## Abstract

Motivated by the Hopf algebra structures established on free commutative Rota-Baxter algebras, we explore Hopf algebra related structures on free commutative Nijenhuis algebras. Applying a cocycle condition, we first prove that a free commutative Nijenhuis algebra on a left counital bialgebra (in the sense that the right-sided counicity needs not hold) can be enriched to a left counital bialgebra. We then establish a general result that a connected graded left counital bialgebra is a left counital right antipode Hopf algebra in the sense that the antipode is also only right-sided. We finally apply this result to show that the left counital bialgebra on a free commutative Nijenhuis algebra on a connected left counital bialgebra is connected and graded, hence is a left counital right antipode Hopf algebra.

Translated title of the contribution | Left counital Hopf algebra structures on free commutative Nijenhuis algebras |
---|---|

Original language | Chinese (Traditional) |

Pages (from-to) | 829-846 |

Number of pages | 18 |

Journal | Scientia Sinica Mathematica |

Volume | 50 |

Issue number | 6 |

DOIs | |

State | Published - Jun 1 2020 |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Keywords

- Bialgebra
- Connected bialgebra
- Left counital bialgebra
- Left counital right antipode Hopf algebra
- Nijenhuis algebra