# Hopf Algebras and Topological Recursion

@article{Esteves2015HopfAA, title={Hopf Algebras and Topological Recursion}, author={Joao N. Esteves}, journal={arXiv: Mathematical Physics}, year={2015} }

We consider a model for topological recursion based on the Hopf Algebra of planar binary trees of Loday and Ronco. We show that extending this Hopf Algebra by identifying pairs of nearest neighbor leaves and producing in this way graphs with loops we obtain the full recursion formula of Eynard and Orantin.

#### Figures from this paper

#### 4 Citations

Hopf Algebraic Structure for Tagged Graphs and Topological Recursion

- Physics, Mathematics
- 2016

Using the shuffle structure of the graphs, we introduce a new kind of the Hopf algebraic structure for tagged graphs with, or without loops. Like a quantum group structure, its product is… Expand

A Quantization of the Loday-Ronco Hopf Algebra

- Mathematics, Physics
- 2021

Abstract. We propose a quantization algebra of the Loday-Ronco Hopf algebra k[Y ], based on the Topological Recursion formula of Eynard and Orantin. We have shown in previous works that the… Expand

Summary of results in topological recursion

- 2018

1 General properties The topological recursion (TR) is an axiomatic construction of a family of correlation functions ωg,n indexed by two integers g, n ≥ 0, from the initial data of a spectral curve… Expand

The geometry of real reducible polarizations in quantum mechanics

- Physics, Mathematics
- 2016

The formulation of Geometric Quantization contains several axioms and assumptions. We show that for real polarizations we can generalize the standard geometric quantization procedure by introducing… Expand

#### References

SHOWING 1-10 OF 27 REFERENCES

QED Hopf algebras on planar binary trees

- Mathematics
- 2001

Abstract In this paper we describe the Hopf algebras on planar binary trees used to renormalize the Feynman propagators of quantum electrodynamics, and the coaction which describes the… Expand

Hopf Algebras, Renormalization and Noncommutative Geometry

- Mathematics, Physics
- 1998

Abstract:We explore the relation between the Hopf algebra associated to the renormalization of QFT and the Hopf algebra associated to the NCG computations of tranverse index theory for foliations.

Incidence Hopf algebras

- Mathematics
- 1994

Abstract We present several results about incidence Hopf algebras of families of partially ordered sets, including a characterization of their algebra structure, a combinatorial technique for finding… Expand

Structure of the Loday–Ronco Hopf algebra of trees

- Mathematics
- 2004

Abstract Loday and Ronco defined an interesting Hopf algebra structure on the linear span of the set of planar binary trees. They showed that the inclusion of the Hopf algebra of non-commutative… Expand

The Incidence Hopf Algebra of Graphs

- Mathematics, Computer Science
- SIAM J. Discret. Math.
- 2012

A new formula is given for the antipode in the graph algebra in terms of acyclic orientations; this formula contains many fewer terms than Schmitt's more general formula for the antithesis in an incidence Hopf algebra. Expand

Les algèbres de Hopf des arbres enracinés décorés, I

- Mathematics
- 2001

Abstract We introduce a Hopf algebra of planar decorated rooted trees H D P,R which is non commutative and non cocommutative and generalizes the Hopf algebra of rooted trees H D R of Connes and… Expand

Hopf Algebra of the Planar Binary Trees

- Mathematics
- 1998

is a sub-Hopf algebra. There is a basis Qn of Soln such that the composite map k[Qn]&Soln k[Sn] has the following property: its linear dual k[Sn] k[Qn] is induced by a set-theoretic map Sn Qn . In… Expand

On the trees of quantum fields

- Physics
- 1999

Abstract. The solution of some equations involving functional derivatives is written as a series indexed by planar binary trees. The terms of the series are given by an explicit recursive formula.… Expand

Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem

- Physics, Mathematics
- 2000

Abstract:This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure… Expand

The spectral curve of the Eynard-Orantin recursion via the Laplace transform

- Mathematics
- 2012

The Eynard-Orantin recursion formula provides an effective tool for certain enumeration problems in geometry. The formula requires a spectral curve and the recursion kernel. We present a uniform… Expand