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Chelsea Walton

Publications and source records attributed to Chelsea Walton.

At least 19 recordsLinked to original sources
arXiv2025

Morita invariants of quasitriangular coideal subalgebras

We use representations of braid groups of Coxeter types BC and D to produce invariants of representation categories of quasitriangular coideal subalgebras. Such categories form a prevalent class of braided module categories. This is analogous to how representations of braid groups of Coxeter type A produce invariants of representation categories of quasitriangular Hopf algebras, a prevalent class of braided monoidal categories. This work also includes concrete examples, and classification results for $K$-matrices of quasitriangular coideal subalgebras.

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arXiv2025

Division algebras in monoidal categories

This work adapts the equivalent definitions of division algebras over a field into multiple types of division algebras in a monoidal category. Examples and consequences of these definitions are then established in various monoidal settings.

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arXiv2024

Nondegenerate module categories

Due to the work of Shimizu (2019), various nondegeneracy conditions for braided finite tensor categories are equivalent. This theory is partially extended to braided module categories here. We introduce when a braided module category is "nondegenerate" and "factorizable", and establish that these properties are equivalent. The proof involves a new monadicity result for module categories. Lastly, we examine the Hopf case, using Kolb's (2020) notion of a quasitriangular comodule algebra to introduce "factorizable" comodule algebras. We then show that the representation category of a quasitriangular comodule algebra is nondegenerate in our sense precisely when the comodule algebra is factorizable. Several examples are provided.

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arXiv2024

On extended Frobenius structures

A classical result in quantum topology is that oriented 2-dimensional topological quantum field theories (2-TQFTs) are fully classified by commutative Frobenius algebras. In 2006, Turaev and Turner introduced additional structure on Frobenius algebras, forming what are called extended Frobenius algebras, to classify 2-TQFTs in the unoriented case. This work provides a systematic study of extended Frobenius algebras in various settings: over a field, in a monoidal category, and in the framework of monoidal functors. Numerous examples, classification results, and general constructions of extended Frobenius algebras are established.

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arXiv2023

Twists of graded algebras in monoidal categories

Zhang twists are a common tool for deforming graded algebras over a field in a way that preserves important ring-theoretic properties. We generalize Zhang twists to the setting of closed monoidal categories equipped with their self-enriched structure. Along the way, we prove several key results about algebraic structures in closed monoidal categories missing from the literature. We use these to ultimately prove Morita-type results, showcasing when graded algebras with equivalent categories of graded modules can be related by Zhang twists.

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arXiv2023

Reflective centers of module categories and quantum K-matrices

Our work is motivated by obtaining solutions to the quantum reflection equation (qRE) by categorical methods. To start, given a braided monoidal category $\mathcal{C}$ and $\mathcal{C}$-module category $\mathcal{M}$, we introduce a version of the Drinfeld center $\mathcal{Z}(\mathcal{C})$ of $\mathcal{C}$ adapted for $\mathcal{M}$; we refer to this category as the "reflective center" $\mathcal{E}_{\mathcal{C}}(\mathcal{M})$ of $\mathcal{M}$. Just like $\mathcal{Z}(\mathcal{C})$ is a canonical braided monoidal category attached to $\mathcal{C}$, we show that $\mathcal{E}_{\mathcal{C}}(\mathcal{M})$ is a canonical braided module category attached to $\mathcal{M}$; its properties are investigated in detail. Our second goal pertains to when $\mathcal{C}$ is the category of modules over a quasitriangular Hopf algebra $H$, and $\mathcal{M}$ is the category of modules over an $H$-comodule algebra $A$. We show that the reflective center $\mathcal{E}_{\mathcal{C}}(\mathcal{M})$ here is equivalent to a category of modules over an explicit algebra, denoted by $R_H(A)$, which we call the "reflective algebra" of $A$. This result is akin to $\mathcal{Z}(\mathcal{C})$ being represented by the Drinfeld double Drin($H$) of $H$. We also study the properties of reflective algebras. Our third set of results is also in the Hopf setting above. We show that reflective algebras are quasitriangular $H$-comodule algebras, and examine their corresponding quantum $K$-matrices; this yields solutions to the qRE. We also establish that the reflective algebra $R_H(\Bbbk)$ is an initial object in the category of quasitriangular $H$-comodule algebras, where $\Bbbk$ is the ground field. The case when $H$ is the Drinfeld double of a finite group is illustrated.

