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Julia Plavnik

Publications and source records attributed to Julia Plavnik.

At least 19 recordsLinked to original sources
arXiv2025

Pivotal Brauer-Picard groupoids and graded extensions

We develop pivotal and spherical versions of graded extension theory. We define the corresponding analogues of Brauer-Picard $2$-categorical groups and realize them as fixed points of natural $\mathbb{Z}$ and $\mathbb{Z}/2\mathbb{Z}$ $2$-categorical actions. We classify graded extensions of a pivotal tensor category by monoidal $2$-functors into the pivotal Brauer-Picard $2$-categorical group. A similar statement is proven for spherical (unimodular) tensor categories. We also develop an obstruction theory for determining when pivotal and spherical structures can be extended.

math.QA↗
arXiv2025

Relation between bicrossed products and crossed extensions of fusion categories

We show that all crossed extensions defined by Natale can be recovered as duals of bicrossed products of fusion categories. As an application, we prove that any exact factorization between a pointed fusion category $\operatorname{vec}_G$ and a fusion category $\mathcal{C}$ can be realized as a bicrossed product $\operatorname{vec}_G\bowtie \mathcal{C}$.

math.QA↗
arXiv2025

Egyptian fractions for few primes

We study solutions to the Egyptian fractions equation with the prime factors of the denominators constrained to lie in a fixed set of primes. We evaluate the effectiveness of the greedy algorithm in establishing bounds on such solutions. Additionally, we present improved algorithms for generating low-rank solutions and solutions restricted to specific prime sets. Computational results obtained using these algorithms are provided, alongside a discussion on their performance.

math.NT↗
arXiv2025

Relating insplittings of 2-graphs and of textile systems

The graphical operation of insplitting is key to understanding conjugacy of shifts of finite type (SFTs) in both one and two dimensions. In this paper, we consider two approaches to studying 2-dimensional SFTs: textile systems and rank-2 graphs. Nasu's textile systems describe all two-sided 2D SFTs up to conjugacy, whereas the 2-graphs (higher-rank graphs of rank 2) introduced by Kumjian and Pask yield associated C*-algebras. Both models have a naturally-associated notion of insplitting. We show that these notions do not coincide, raising the question of whether insplitting a 2-graph induces a conjugacy of the associated one-sided 2-dimensional SFTs. Our first main result shows how to reconstruct 2-graph insplitting using textile-system insplits and inversions, and consequently proves that 2-graph insplitting induces a conjugacy of dynamical systems. We also present several other facets of the relationship between 2-graph insplitting and textile-system insplitting. Incorporating an insplit of the bottom graph of the textile system turns out to be key to this relationship. By articulating the connection between operator-algebraic and dynamical notions of insplitting in two dimensions, this article lays the groundwork for a C*-algebraic framework for classifying one-sided conjugacy in higher-dimensional SFTs.

math.OA↗
arXiv2025

Generalised Orbifolds and G-equivariantisation

In a construction motivated by topological field theory, a so-called orbifold datum $\mathbb{A}$ in a ribbon category $C$ allows one to define a new ribbon category $C_{\mathbb{A}}$. If $C$ is the neutral component of a $G$-crossed ribbon category $B$, and $\mathbb{A}$ is an orbifold datum in $C$ defined in terms of $B$, one finds that $C_{\mathbb{A}}$ is equivalent to the equivariantisation $B^G$ of $B$ as a ribbon category. We give a constructive proof of this equivalence.

math.QA↗
arXiv2025

On the finite generation of the cohomology of bosonizations

We use deformation sequences of (Hopf) algebras, extending the results of Negron and Pevtsova, to show that bosonizations of some suitable braided Hopf algebras by some suitable finite-dimensional Hopf algebras have finitely generated cohomology. In fact, our results are shown in more generality for smash products. As applications, we prove the bosonizations of some Nichols algebras (such as Nichols algebras of diagonal type, the restricted Jordan plane, Nichols algebras of direct sums of Jordan blocks plus points labeled with 1), by some suitable finite-dimensional Hopf algebras, have finitely generated cohomology, recovering some known results as well as providing new examples.

math.QA↗
arXiv2024

The Classification of Fusion 2-Categories

We classify (multi)fusion 2-categories in terms of braided fusion categories and group cohomological data. This classification is homotopy coherent -- we provide an equivalence between the 3-groupoid of (multi)fusion 2-categories up to monoidal equivalences and a certain 3-groupoid of commuting squares of $\mathrm{B}\mathbb{Z}/2$-equivariant spaces. Rank finiteness and Ocneanu rigidity for fusion 2-categories are immediate corollaries of our classification.

math.CT↗
arXiv2024

The Condensed Fiber Product and Zesting

We introduce the condensed fiber product of two $G$-crossed braided fusion categories, generalizing existing constructions in the literature. We show that this product is closely related to the cohomological construction known as zesting. Furthermore, the condensed fiber product defines a monoidal structure on the 2-category of $G$-crossed and braided extensions of braided categories with a fixed transparent symmetric pointed subcategory.

