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The Hopf envelope of a bialgebra is the free Hopf algebra generated by the given bialgebra. Its existence, as well as that of the cofree Hopf algebra, is a well-known fact in Hopf algebra theory, but their construction is not particularly…
A quantum symmetric pair consists of a quantum group $\widetilde{\mathbf{U}}$ and its coideal subalgebra $\widetilde{\mathbf{U}}^\imath$. The Hall algebra constructions of $\widetilde{\mathbf{U}}$ and $\widetilde{\mathbf{U}}^\imath$ are…
We investigate invariants of 4-dimensional 2-handlebodies associated to the Temperley-Lieb category in characteristic $p>2$ and at a primitive fourth root of unity. These invariants depend additionally on a height parameter $n$, and we…
We give a cabling formula for the Links--Gould polynomial of knots colored with a $4n$-dimensional irreducible representation of $\mathrm{U}^H_q\mathfrak{sl}(2|1)$ and identify them with the $V_n$-polynomial of knots for $n=2$. Using the…
We introduce a Drinfeld presentation for the super-Yangian $\mathrm{Y}(\mathfrak{q}_n)$ associated with the queer Lie superalgebra $\mathfrak{q}_n$. The Drinfeld generators of $\mathrm{Y}(\mathfrak{q}_n)$ are obtained through a block Gauss…
In this paper, we extend a classical vanishing result of Burnside from the character tables of finite groups to the character tables of commutative fusion rings, or more generally to a certain class of abelian normalizable hypergroups. We…
This paper presents an abstract Isaacs property that involves the Fourier transform for fusion rings, which may be non-commutative, thus expanding upon the commutative version described in [12]. A categorical version of this property was…
We introduce the triangular prism equations (TPE) for fusion categories, obtained by evaluating triangular prisms in terms of tetrahedra. Using an oriented graphical calculus, we show that the geometric symmetries of the regular tetrahedron…
We study the reduced coaction Lie algebra $\mathfrak{rc}_0$, which is defined by an algebraic equation satisfied by the reduced coaction (an upgraded version of the necklace cobracket) and the skew-symmetric condition. We prove that the…
Consider a Lie subalgebra $\mathfrak{l} \subset \mathfrak{g}$ and an $\mathfrak{l}$-invariant open submanifold $V \subset \mathfrak{l}^{\ast}$. We demonstrate that any smooth dynamical twist on $V$, valued in $U(\mathfrak{g}) \otimes…
Vafa and Warner observed that the Landau-Ginzburg model associated to the potential $E_6$ (resp. $E_8$) is a product of two other models, associated to the potentials $A_2$ and $ A_3$ (resp. $A_2 $ and $ A_4$). We translate this along the…
This is a contribution to the problem of classifying all deformations - a. k. a. liftings - of the bosonization of a Nichols algebra $\mathfrak{B}(V)$ over a cosemisimple and non-semisimple Hopf algebra $H$. Such a situation arises when the…
This work initiates a systematic study of the class of quasi bijective and quasi non-degenerate solutions to the set-theoretic Yang-Baxter equation. The motivation stems from the observation that solutions that arise from dual weak braces…
We obtain two related characterizations of discrete quantum groups and discrete quantum groups of Kac type as allegorical group objects in the symmetric monoidal dagger category of quantum sets and relations, of interest to quantum…
We introduce uniparametric and multiparametric quantisations of the general linear supergroup, in the form of "quantised function algebras", both in a formal setting - yielding "quantum formal series Hopf superalgebras", a` la Drinfeld -…
Topological quantum field theories (TQFTs) are symmetric monoidal functors out of cobordism categories. In dimension two, oriented TQFTs are famously classified by commutative Frobenius algebras. In the unoriented setting, the…
In this paper, we will characterize the spectrum of two non--commutative U(1)-gauge Laplacians on the upper sheet of a two--sheet quantum hyperboloid.
The Connes-Kreimer Hopf algebra of rooted trees is an operated Hopf algebra whose coproduct satisfies the classical Hochschild 1-cocycle condition. In this paper, we extend the setting from rooted trees to the space $H_{\rm RT}(X,\Omega)$…
Given a bialgebra $H$ such that the associated trivial topological bialgebra $H[[\hbar]]$ admits a quasitriangular structure $\tilde{\mathcal{R}}=\mathcal{R}(1\otimes 1+\hbar\chi+\mathcal{O}(\hbar^2))$, one gets a distinguished element…
Let $V$ be a vertex operator algebra, $g$ be an automorphism of $V$ of order $T$, and $m, n \in (1/T)\mathbb{N}$. In~\cite{HX2} and~\cite{HXX1}, it was shown respectively that the associative algebra $A_{g,n}(V)$ constructed by Dong, Li,…