2023/01/05 by G. -S. Zhou, Zhou, G. -S. · 2 citations
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.2301.02139
openalex publication_date 2023/01/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let H be a pointed Hopf algebra with abelian coradical. Let A⊇ B be left (or right) coideal subalgebras of H that contain the coradical of H. We show that A has a PBW basis over B, provided that H satisfies certain mild conditions. In the case that H is a connected graded Hopf algebra of characteristic zero and A and B are both homogeneous of finite Gelfand-Kirillov dimension, we show that A is a graded iterated Ore extension of B. These results turn out to be conceptual consequences of a structure theorem for each pair S⊇ T of homogeneous coideal subalgebras of a connected graded braided bialgebra R with braiding satisfying certain mild conditions. The structure theorem claims the existence of a well-behaved PBW basis of S over T. The approach to the structure theorem is constructive by means of a combinatorial method based on Lyndon words and braided commutators, which is originally developed by V. K. Kharchenko for primitively generated braided Hopf algebras of diagonal type. Since in our context we don't priorilly assume R to be primitively generated, new methods and ideas are introduced to handle the corresponding difficulties, among others.