#
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]. |

## References

- Stasheff, J. Differential graded Lie algebras, quasi-Hopf algebras and higher homotopy algebras. In Quantum Groups; Springer: Berlin/Heidelberg, Germany, 1992; pp. 120–137. [Google Scholar]
- Zwiebach, B. Closed String Field Theory: Quantum action and the B-V master equation. Nucl. Phys. B
**1993**, 390, 33. [Google Scholar] [CrossRef][Green Version] - Lada, T.; Stasheff, J. Introduction to SH Lie algebras for physicists. Int. J. Theor. Phys.
**1993**, 32, 1087. [Google Scholar] [CrossRef][Green Version] - Borsten, L.; Jurčo, B.; Kim, H.; Macrelli, T.; Saemann, C.; Wolf, M. Double Copy from Homotopy Algebras. Fortsch. Phys.
**2021**, 69, 2100075. [Google Scholar] [CrossRef] - Bayen, F.; Flato, M.; Fronsdal, C.; Lichnerowicz, A.; Sternheimer, D. Deformation Theory and Quantization. 1. Deformations of Symplectic Structures. Ann. Phys.
**1978**, 111, 61. [Google Scholar] [CrossRef] - Kontsevich, M. Deformation quantization of Poisson manifolds, I. Lett. Math. Phys.
**2003**, 66, 157–216. [Google Scholar] [CrossRef][Green Version] - Cattaneo, A.S.; Felder, G. A path integral approach to the Kontsevich quantization formula. Commun. Math. Phys.
**2000**, 212, 591–611. [Google Scholar] [CrossRef][Green Version] - Ikeda, N. Two-Dimensional Gravity and Nonlinear Gauge Theory. Ann. Phys.
**1994**, 235, 435–464. [Google Scholar] [CrossRef][Green Version] - Schaller, P.; Strobl, T. Poisson Structure Induced (Topological) Field Theories. Mod. Phys. Lett. A
**1994**, 9, 3129–3136. [Google Scholar] [CrossRef][Green Version] - Blumenhagen, R.; Brunner, I.; Kupriyanov, V.; Lüst, D. Bootstrapping Non-commutative Gauge Theories from L
_{∞}algebras. J. High Energy Phys.**2018**, 5, 97. [Google Scholar] [CrossRef][Green Version] - Drinfel’d, V. Hopf Algebras and the Quantum Yang-Baxter Equation. Sov. Math. Dokl.
**1985**, 32, 254. [Google Scholar] - Oeckl, R. Untwisting Noncommutative R
^{d}and the Equivalence of Quantum Field Theories. Nucl. Phys. B**2000**, 581, 559–574. [Google Scholar] [CrossRef][Green Version] - Aschieri, P.; Dimitrijević, M.; Meyer, F.; Schraml, S.; Wess, J. Twisted Gauge Theories. Lett. Math. Phys.
**2006**, 78, 61–71. [Google Scholar] [CrossRef][Green Version] - Vassilevich, D.V. Twist to close. Mod. Phys. Lett. A
**2006**, 21, 1279–1284. [Google Scholar] [CrossRef][Green Version] - Chaichian, M.; Tureanu, A. Twist Symmetry and Gauge Invariance. Phys. Lett. B
**2006**, 637, 199–202. [Google Scholar] [CrossRef][Green Version] - Dimitrijević Ćirić, M.; Giotopoulos, G.; Radovanović, V.; Szabo, R.J. Braided L
_{∞}-Algebras, Braided Field Theory and Noncommutative Gravity. Lett. Math. Phys.**2021**, 111, 148. [Google Scholar] [CrossRef] - Giotopoulos, G.; Szabo, R.J. Braided Symmetries in Noncommutative Field Theory. arXiv
**2022**, arXiv:2112.00541. [Google Scholar] [CrossRef] - Jurčo, B.; Raspollini, L.; Sämann, C.; Wolf, M. L
_{∞}-Algebras of Classical Field Theories and the Batalin-Vilkovisky Formalism. Fortsch. Phys.**2019**, 67, 1900025. [Google Scholar] [CrossRef][Green Version] - Lada, T.; Markl, M. Strongly homotopy Lie algebras. Comm. Algebra
**1995**, 23, 2147. [Google Scholar] [CrossRef] - Getzler, E.; Jones, J.D.S. A
_{∞}-algebras and the cyclic bar complex. Illinois J. Math.**1990**, 34, 256–283. [Google Scholar] [CrossRef] - Getzler, E.; Jones, J.D.S.; Petrack, S. Differential forms on loop spaces and the cyclic bar complex. Topology
**1991**, 30, 339–371. [Google Scholar] [CrossRef][Green Version] - Schupp, P. Quantum Groups, Noncommutative Differential Geometry and Applications. Ph.D. Thesis, University of California, Berkeley, CA, USA, 1993. [Google Scholar]
- Majid, S. Foundations of Quantum Group Theory; Cambridge University Press: Cambridge, UK; New York, NY, USA, 1995. [Google Scholar]
- Aschieri, P.; Dimitrijević, M.; Meyer, F.; Wess, J. Noncommutative Geometry and Gravity. Class. Quant. Grav.
**2006**, 23, 1883–1912. [Google Scholar] [CrossRef][Green Version] - Aschieri, P.; Dimitrijević, M.; Kulish, P.; Lizzi, F.; Wess, J. Noncommutative Spacetimes: Symmetries in Noncommutative Geometry and Field Theory. Lect. Notes Phys.
**2009**, 774, 1–199. [Google Scholar] - Schenkel, A. Noncommutative Gravity and Quantum Field Theory on Noncommutative Curved Spacetimes. Ph.D. Thesis, Würzburg University, Würzburg, Germany, 2012. [Google Scholar]
- Kajiura, H.; Stasheff, J. Homotopy algebras inspired by classical open-closed string field theory. Commun. Math. Phys.
**2006**, 263, 553–581. [Google Scholar] [CrossRef][Green Version] - Esposito, C.; de Kleijn, N. Universal deformation formula, formality and actions. arXiv
**2017**, arXiv:1704.07054. [Google Scholar] - Esposito, C.; de Kleijn, N. L
_{∞}-resolutions and twisting in the curved context. Rev. Mat. Iberoam.**2020**, 37, 1581–1598. [Google Scholar] [CrossRef] - Mehta, R.; Zambon, M. L
_{∞}algebra actions. Diff. Geom. Appl.**2012**, 30, 576–587. [Google Scholar] [CrossRef][Green Version] - Nguyen, H.; Schenkel, A.; Szabo, R.J. Batalin-Vilkovisky quantization of fuzzy field theories. Lett. Math. Phys.
**2021**, 111, 149. [Google Scholar] [CrossRef] - Arvanitakis, A.S.; Hohm, O.; Hull, C.; Lekeu, V. Homotopy Transfer and Effective Field Theory I: Tree-level. arXiv
**2007**, arXiv:2007.07942. [Google Scholar] [CrossRef] - Costello, K. A geometric construction of the Witten genus, II. arXiv
**2010**, arXiv:1112.0816. [Google Scholar] - Getzler, E. Covariance in the Batalin-Vilkovisky formalism and the Maurer-Cartan equation for curved Lie algebras. Lett. Math. Phys.
**2019**, 109, 187–224. [Google Scholar] [CrossRef][Green Version] - Grewcoe, C.J. Geometric Structure of Generalised Gauge Field Theories. Ph.D. Thesis, University of Zagreb, Zagreb, Croatia, 2021. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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