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Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

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 second part of two papers, we give the general categorical formulation.

Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation

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 second part of two papers, we give the general categorical formulation.

Paper Structure

This paper contains 104 sections, 44 theorems, 55 equations, 7 figures, 3 tables.

Table of Contents

  1. The general case: Bi-- and Hopf algebras from Feynman categories
  2. Preview
  3. Algebra and co-algebra structures for morphisms
  4. Isomorphism classes
  5. Algebra of morphisms of a (strict) monoidal category
  6. The decomposition co--product
  7. Co--product of the identity morphisms
  8. Co--algebra on isomorphisms classes
  9. Direct formula for $\Delta^{iso}$
  10. Bi--algebra structure conditions
  11. Feynman categories and bi--algebra structures
  12. Definition of a Feynman category
  13. Non-symmetric version
  14. Strict Feynman categories
  15. Bi--algebra structure for non--$\Sigma$ Feynman categories
  16. ...and 89 more sections

Key Result

Theorem 1.1

Let $\mathfrak F$ be a non--$\Sigma$ decomposition finite strict monoidal Feynman category. Set $\mathscr{B}={\mathbb Z}\it Mor({\mathcal{F}})$. Let $\mu=\otimes$, $\eta(1)=id_{\Eins}$, set $\Delta(\phi)=$$\sum_{(\phi_0,\phi_1): \phi=\phi_0\circ \phi_1}\phi_0\otimes \phi_1$ and define $\epsilon(\phi

Figures (7)

  • Figure 1: Marking a corolla by a simplex in $N_{\bullet}({\mathcal{C}})$. The morphisms decorate the ends of the tree, while the objects decorate the angles which correspond to the marks on the half circle
  • Figure 2: An example of a factorization in three--valent graphs aka. $\phi^3$. Alternatively the top graph $\mathbbnew{\Gamma}$ results from inserting the left graph $\mathbbnew{\Gamma}_1$, which has three components, into the right graph according $\mathbbnew{\Gamma}_0$ to the vertex map $\{u,v,w\}\mapsto r, p\mapsto p, q\mapsto q$, viz. $\mathbbnew{\Gamma}=\mathbbnew{\Gamma}_0\circ\mathbbnew{\Gamma}_1$, or the left graph is a subgraph of the top graph $\mathbbnew{\Gamma}_1\subset \mathbbnew{\Gamma}$ and the right graph $\mathbbnew{\Gamma}_0$ is the quotient graph. $\mathbbnew{\Gamma}_0=\mathbbnew{\Gamma}/\mathbbnew{\Gamma}_1$
  • Figure 3: The co--product of a graph. The factor of 2 is there, since there are two distinct subgraphs ---given by the two distinct edges--- which give rise to two factorizations whose abstract graphs coincide
  • Figure 4: One decomposition. To fix $\phi$ we specify $\phi^F(1)=1,\phi^F(2)=1'$, to fix $\phi_1$, we set $\phi_1^F(1)=1,\phi_1^F(2)=1,\phi_1^F(3)=1',\phi_1^F(4)=2'$ and to fix $\phi_0$ we fix $\phi_0^F(1)=1,\phi_0^F(2)=2$. There is no choice for the vertex maps and the involution is the one given by the ghost graph.
  • Figure 5: The interval injection $[1]\to [n]$ on the left, the surjection $\underline{n}\to \underline{1}$ on the right and and Joyal duality in the middle. Here reading the morphism upwards yields the double base point preserving injection, while reading it downward the surjection.
  • ...and 2 more figures

Theorems & Definitions (125)

  • Theorem 1.1
  • Proposition 1.2
  • Remark 1.3
  • Lemma 1.4
  • Proposition 1.5
  • proof
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • ...and 115 more