Julia Plavnik

Given an exact module category over a finite tensor category with finitely generated cohomology, we show that if there exists an object of complexity at least three, then the category is of wild representation type. In particular, if the Krull dimension of the cohomology ring of the finite tensor category is at least three, then this category is wild.

The Yang-Baxter equation is a famous equation in mathematics and mathematical physics. It plays a central role in several areas of mathematics, including algebra, topology, and quantum field theory. The aim of the workshop is to review recent developments in areas where the Yang-Baxter equation is crucial to discuss new research directions and ideas for addressing open problems.

Tambara and Yamagami investigated a simple set of fusion rules with only one noninvertible object, and proved under which circumstances those rules could be given a coherent associator. They also classified all of the resulting fusion categories up to monoidal equivalence. 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 simple objects are either real, complex, or quaternionic. In this context, we prove a similar categorification result to the one of Tambara and Yamagami.

We contribute to the classification of modular categories C \mathcal {C} with F P d i m ( C ) ≡ 2 ( mod 4 ) FPdim(\mathcal {C})\equiv 2 \pmod 4 . We prove that such categories have group of invertibles of even order and that they factorize as C ≅ C ~ ⊠ P \mathcal C\cong \widetilde {\mathcal C} \boxtimes \mathcal P , where C ~ \widetilde {\mathcal C} is an odd-dimensional modular category and P \mathcal P is a pointed modular category of dimension 2. This reduces the classification of these categories to the classification of odd-dimensional modular categories. It follows that modular categories C \mathcal {C} with F P d i m ( C ) ≡ 2 ( mod 4 ) FPdim(\mathcal {C})\equiv 2 \pmod 4 of rank up to 46 (except for 34) are pointed. More generally, we prove that if C \mathcal {C} is a weakly integral modular category and p p is an odd prime dividing the order of the group of invertibles that has multiplicity one in F P d i m ( C ) FPdim(\mathcal {C}) , then we have a factorization C ≅ C ~ ⊠ P \mathcal {C} \cong \tilde {\mathcal {C}} \boxtimes \mathcal {P} , for C ~ \tilde {\mathcal {C}} a modular category with dimension not divisible by p p and P \mathcal {P} a pointed modular category of dimension p p .

Given an exact module category over a finite tensor category with finitely generated cohomology, we show that if there exists an object of complexity at least three, then the category is of wild representation type. In particular, if the Krull dimension of the cohomology ring of the finite tensor category is at least three, then this category is wild.

We show that the ribbon zesting construction can produce modular isotopes —different modular fusion categories with the same modular data. The result relies on the observation that the Reshetikhin–Turaev invariants of framed links associated to a ribbon fusion category satisfy a factorization property under zesting. This gives a new perspective on using topological invariants to classify topological order in light of modular data not being a complete invariant.

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.

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.