math.QA↗
arXiv2024

On bicrossed product of fusion categories and exact factorizations

We introduce the notion of a matched pair of fusion rings and fusion categories, generalizing the one for groups. Using this concept, we define the bicrossed product of fusion rings and fusion categories and we construct exact factorizations for them. This concept generalizes the bicrossed product, also known as external Zappa-Sz\'ep product, of groups. We also show that every exact factorization of fusion rings can be presented as a bicrossed product. With this characterization, we describe the adjoint subcategory and universal grading group of an exact factorization of fusion categories. We give explicit fusion rules and associativity constraints for examples of fusion categories arising as a bicrossed product of combinations of Tambara-Yamagami categories and pointed fusion categories. These examples are new to the best of the knowledge of the authors.

math.QA↗
arXiv2024

On Frobenius-Schur exponent bounds

Here we study bounds on the Frobenius-Schur exponent of spherical fusion categories based on their global dimension generalizing bounds from the representation theory of finite-dimensional quasi-Hopf algebras. Our main result is that if the Frobenius-Schur exponent of a modular fusion category is a prime power for some prime integer $p$, then it is bounded by the norm of its global dimension when $p$ is odd, and four times the norm of its global dimension when $p=2$; these bounds are optimal. If one assumes in addition pseudounitarity, the categories achieving the optimal bound are completely described and we attain similar bounds and classifications for arbitrary spherical fusion categories. This proof includes an explicit classification of modular fusion categories of Frobenius-Perron dimension $p^5$; all examples which are not pointed are constructed explicitly from the Ising categories and the representation theory of extraspecial $p$-groups.

math.QA↗
arXiv2024

On odd-dimensional modular tensor categories

We study odd-dimensional modular tensor categories and maximally non-self dual (MNSD) modular tensor categories of low rank. We give lower bounds for the ranks of modular tensor categories in terms of the rank of the adjoint subcategory and the order of the group of invertible objects. As an application of these results, we prove that MNSD modular tensor categories of ranks 13 and 15 are pointed. In addition, we show that MNSD tensor categories of ranks 17, 19, 21 and 23 are either pointed or perfect.

math.QA↗
arXiv2024

Contragredient Lie algebras in symmetric categories

We define contragredient Lie algebras in symmetric categories, generalizing the construction of Lie algebras of the form $\mathfrak{g}(A)$ for a Cartan matrix $A$ from the category of vector spaces to an arbitrary symmetric tensor category. The main complication resides in the fact that, in contrast to the classical case, a general symmetric tensor category can admit tori (playing the role of Cartan subalgebras) which are non-abelian and have a sophisticated representation theory. Using this construction, we obtain and describe new examples of Lie algebras in the universal Verlinde category in characteristic $p\geq5$. We also show that some previously known examples can be obtained with our construction.

math.QA↗
arXiv2023

Classification of low-rank odd-dimensional modular categories

We prove that any odd-dimensional modular category of rank at most 23 is pointed. We also show that an odd-dimensional modular category of rank 25 is either pointed, perfect, or equivalent to $\operatorname{Rep}(D^ω(\mathbb Z_7\rtimes \mathbb Z_3))$. Finally, we give partial classification results for modular categories of rank up to 73.

math.QA↗
arXiv2023

Semisimplification of contragredient Lie algebras

We describe the structure and different features of Lie algebras in the Verlinde category, obtained as semisimplification of contragredient Lie algebras in characteristic $p$ with respect to the adjoint action of a Chevalley generator. In particular, we construct a root system for these algebras that arises as a parabolic restriction of the known root system for the classical Lie algebra. This gives a lattice grading with simple homogeneous components and a triangular decomposition for the semisimplified Lie algebra. We also obtain a non-degenerate invariant form that behaves well with the lattice grading. As an application, we exhibit concrete new examples of Lie algebras in the Verlinde category.

math.RT↗
arXiv2023

On modular categories with Frobenius-Perron dimension congruent to 2 modulo 4

We contribute to the classification of modular categories $\mathcal{C}$ with $\operatorname{FPdim}(\mathcal{C})\equiv 2 \pmod 4$. We prove that such categories have group of invertibles of even order, and that they factorize as $\mathcal C\cong \widetilde{\mathcal C} \boxtimes \operatorname{sem}$, where $\widetilde{\mathcal C}$ is an odd-dimensional modular category and $\operatorname{sem}$ is the rank 2 pointed modular category. This reduces the classification of these categories to the classification of odd-dimensional modular categories. It follows that modular categories $\mathcal C$ with $\operatorname{FPdim}(\mathcal{C})\equiv 2 \pmod 4$ of rank up to 46 are pointed. More generally, we prove that if $\mathcal C$ is a weakly integral MTC and $p$ is an odd prime dividing the order of the group of invertibles that has multiplicity one in $\operatorname{FPdim}(\mathcal C)$, then we have a factorization $\mathcal C \cong \widetilde{\mathcal C} \boxtimes \operatorname{Vec}_{\mathbb Z_p}^χ,$ for $\widetilde{\mathcal C}$ an MTC with dimension not divisible by $p$.

math.QA↗
arXiv2023

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 $Φ$ 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 $Φ$. 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.

math.QA↗
arXiv2023

Tambara-Yamagami Categories over the Reals: The Non-Split Case

Tambara and Yamagami investigated a simple set of fusion rules with only one non-invertible object, and proved under which circumstances those rules could be given a coherent associator. We consider a generalization of such fusion rules to the setting where simple objects are no longer required to be split simple. Over the real numbers, this means that objects are either real, complex, or quaternionic. In this context, we prove a similar categorification result to the one of Tambara and Yamagami.

math.QA↗