#
From Hopf Algebra to Braided L_{∞}-Algebra

^{*}

## Abstract

**:**

## 1. Introduction

## 2. ${\mathbf{L}}_{\infty}$ as Hopf Algebra

**Theorem**

**1.**

**Proof.**

## 3. Braided ${L}_{\infty}$-Algebra from a Drinfel’d Twist

## 4. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. On L_{∞}-Algebras

**Definition**

**A1**

## Appendix B. On Hopf Algebras

**Definition**

**A2 (Bialgebra).**

**Definition**

**A3 (Hopf algebra).**

## Notes

1 | The suspension map is also called a shift isomorphism [18]; see Appendix A for more details. |

2 | In the rest of the paper, we work in the coalgebraic picture and denote with X the underlying graded vector space in order to simplify the notation. |

3 | We shall use the term ${L}_{\infty}$-algebra for both cases when the distinction is not relevant. |

4 | The twist generators commute in the case of an Abelian twist. |

5 | Only when acting on explicit elements of the tensor algebra does one have to take into account the braided transposition map. In that case, the first difference with respect to the untwisted algebra appears when acting on three or more elements, as shown in [16]. |

6 | In the infinite-dimensional case, one either restricts ${X}^{\U0001f7c9}$ to the space spanned by $\left\{{z}^{\alpha}\right\}$, or considers continuous duals in infinite-dimensional topological vector spaces; see discussion in [32]. |

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**MDPI and ACS Style**

Grewcoe, C.J.; Jonke, L.; Kodžoman, T.; Manolakos, G.
From Hopf Algebra to Braided *L*_{∞}-Algebra. *Universe* **2022**, *8*, 222.
https://doi.org/10.3390/universe8040222

**AMA Style**

Grewcoe CJ, Jonke L, Kodžoman T, Manolakos G.
From Hopf Algebra to Braided *L*_{∞}-Algebra. *Universe*. 2022; 8(4):222.
https://doi.org/10.3390/universe8040222

**Chicago/Turabian Style**

Grewcoe, Clay James, Larisa Jonke, Toni Kodžoman, and George Manolakos.
2022. "From Hopf Algebra to Braided *L*_{∞}-Algebra" *Universe* 8, no. 4: 222.
https://doi.org/10.3390/universe8040222