Noetherian pointed Hopf algebras are affine

2025/11/24 by Jia, Huan, Zhang, Yinhuo
Computer Science · Mathematics · #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2511.19293

openalex publication_date 2025/11/24 · openalex created_date 2025/11/27 · openalex updated_date 2026/07/28

Abstract

Let k be a field. In this paper, we introduce the notions of reduction order and reduction-factorization on words, and use them to show that any right or left Noetherian pointed Hopf algebra over k is affine. This result offers a partial affirmative answer to the classical affineness question for Noetherian Hopf algebras posed by Wu and Zhang \citeWZ2003. For a pointed Hopf algebra H over k, we construct a well-ordered set X such that: (1) H is generated, as an algebra, by the subset X_I of irreducible letters (with respect to the reduction order); and (2) X_I is finite whenever H is right Noetherian.

Citations

• On infinite-dimensional Hopf algebras
• Coideal subalgebras of pointed and connected Hopf algebras
• Survey on Hopf algebras of GK-dimension 1 and 2
• The structure of connected (graded) Hopf algebras
• Non-affine Hopf algebra domains of Gelfand-Kirillov dimension two
• On finite GK-dimensional Nichols algebras over abelian groups
• Connected (graded) Hopf algebras
• Primitive Cohomology of Hopf algebras
• Hopf algebras under finiteness conditions
• Connected Hopf algebras of Gelfand-Kirillov dimension four
• Properties of pointed and connected Hopf algebras of finite Gelfand-Kirillov dimension
• Noetherian Hopf algebras
• PBW bases for a class of braided Hopf algebras

Related

• Noetherian Hopf algebras
• Affineness on Noetherian graded rings, algebras and Hopf algebras
• Noetherian Hopf algebras
• Pointed Hopf algebras, the Dixmier-Moeglin Equivalence and Noetherian group algebras
• Flatness of Noetherian Hopf algebras over coideal subalgebras