Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

Imma Gálvez-Carrillo, Ralph M. Kaufmann, Andrew Tonks

Abstract

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework, which is presented step-by-step with examples throughout. In this first part of two papers, we concentrate on the simplicial and operadic aspects.

Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects

Abstract

We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes-Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework, which is presented step-by-step with examples throughout. In this first part of two papers, we concentrate on the simplicial and operadic aspects.

Paper Structure

This paper contains 90 sections, 75 theorems, 109 equations, 10 figures.

Table of Contents

  1. Preface: Three Hopf algebras
  2. Multiple zeta values
  3. Formal symbols
  4. Goncharov's first Hopf algebra
  5. Goncharov's second Hopf algebra and the version of Brown
  6. Discussion
  7. Connes--Kreimer
  8. Rooted forests without tails
  9. Connes--Kreimer's Hopf algebra of rooted forests
  10. Other variants
  11. Discussion
  12. Baues' Hopf algebra for double loop spaces
  13. Discussion
  14. Bi-- and Hopf Algebras from (Co)-Operads
  15. Connes--Kreimer trees as a road map
  16. ...and 75 more sections

Key Result

Theorem 1.2

Gont If we assign $\hat{I}(a_0;a_1,\dots,a_{m};a_{m+1})$ degree $m$ then $\mathscr H_G$ with the co--product HopfPart1:gontcoprod (and the unique antipode) is a connected graded Hopf algebra.

Figures (10)

  • Figure 1: Grafting trees with labeled leaves. The tree is grafted onto the leaf number 2.
  • Figure 2: A cut in the non--symmetric planar case. Then the right side is an ordered tensor product. Alternatively, the same cut can be seen as a term on the full coinvariants, viz. the unlabeled version where on the right side, the forest is a symmetric product.
  • Figure 3: One term of the dual to $\gamma$ as given by a cut in the labeled case. The labels $i,j,k,l,m$ indicate the parring of the half edges after severing the edges.
  • Figure 4: An example of a cut for the coinvariants yielding the relative tensor product according to Remark \ref{['HopfPart1:coinvrmk']} part (2).
  • Figure 5: Co--product for the amputated version. The same example for the amputated version: First all tails are removed. After cutting all newly formed tails are amputated and empty trees/forests are represented by $1=1_K$. Notice that indeed $||$ from Figure \ref{['HopfPart1:fig:coproducts2a']} is set to $1_K$ as is done in the Hopf quotient.
  • ...and 5 more figures

Theorems & Definitions (207)

  • Example 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 2.1
  • Example 2.2
  • ...and 197 more