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arXiv2022

On non-counital Frobenius algebras

A Frobenius algebra is a finite-dimensional algebra $A$ which comes equipped with a coassociative, counital comultiplication map $\Delta$ that is an $A$-bimodule map. Here, we examine comultiplication maps for generalizations of Frobenius algebras: finite-dimensional self-injective (quasi-Frobenius) algebras. We show that large classes of such algebras, including finite-dimensional weak Hopf algebras, come equipped with a nonzero map $\Delta$ as above that is not necessarily counital. We also conjecture that this comultiplicative structure holds for self-injective algebras in general.

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arXiv2022

Constructing non-semisimple modular categories with local modules

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 A. Kirillov, Jr. and V. Ostrik [Adv. Math. 171 (2002), no. 2] in the semisimple setup. Examples of non-semisimple modular categories via local modules, as well as connections to the authors' prior work on relative monoidal centers, are provided. In particular, we classify rigid Frobenius algebras in Drinfeld centers of module categories over group algebras, thus generalizing the classification by A. Davydov [J. Algebra 323 (2010), no. 5] to arbitrary characteristic.

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arXiv2021

Filtered Frobenius algebras in monoidal categories

We develop filtered-graded techniques for algebras in monoidal categories with the main goal of establishing a categorical version of Bongale's 1967 result: A filtered deformation of a Frobenius algebra over a field is Frobenius as well. Towards the goal, we first construct a monoidal associated graded functor, building on prior works of Ardizzoni-Menini, of Galatius et al., and of Gwillian-Pavlov. Next, we produce equivalent conditions for an algebra in a rigid monoidal category to be Frobenius in terms of the existence of categorical Frobenius form; this builds on work of Fuchs-Stigner. These two results of independent interest are then used to achieve our goal. As an application of our main result, we show that any exact module category over a symmetric finite tensor category $\mathcal{C}$ is represented by a Frobenius algebra in $\mathcal{C}$. Several directions for further investigation are also proposed.

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arXiv2021

Algebraic properties of face algebras

Prompted an inquiry of Manin on whether a coacting Hopf-type structure $H$ and an algebra $A$ that is coacted upon share algebraic properties, we study the particular case of $A$ being a path algebra $\Bbbk Q$ of a finite quiver $Q$ and $H$ being Hayashi's face algebra $\mathfrak{H}(Q)$ attached to $Q$. This is motivated by the work of Huang, Wicks, Won, and the second author, where it was established that the weak bialgebra coacting universally on $\Bbbk Q$ (either from the left, right, or both sides compatibly) is $\mathfrak{H}(Q)$. For our study, we define the Kronecker square $\widehat{Q}$ of $Q$, and show that $\mathfrak{H}(Q) \cong \Bbbk \widehat{Q}$ as unital algebras. Then we obtain ring-theoretic and homological properties of $\mathfrak{H}(Q)$ in terms of graph-theoretic properties of $Q$ by way of $\widehat{Q}$.

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arXiv2020

Constructing non-semisimple modular categories with relative monoidal centers

This paper is a contribution to the construction of non-semisimple modular categories. We establish when M\"uger centralizers inside non-semisimple modular categories are also modular. As a consequence, we obtain conditions under which relative monoidal centers give (non-semisimple) modular categories, and we also show that examples include representation categories of small quantum groups. We further derive conditions under which representations of more general quantum groups, braided Drinfeld doubles of Nichols algebras of diagonal type, give (non-semisimple) modular categories.

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arXiv2020

Universal quantum semigroupoids

We introduce the concept of a universal quantum linear semigroupoid (UQSGd), which is a weak bialgebra that coacts on a (not necessarily connected) graded algebra $A$ universally while preserving grading. We restrict our attention to algebraic structures with a commutative base so that the UQSGds under investigation are face algebras (due to Hayashi). The UQSGd construction generalizes the universal quantum linear semigroups introduced by Manin in 1988, which are bialgebras that coact on a connected graded algebra universally while preserving grading. Our main result is that when $A$ is the path algebra $\Bbbk Q$ of a finite quiver $Q$, each of the various UQSGds introduced here is isomorphic to the face algebra attached to $Q$. The UQSGds of preprojective algebras and of other algebras attached to quivers are also investigated.

