Noetherian connected braided Hopf algebras are affine
Year: 2026
Type: preprint
Abstract: The well-known Wu--Zhang question asks whether every Noetherian Hopf algebra over a field is affine (finitely generated as an algebra). In this paper, we extend this fundamental problem to the braided setting within the monoidal category ${}_{H}^{H}\mathcal{YD}$ of left Yetter--Drinfel'd modules over a Hopf algebra $H$. By developing term-rewriting methods based on reduction orderings and prime-word factorizations in free prebraided algebras, we establish explicit expansions for YD-triangular comultiplications. We prove that if $(\mathbb{K}\langle X \rangle, ε)$ is an augmented free algebra in ${}_{H}^{H}\mathcal{YD}$ and $I \subseteq \ker ε$ is an ideal invariant under a YD-triangular comultiplication, then every $I$-reducible letter---and consequently every non-empty word in $\langle X \rangle$---can be expressed modulo $I$ as a linear combination of products of $I$-irreducible letters strictly smaller with respect to the reduction order. As a primary consequence, we prove that any right Noetherian connected prebraided Hopf algebra $B$ in ${}_{H}^{H}\mathcal{YD}$ is affine, generated by the image of a finite set $X_{I}$ of $I$-irreducible letters. As an application, we show that the braided component $B = A^{\mathrm{co}\,π}$ of a right Noetherian Radford biproduct $A = B \sharp H$ is affine whenever $A_{(0)} \subseteq \jmath(H)$, reducing the affineness of $A$ entirely to that of $H$.
Source: arXiv (Cornell University)
Authors: Jia, Huan, Zhang, Yinhuo
Language: English
Topic: Advanced Topics in Algebra
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225831070
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