz, R. U. (2024). The Braids on Your Blanket. Journal of Humanistic Mathematics, 14(2), 286-337. https://doi.org/10.5642/jhummath.YMZO2460

In this expositional essay, we introduce some elements of the study of groups by analysing the braid pattern on a knitted blanket. We determine that the blanket features pure braids with a minimal number of crossings. Moreover, we determine polynomia... Read More about The Braids on Your Blanket.

Planar diagrammatics of self-adjoint functors and recognizable tree series (2023)
Journal Article
Khovanov, M., & Laugwitz, R. (2023). Planar diagrammatics of self-adjoint functors and recognizable tree series. Pure and Applied Mathematics Quarterly, 19(5), 2409-2499. https://doi.org/10.4310/pamq.2023.v19.n5.a4

A pair of biadjoint functors between two categories produces a collection of elements in the centers of these categories, one for each isotopy class of nested circles in the plane. If the centers are equipped with a trace map into the ground field, t... Read More about Planar diagrammatics of self-adjoint functors and recognizable tree series.

Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras (2023)
Journal Article
Hannah, S., Laugwitz, R., & Ros Camacho, A. (2023). Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras. Symmetry, Integrability and Geometry: Methods and Applications, 19, Article 075. https://doi.org/10.3842/sigma.2023.075

We construct a separable Frobenius monoidal functor from Z Vect ω| H H to Z Vect ω G for any subgroup H of G which preserves braiding and ribbon structure. As an application, we classify rigid Frobenius algebras in Z Vect ω G , recovering the classif... Read More about Frobenius Monoidal Functors of Dijkgraaf-Witten Categories and Rigid Frobenius Algebras.

Constructing Non-semisimple Modular Categories with Local Modules (2023)
Journal Article
Laugwitz, R., & Walton, C. (2023). Constructing Non-semisimple Modular Categories with Local Modules. Communications in Mathematical Physics, 403, 1363-1409. https://doi.org/10.1007/s00220-023-04824-4

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again, modular. This generalizes previous work of Kirillov and Ostrik (Adv Math 171(2):183–227, 2002) in the... Read More about Constructing Non-semisimple Modular Categories with Local Modules.

Indecomposable objects in Khovanov–Sazdanovic's generalizations of Deligne's interpolation categories (2023)
Journal Article
Flake, J., Laugwitz, R., & Posur, S. (2023). Indecomposable objects in Khovanov–Sazdanovic's generalizations of Deligne's interpolation categories. Advances in Mathematics, 415, Article 108892. https://doi.org/10.1016/j.aim.2023.108892

Khovanov and Sazdanovic recently introduced symmetric monoidal categories parameterized by rational functions and given by quotients of categories of two-dimensional cobordisms. These categories generalize Deligne's interpolation categories of repres... Read More about Indecomposable objects in Khovanov–Sazdanovic's generalizations of Deligne's interpolation categories.

A categorification of cyclotomic rings (2023)
Journal Article
Laugwitz, R., & Qi, Y. (2023). A categorification of cyclotomic rings. Quantum Topology, 13(3), 539-577. https://doi.org/10.4171/qt/172

For any natural number n≥2, we construct a triangulated monoidal category whose Grothendieck ring is isomorphic to the ring of cyclotomic integers On​. This construction provides an affirmative resolution to a problem raised by Khovanov in 2005.

The indecomposable objects in the center of Deligne's category Rep St (2023)
Journal Article
Flake, J., Harman, N., & Laugwitz, R. (2023). The indecomposable objects in the center of Deligne's category Rep St. Proceedings of the London Mathematical Society, 126(4), 1134-1181. https://doi.org/10.1112/plms.12509

We classify the indecomposable objects in the monoidal center of Deligne's interpolation category Rep St by viewing Rep St as a model‐theoretic limit in rank and characteristic. We further prove that the center of Rep St is semisimple if and only if... Read More about The indecomposable objects in the center of Deligne's category Rep St.

Constructing Non-Semisimple Modular Categories with Relative Monoidal Centers (2021)
Journal Article
Laugwitz, R., & Walton, C. (2022). Constructing Non-Semisimple Modular Categories with Relative Monoidal Centers. International Mathematics Research Notices, 2022(20), 15826-15868. https://doi.org/10.1093/imrn/rnab097

This paper is a contribution to the construction of non-semisimple modular categories. We establish when Müger centralizers inside non-semisimple modular categories are also modular. As a consequence, we obtain conditions under which relative monoida... Read More about Constructing Non-Semisimple Modular Categories with Relative Monoidal Centers.

On the monoidal center of Deligne's category Re̲p(St) (2020)
Journal Article
Flake, J., & Laugwitz, R. (2021). On the monoidal center of Deligne's category Re̲p(St). Journal of the London Mathematical Society, 103(3), 1153-1185. https://doi.org/10.1112/jlms.12403

We explicitly compute a monoidal subcategory of the monoidal center of Deligne’s interpolation category Rep(St), for t not necessarily a natural number, and we show that this subcategory is a ribbon category. For t = n, a natural number, there exists... Read More about On the monoidal center of Deligne's category Re̲p(St).

Braided commutative algebras over quantized enveloping algebras (2020)
Journal Article
Laugwitz, R., & Walton, C. (2021). Braided commutative algebras over quantized enveloping algebras. Transformation Groups, 26, 957-993. https://doi.org/10.1007/s00031-020-09599-9

We produce braided commutative algebras in braided monoidal categories by generalizing Davydov's full center construction of commutative algebras in centers of monoidal categories. Namely, we build braided commutative algebras in relative monoidal ce... Read More about Braided commutative algebras over quantized enveloping algebras.

Noncommutative Shifted Symmetric Functions (2020)
Journal Article
Laugwitz, R., & Retakh, V. (2020). Noncommutative Shifted Symmetric Functions. Moscow Mathematical Journal, 20(1), 93-126

We introduce a ring of noncommutative shifted symmetric functions based on an integer-indexed sequence of shift parameters. Using generating series and quasideterminants, this multiparameter approach produces deformations of the ring of noncommutativ... Read More about Noncommutative Shifted Symmetric Functions.

Cell 2-Representations and Categorification at Prime Roots of Unity (2019)
Journal Article
Laugwitz, R., & Miemietz, V. (2020). Cell 2-Representations and Categorification at Prime Roots of Unity. Advances in Mathematics, 361, Article 106937. https://doi.org/10.1016/j.aim.2019.106937

Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2- categories, enriched with a p-differential, which satisfy finiteness conditions analogous to those of fini... Read More about Cell 2-Representations and Categorification at Prime Roots of Unity.

The relative monoidal center and tensor products of monoidal categories (2019)
Journal Article
Laugwitz, R. (2020). The relative monoidal center and tensor products of monoidal categories. Communications in Contemporary Mathematics, 22(8), Article 1950068. https://doi.org/10.1142/s0219199719500688

This paper develops a theory of monoidal categories relative to a braided monoidal category, called augmented monoidal categories. For such categories, balanced bimodules are defined using the formalism of balanced functors. It is shown that there ex... Read More about The relative monoidal center and tensor products of monoidal categories.

Comodule algebras and 2-cocycles over the (Braided) Drinfeld double (2019)
Journal Article
Laugwitz, R. (2019). Comodule algebras and 2-cocycles over the (Braided) Drinfeld double. Communications in Contemporary Mathematics, 21(04), Article 1850045. https://doi.org/10.1142/S0219199718500451

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 objec... Read More about Comodule algebras and 2-cocycles over the (Braided) Drinfeld double.