English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Let $\mathbb{K}$ be a field. We study the free bialgebra $\mathcal{T}$generated by the coalgebra $C=\mathbb{K} g \oplus \mathbb{K} h$ and itsquotient bialgebras (or Hopf algebras) over $\mathbb{K}$. We show that the freenoncommutative Fa\`a di Bruno bialgebra is a sub-bialgebra of $\mathcal{T}$,and the quotient bialgebra$\overline{\mathcal{T}}:=\mathcal{T}/(E_{\alpha}|~\alpha(g)\ge 2)$ is an Oreextension of the well-known Fa\`a di Bruno bialgebra. The image of the freenoncommutative Fa\`a di Bruno bialgebra in the quotient$\overline{\mathcal{T}}$ gives a more reasonable non-commutative version of thecommutative Fa\`a di Bruno bialgebra from the PBW basis point view. If char$\mathbb{K}=p>0$, we obtain a chain of quotient Hopf algebras of$\overline{\mathcal{T}}$: $\overline{\mathcal{T}} \twoheadrightarrow\overline{\mathcal{T}}_{n}\twoheadrightarrow\overline{\mathcal{T}}_{n}'(p)\twoheadrightarrow\overline{\mathcal{T}}_{n}(p)\twoheadrightarrow\overline{\mathcal{T}}_{n}(p;d_{1}) \twoheadrightarrow \ldots\twoheadrightarrow \overline{\mathcal{T}}_{n}(p;d_{j},d_{j-1},\ldots,d_{1})\twoheadrightarrow \ldots \twoheadrightarrow\overline{\mathcal{T}}_{n}(p;d_{p-2},d_{p-3},\ldots,d_{1})$ with finiteGK-dimensions. Furthermore, we study the homological properties and thecoradical filtrations of those quotient Hopf algebras.

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