In this paper we give a topological interpretation and diagrammatic calculus for the rank (n − 2) Askey-Wilson algebra by proving there is an explicit isomorphism with the Kauffman bracket skein algebra of the (n + 1)-punctured sphere. To do this we consider the Askey-Wilson algebra in the braided tensor product of n copies of either the quantum group Uq(sl 2 ) or the reflection equation algebra. We then use the isomorpism of the Kauffman bracket skein algebra of the (n + 1)-punctured sphere with the Uq(sl 2 ) invariants of the Aleeksev moduli algebra to complete the correspondence. We also find the graded vector space dimension of the Uq(sl 2 ) invariants of the Aleeksev moduli algebra and apply this to finding a presentation of the skein algebra of the five-punctured sphere and hence also find a presentation for the rank 2 Askey-Wilson algebra. Contents 1. Introduction 1 2. Askey-Wilson Algebras 5 3. Braided Tensor Product of copies of the locally finite part of U q (sl 2 ) 8 4. Moduli algebras 14 5. Skein Algebras 17 6. Commutator Relations 20 7. Action of the braid group 21 8. Graded Dimensions 23 9. Presentation of the Skein Algebra of the Five-Punctured Sphere 25 A. Appendix 34 References 37

Given a slightly degenerate braided pivotal fusion category $\mathscr{C}$, we explain how it naturally gives rise to a $\mathbb{Z}$-modular data. We do not restrict to spherical categories and work with pivotal categories. Finally, we give an interpretation in this framework of the Bonnafé–Rouquier categorification of the $\mathbb{Z}$-modular datum associated to nontrivial family of the cyclic complex reflection group.

We construct a dg-enhancement of KLRW algebras that categorifies the tensor product of a universal \mathfrak{sl}_2 Verma module and several integrable irreducible modules. When the integrable modules are two-dimensional, we construct a categorical action of the blob algebra on derived categories of these dg-algebras which intertwines the categorical action of \mathfrak{sl}_2 . From the above we derive a categorification of the blob algebra.

We construct a dg-enhancement of Webster's tensor product algebras that categorifies the tensor product of a universal sl 2 Verma module and several integrable irreducible modules. We show that the blob algebra acts via endofunctors on derived categories of such dg-enhanced algebras in the case when the integrable modules are twodimensional. This action intertwines with the categorical action of sl 2 . From the above we derive a categorification of the blob algebra.

Generalizing Lusztig’s work, Malle has associated to some imprimitive complex reflection group W a set of “unipotent characters”, which are in bijection of the usual unipotent characters of the associated finite reductive group if W is a Weyl group. He also obtained a partition of these characters into families and associated to each family a \mathbb Z -modular datum. We construct a categorification of some of these data, by studying the category of tilting modules of the Drinfeld double of the quantum enveloping algebra of the Borel of a simple complex Lie algebra. As an application, we obtain a proof of a conjecture by Cuntz at the decategorified level.