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arXiv2020

Algebraic structures in group-theoretical fusion categories

It was shown by Ostrik (2003) and Natale (2017) that a collection of twisted group algebras in a pointed fusion category serve as explicit Morita equivalence class representatives of indecomposable, separable algebras in such categories. We generalize this result by constructing explicit Morita equivalence class representatives of indecomposable, separable algebras in group-theoretical fusion categories. This is achieved by providing the free functor $\Phi$ from fusion category to a category of bimodules in the original category with a (Frobenius) monoidal structure. Our algebras of interest are then constructed as the image of twisted group algebras under $\Phi$. We also show that twisted group algebras admit the structure of Frobenius algebras in a pointed fusion category, and as a consequence, our algebras are Frobenius algebras in a group-theoretical fusion category. They also enjoy several good algebraic properties.

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arXiv2019

Algebraic structures in comodule categories over weak bialgebras

For a bialgebra $L$ coacting on a $\Bbbk$-algebra $A$, a classical result states that $A$ is a right $L$-comodule algebra if and only if $A$ is an algebra in the monoidal category $\mathcal{M}^{L}$ of right $L$-comodules; the former notion is formulaic while the latter is categorical. We generalize this result to the setting of weak bialgebras $H$. The category $\mathcal{M}^H$ admits a monoidal structure by work of Nill and B\"{o}hm-Caenepeel-Janssen, but the algebras in $\mathcal{M}^H$ are not canonically $\Bbbk$-algebras. Nevertheless, we prove that there is an isomorphism between the category of right $H$-comodule algebras and the category of algebras in $\mathcal{M}^H$. We also recall and introduce the formulaic notion of $H$ coacting on a $\Bbbk$-coalgebra and on a Frobenius $\Bbbk$-algebra, respectively, and prove analogous category isomorphism results. Our work is inspired by the physical applications of Frobenius algebras in tensor categories and by symmetries of algebras with a base algebra larger than the ground field (e.g. path algebras). We produce examples of the latter by constructing a monoidal functor from a certain corepresentation category of a bialgebra $L$ to the corepresentation category of a weak bialgebra built from $L$ (a "quantum transformation groupoid"), thereby creating weak quantum symmetries from ordinary quantum symmetries.

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arXiv2019

Tensor algebras in finite tensor categories

This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers, and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite (multi-)tensor categories $\mathcal{C}$, classifying tensor algebras in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. We obtain two classification results for actions of semisimple Hopf algebras: the first for actions which preserve the ascending filtration on tensor algebras, and the second for actions which preserve the descending filtration on completed tensor algebras. Extending to more general fusion categories, we illustrate our classification result for tensor algebras in the pointed fusion categories ${\sf Vec}_{G}^{\omega}$ and in group-theoretical fusion categories, especially for the representation category of the Kac-Paljutkin Hopf algebra.

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arXiv2019

Braided commutative algebras over quantized enveloping algebras

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 centers $\mathcal{Z}_{\mathcal{B}}(\mathcal{C})$ from algebras in $\mathcal{B}$-central monoidal categories $\mathcal{C}$, where $\mathcal{B}$ is an arbitrary braided monoidal category; Davydov's (and previous works of others) take place in the special case when $\mathcal{B}$ is the category of vector spaces $\mathbf{Vect}$ over a field $\Bbbk$. Since key examples of relative monoidal centers are suitable representation categories of quantized enveloping algebras, we supply braided commutative module algebras over such quantum groups. One application of our work is that we produce Morita invariants for algebras in $\mathcal{B}$-central monoidal categories. Moreover, for a large class of $\mathcal{B}$-central monoidal categories, our braided commutative algebras arise as a braided version of centralizer algebras. This generalizes the fact that centers of algebras in $\mathbf{Vect}$ serve as Morita invariants. Many examples are provided throughout.

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arXiv2018

Noncommutative Kn\"{o}rrer periodicity and noncommutative Kleinian singularities

We establish a version of Kn\"{o}rrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let $A$ be a left noetherian AS-regular algebra, let $f$ be a normal and regular element of $A$ of positive degree, and take $B=A/(f)$. Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over $B$ and those over (a noncommutative analog of) its second double branched cover $(B^\#)^\#$. Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with Cassidy. These results are applied to the noncommutative Kleinian singularities studied by the second and fourth authors with Chan and Zhang.

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