All Outputs (17)

The braids on your blanket (2024)
Journal Article
Cheng, M., & Laugwitz, R. (in press). The braids on your blanket. Journal of Humanistic Mathematics,

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.

Algebras of quasi-Plücker coordinates are Koszul (2017)
Journal Article
Laugwitz, R., & Retakh, V. (2018). Algebras of quasi-Plücker coordinates are Koszul. Journal of Pure and Applied Algebra, 222(9), 2810-2822. https://doi.org/10.1016/j.jpaa.2017.11.001

Motivated by the theory of quasi-determinants, we study non-commutative algebras of quasi-Plücker coordinates. We prove that these algebras provide new examples of non-homogeneous quadratic Koszul algebras by showing that their quadratic duals have q... Read More about Algebras of quasi-Plücker coordinates are Koszul.

Pointed Hopf Algebras with Triangular Decomposition: A Characterization of Multiparameter Quantum Groups (2016)
Journal Article
Laugwitz, R. (2016). Pointed Hopf Algebras with Triangular Decomposition: A Characterization of Multiparameter Quantum Groups. Algebras and Representation Theory, 19(3), 547-578. https://doi.org/10.1007/s10468-015-9588-x

© 2016, The Author(s). In this paper, we present an approach to the definition of multiparameter quantum groups by studying Hopf algebras with triangular decomposition. Classifying all of these Hopf algebras which are of what we call weakly separable... Read More about Pointed Hopf Algebras with Triangular Decomposition: A Characterization of Multiparameter Quantum Groups.

Braided Drinfeld and Heisenberg doubles (2015)
Journal Article
Laugwitz, R. (2015). Braided Drinfeld and Heisenberg doubles. Journal of Pure and Applied Algebra, 219(10), 4541-4596. https://doi.org/10.1016/j.jpaa.2015.02.031

© 2015 Elsevier B.V. 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... Read More about Braided Drinfeld and Heisenberg doubles.