No braiding for quantum sl2 reps - A short proof

Let $V$ be a strongly rational vertex operator algebra and $G\leq\operatorname{Aut}(V)$ a finite group. We prove Huang's conjecture that the categories of $g$-twisted $V$-modules, for $g\in G$, form a $G$-crossed braided fusion category. This category is faithfully $G$-graded, with tensor products representing the complete twisted intertwining operators of Du-Huang, the canonical pullback $G$-action, and crossed braiding. In particular, the associativity, coherence and rigidity required by the conjecture all hold.

math.QA↗

arXiv2026

Extending Doi--Takeuchi correspondence to bimodules and noncommutative covers

Doi and Takeuchi, and independently Blattner, Cohen and Montgomery, have exhibited a correspondence between cleft extensions of an algebra $U$ by a Hopf algebra $H$ and $H$-measurings on $U$ equipped with a compatible normalized 2-cocycle. Each cleft extension is isomorphic as an algebra, a left $U$-module and a right $H$-comodule to the crossed product algebra constructed from the measuring and the cocycle. Cleft $H$-extensions are a special case of $H$-Hopf-Galois extensions, describing noncommutative principal $H$-bundles with affine total and affine base space. Corings are used to describe noncommutative space covers, allowing a description of possibly nonaffine noncommutative spaces. We propose to describe the total space of a noncommutative $H$-principal bundle by an internal $E$-coring in $H$-comodules where $E$ is an $H$-comodule algebra which is itself $H$-Hopf-Galois. This article focuses on the cleft case. As corings involve bimodules and morphisms of bimodules we extend the Doi--Takeuchi correspondence to bimodule $H$-comodules between cleft extensions on one side and bimodules of $H$-coinvariants equipped with bimodule $H$-measurings compatible with 2-cocycles on the other side. In fact, the correspondences for algebras and for bimodules are parts of a biequivalence of suitable bicategories defined in this article. As a corollary one obtains a correspondence between cleft $H$-extensions of corings and corings equipped with a nonabelian Čech 1-cocycle covalued in $H$.

math.QA↗

arXiv2026

Twisting of affine algebraic groups, III

Let $G$ be an affine algebraic group over $\C$ and let $J$ be a Hopf $2$-cocycle for $\Os(G)$. We prove that one-sided cocycle twisting preserves Gelfand--Kirillov dimension and that $\Os(G)_J$ is simple if and only if $J$ is minimal. If $H$ is the support of $J$, we identify the primitive and maximal spectra of $\Os(G)_J$ with the closed points of $G/H$ and show that all primitive quotients are isomorphic to the minimal support algebra $\Os(H)_J$. In particular, the prime and Jacobson radicals vanish, and finite-dimensional irreducible modules exist exactly when $H$ is finite. For connected solvable $G$, we construct an iterated differential Ore presentation over an algebraic quantum torus and deduce that $\Os(G)_J$ is an affine Noetherian domain. We describe the torus-trivial Hopf $2$-cocycles on $\Os(G)$ using nilpotent solutions of the classical Yang--Baxter equation and identify their supports with the algebraic hulls of their carriers. We also separate the centrally supported toral sector and formulate a relative Hopf $2$-cocycle problem for the remaining classes. Finally, we compute multi-extended Jordanian examples, study a McConnell-type family of quantum differential algebras, and establish a conditional birational deletion criterion and a normal form when the unipotent radical has dimension at most one.

math.QA↗