Mathematics

We introduce monoidal categories whose monoidal products of any positive number of factors are lax coherent and whose nullary products are oplax coherent. We call them Lax + Oplax 0 -monoidal. Dually, we consider Lax 0 Oplax + -monoidal categories which are oplax coherent for positive numbers of factors and lax coherent for nullary monoidal products. We define Lax + 0 Oplax 0 + -duoidal categories with compatible Lax + Oplax 0 - and Lax 0 Oplax + -monoidal structures. We introduce comonoids in Lax + Oplax 0 -monoidal categories, monoids in Lax 0 Oplax + -monoidal categories and bimonoids in Lax + 0 Oplax 0 + - duoidal categories. Motivation for these notions comes from a generalization of a construction due to Caenepeel and Goyvaerts. This assigns a Lax + 0 Oplax 0 + -duoidal category D to any symmetric monoidal category V . The unital BiHom -monoids, counital BiHom -comonoids, and unital and counital BiHom -bimonoids in V are identified with the monoids, comonoids and bimonoids in D , respectively.

We provide a description of adequate categorical data to give a Turaev-Viro type state-sum construct of invariants of 3-manifolds with a system of defects, generalizing the Dijkgraaf-Witten type invariants of our earlier work. We term the defects in our construction defects-with-structure because algebraic data associated to them is in general richer than a module category over the spherical fusion category from which the theory is constructed when no defect is present.

The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. It is semisimple for generic values of the parameter t while producing categories of representations of the symmetric group when modded out by the ideal of negligible morphisms when t is a non-negative integer. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. The Deligne category and its semisimple quotients admit similar interpretations. This viewpoint coupled to the universal construction of two-dimensional topological theories leads to multi-parameter monoidal generalizations of the partition and the Deligne categories, one for each rational function in one variable.

We introduce bivariate versions of the continuous q-Hermite polynomials. We obtain algebraic properties for them (generating function, explicit expressions in terms of the univariate ones, backward difference equations and recurrence relations) and analytic properties (determining the orthogonality measure). We find a direct link between bivariate continuous q-Hermite polynomials and the star product method of [Kolb and Yakimov, Adv. Math. 2020] for quantum symmetric pairs to establish deformed quantum Serre relations for quasi-split quantum symmetric pairs of Kac-Moody type. We prove that these defining relations are obtained from the usual quantum Serre relations by replacing all monomials by multivariate orthogonal polynomials.

The rank 1 bosonic ghost vertex algebra, also known as the βγ ghosts, symplectic bosons or Weyl vertex algebra, is a simple example of a conformal field theory which is neither rational, nor C 2 -cofinite. We identify a module category, denoted category F , which satisfies three necessary conditions coming from conformal field theory considerations: closure under restricted duals, closure under fusion and closure under the action of the modular group on characters. We prove the second of these conditions, with the other two already being known. Further, we show that category F has sufficiently many projective and injective modules, give a classification of all indecomposable modules, show that fusion is rigid and compute all fusion products. The fusion product formulae turn out to perfectly match a previously proposed Verlinde formula, which was computed using a conjectured generalisation of the usual rational Verlinde formula, called the standard module formalism. The bosonic ghosts therefore exhibit essentially all of the rich structure of rational theories despite satisfying none of the standard rationality assumptions such as C 2 -cofiniteness, the vertex algebra being isomorphic to its restricted dual or having a one-dimensional conformal weight 0 space. In particular, to the best of the authors' knowledge this is the first example of a proof of rigidity for a logarithmic non- C 2 -cofinite vertex algebra.

In the present paper we construct braid group actions on quantum symmetric pair coideal subalgebras of type AIII/AIV. This completes the proof of a conjecture by Kolb and Pellegrini in the case where the underlying Lie algebra is sl n . The braid group actions are defined on the generators of the coideal subalgebras and the defining relations and braid relations are verified by explicit calculations.

We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. Our results hint at a relationship between the braidings on the G -gaugings of a pointed modular category C(A,Q) and that of C(A,Q) itself.

We construct a noncommutative Cartan calculus on any braided commutative algebra and study its applications in noncommutative geometry. The braided Lie derivative, insertion and de Rham differential are introduced and related via graded braided commutators, also incorporating the braided Schouten-Nijenhuis bracket. The resulting braided Cartan calculus generalizes the Cartan calculus on smooth manifolds and the twisted Cartan calculus. While it is a necessity of derivation based Cartan calculi on noncommutative algebras to employ central bimodules our approach allows to consider bimodules over the full underlying algebra. Furthermore, equivariant covariant derivatives and metrics on braided commutative algebras are discussed. In particular, we prove the existence and uniqueness of an equivariant Levi-Civita covariant derivative for any fixed non-degenerate equivariant metric. Operating in a symmetric braided monoidal category we argue that Drinfel'd twist deformation corresponds to gauge equivalences of braided Cartan calculi. The notions of equivariant covariant derivative and metric are compatible with the Drinfel'd functor as well. Moreover, we project braided Cartan calculi to submanifold algebras and prove that this process commutes with twist deformation.

In this thesis we give obstructions for Drinfel'd twist deformation quantization on several classes of symplectic manifolds. Motivated from this quantization procedure, we further construct a noncommutative Cartan calculus on any braided commutative algebra, as well as an equivariant Levi-Civita covariant derivative for any non-degenerate equivariant metric. This generalizes and unifies the Cartan calculus on a smooth manifold and the Cartan calculus on twist star product algebras. We prove that the Drinfel'd functor leads to equivalence classes in braided commutative geometry and commutes with submanifold algebra projection.

We classify various types of graded extensions of a finite braided tensor category B in terms of its 2 -categorical Picard groups. In particular, we prove that braided extensions of B by a finite group A correspond to braided monoidal 2 -functors from A to the braided 2 -categorical Picard group of B (consisting of invertible central B -module categories). Such functors can be expressed in terms of the Eilnberg-Mac~Lane cohomology. We describe in detail braided 2 -categorical Picard groups of symmetric fusion categories and of pointed braided fusion categories.