Quotient Hopf algebras of the free bialgebra with PBW bases and GK-dimensions

2023/04/16 by Huan Jia, Naihong Hu, Jia, Huan +5
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Nonlinear Waves and Solitons

paper · pdf · doi:10.48550/arxiv.2304.07755

Abstract

Let \mathbbK be a field. We study the free bialgebra T generated by the coalgebra C=\mathbbK g ⊕ \mathbbK h and its quotient bialgebras (or Hopf algebras) over \mathbbK. We show that the free noncommutative Faà di Bruno bialgebra is a sub-bialgebra of T, and the quotient bialgebra T:=T/(E_α|~α(g)≥ 2) is an Ore extension of the well-known Faà di Bruno bialgebra. The image of the free noncommutative Faà di Bruno bialgebra in the quotient T gives a more reasonable non-commutative version of the commutative Faà di Bruno bialgebra from the PBW basis point view. If char \mathbbK=p>0, we obtain a chain of quotient Hopf algebras of T: T \twoheadrightarrow T_n\twoheadrightarrow T_n'(p)\twoheadrightarrow T_n(p)\twoheadrightarrow T_n(p;d₁) \twoheadrightarrow … \twoheadrightarrow T_n(p;d_j,d_j-1,…,d₁) \twoheadrightarrow … \twoheadrightarrow T_n(p;d_p-2,d_p-3,…,d₁) with finite GK-dimensions. Furthermore, we study the homological properties and the coradical filtrations of those quotient Hopf algebras.

Related

• Makar-Limanov's conjecture on free subalgebras
• Free bicommutative superalgebras
• On the Hopf envelope of finite-dimensional bialgebras
• On the structure of abelian Hopf algebras
• The structure of iterated Hopf Ore extensions