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Commutative B\_infty -algebras are shuffle algebras

Loïc Foissy, Frédéric Patras

Abstract

We here construct an explicit isomorphism between any commutative Hopf algebra which underlying coalgebra is the tensor coalgebra of a space $V$ and the shuffle algebra based on the same space. This isomorphism uses the commutative $B_\infty$ structure that governs the product and the eulerian idempotent, as well as the canonical projection on the space $V$. This generalizes Homan's isomorphism between commutative quasi-shuffle and shuffle algebras, which correspond to the case when the $B_\infty$ structure is given by an associative and commutative product. We develop several examples in details, including the Hopf algebra of finite topologies.

Commutative B\_infty -algebras are shuffle algebras

Abstract

We here construct an explicit isomorphism between any commutative Hopf algebra which underlying coalgebra is the tensor coalgebra of a space and the shuffle algebra based on the same space. This isomorphism uses the commutative structure that governs the product and the eulerian idempotent, as well as the canonical projection on the space . This generalizes Homan's isomorphism between commutative quasi-shuffle and shuffle algebras, which correspond to the case when the structure is given by an associative and commutative product. We develop several examples in details, including the Hopf algebra of finite topologies.
Paper Structure (10 sections, 31 theorems, 95 equations)

This paper contains 10 sections, 31 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Filtered-graded coalgebras
  3. Cofree filtered-graded coalgebras
  4. $B_\infty$-algebras
  5. Graded $B_\infty$-algebras
  6. Fundamental examples
  7. The structure of graded commutative $B_\infty$-algebras
  8. The structure of commutative $B_\infty$-algebras
  9. A $B_\infty$ structure on finite topologies
  10. Finite topologies: the double bialgebra structure

Key Result

Lemma 2.2

The coproduct $\Delta$ and the coalgebra $C$ are called conilpotent if $\bar{C}=\bigcup\limits_{n\geq 1} F_n$, that is if for every $c\in \bar{C}$ there exists an integer $n\geq 2$ such that $\overline \Delta_n(c)=0$. When the coalgebra $C$ is graded and connected, the coproduct is automatically co

Theorems & Definitions (79)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Definition 3.2: Cofree filtered-graded coalgebras
  • Remark 3.1
  • Lemma 3.3
  • proof
  • Remark 3.2
  • ...and 69 more