Mathematics

The B p -algebras are a family of vertex operator algebras parameterized by p∈ Z ≥2 . They are important examples of logarithmic CFTs and appear as chiral algebras of type ( A 1 , A 2p−3 ) Argyres-Douglas theories. The first member of this series, the B 2 -algebra, are the well-known symplectic bosons also often called the βγ vertex operator algebra. We study categories related to the B p vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of sl 2 . These categories are braided, rigid and non semi-simple tensor categories. We list their simple and projective objects, their tensor products and their Hopf links. The latter are successfully compared to modular data of characters thus confirming a proposed Verlinde formula of David Ridout and the second author.

This paper introduces the braidings of dendriform algebras and tridendriform algebras. By studying free braided dendriform algebras, we obtain braidings of the Hopf algebras of Loday and Ronco of planar binary rooted trees. We also give a variation of the braiding of Foissy for the noncommutative Connes-Kreimer (a.k.a the Foissy-Holtkamp) Hopf algebra of planar rooted forests so that the well-known isomorphism between this Hopf algebra and the Loday-Ronco Hopf algebra is extended to the braided context. As free braided tridendriform algebras, we also give braided extension of the Hopf algebra of Loday and Ronco on planar rooted trees.

We give a rigorous development of the construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. Here we provide a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples.

We present an explicit formula for the transition matrix C from the type B C n Koornwinder polynomials P ( 1 r ) (x|a,b,c,d|q,t) with one column diagrams, to the type B C n monomial symmetric polynomials m ( 1 r ) (x) . The entries of the matrix C enjoy a set of four terms recursion relations. These recursions provide us with the branching rules for the Koornwinder polynomials with one column diagrams, namely the restriction rules from B C n to B C n−1 . To have a good description of the transition matrices involved, we introduce the following degeneration scheme of the Koornwinder polynomials: P ( 1 r ) (x|a,b,c,d|q,t)⟷ P ( 1 r ) (x|a,−a,c,d|q,t)⟷ P ( 1 r ) (x|a,−a,c,−c|q,t)⟷ P ( 1 r ) (x| t 1/2 c,− t 1/2 c,c,−c|q,t)⟷ P ( 1 r ) (x| t 1/2 ,− t 1/2 ,1,−1|q,t) . We prove that the transition matrices associated with each of these degeneration steps are given in terms of the matrix inversion formula of Bressoud. As an application, we give an explicit formula for the Kostka polynomials of type B n , namely the transition matrix from the Schur polynomials P ( B n , B n ) ( 1 r ) (x|q;q,q) to the Hall-Littlewood polynomials P ( B n , B n ) ( 1 r ) (x|t;0,t) . We also present a conjecture for the asymptotically free eigenfunctions of the B n q -Toda operator, which can be regarded as a branching formula from the B n q -Toda eigenfunction restricted to the A n−1 q -Toda eigenfunctions.

We study the differential and Riemannian geometry of algebras A endowed with an action of a triangular Hopf algebra H and noncomutativity compatible with the associated braiding. The modules of one forms and of braided derivations are modules in a symmetric ribbon category of H -modules A -bimodules, whose internal morphisms correspond to tensor fields. Different approaches to curvature and torsion are proven to be equivalent by extending the Cartan calculus to left (right) A -module connections. The Cartan structure equations and the Bianchi identities are derived. Existence and uniqueness of the Levi-Civita connection for arbitrary braided symmetric pseudo-Riemannian metrics is proven.

Let U ′ q (g) be a quantum affine algebra of untwisted affine ADE type, and C 0 g the Hernandez-Leclerc category of finite-dimensional U ′ q (g) -modules. For a suitable infinite sequence w ˆ 0 =⋯ s i −1 s i 0 s i 1 ⋯ of simple reflections, we introduce subcategories C [a,b] g of C 0 g for all a≤b∈Z⊔{±∞} . Associated with a certain chain C of intervals in [a,b] , we construct a real simple commuting family M(C) in C [a,b] g , which consists of Kirillov-Reshetikhin modules. The category C [a,b] g provides a monoidal categorification of the cluster algebra K( C [a,b] g ) , whose set of initial cluster variables is [M(C)] . In particular, this result gives an affirmative answer to the monoidal categorification conjecture on C − g by Hernandez-Leclerc since it is C [−∞,0] g , and is also applicable to C 0 g since it is C [−∞,∞] g .

Let U be a quantized enveloping algebra. We consider the adjoint action of an sl 2 -subalgebra of U on a subalgebra of U + that is maximal integrable for this action. We categorify this representation in the context of quiver Hecke algebras. We obtain an action of the 2-category associated with sl 2 on a category of modules over certain quotients of quiver Hecke algebras. Our approach is similar to that of Kang-Kashiwara for categorifications of highest weight modules via cyclotomic quiver Hecke algebras. One of the main new features is a compatibility of the categorical action with the monoidal structure, categorifying the notion of derivation on an algebra. As an application of some of our results, we categorify the higher order quantum Serre relations, extending results of Stošić to the non simply-laced case.

In their paper entitled "Quantum Enhancements and Biquandle Brackets," Nelson, Orrison, and Rivera introduced biquandle brackets, which are customized skein invariants for biquandle-colored links. These invariants generalize the Jones polynomial, which is categorified by Khovanov homology. At the end of their paper, Nelson, Orrison, and Rivera asked if the methods of Khovanov homology could be extended to obtain a categorification of biquandle brackets. We outline herein a Khovanov homology-style construction that is an attempt to obtain such a categorification of biquandle brackets. The resulting knot invariant generalizes Khovanov homology, but the biquandle bracket is not always recoverable, meaning the construction is not a true categorification of biquandle brackets. However, the construction does lead to a definition that gives a "canonical" biquandle 2-cocycle associated to a biquandle bracket, which, to the authors' knowledge, was not previously known.

In this paper, it is established that quantum nilpotent algebras (also known as CGL extensions) are catenary, i.e., all saturated chains of inclusions of prime ideals between any two given prime ideals P⊊Q have the same length. This is achieved by proving that the prime spectra of these algebras have normal separation, and then establishing the mild homological conditions necessary to apply a result of Lenagan and the first author. The work also recovers the Tauvel height formula for quantum nilpotent algebras, a result that was first obtained by Lenagan and the authors through a different approach.

We establish the analogue of the Cayley--Hamilton theorem for the quantum matrix algebras of the symplectic type.