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Cofree Com-PreLie algebras

Loïc Foissy

Abstract

A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie product satisfying some compatibilities with the product and the coproduct. We here give examples of cofree Com-PreLie bialgebras, including all the ones such that the preLie product is homogeneous of degree $\ge$ --1. We also give a graphical description of free unitary Com-PreLie algebras, explicit their canonical bialgebra structure and exhibit with the help of a rigidity theorem certain cofree quotients, including the Connes-Kreimer Hopf algebra of rooted trees. We finally prove that the dual of these bialgebras are also enveloping algebras of preLie algebras, combinatorially described.

Cofree Com-PreLie algebras

Abstract

A Com-PreLie bialgebra is a commutative bialgebra with an extra preLie product satisfying some compatibilities with the product and the coproduct. We here give examples of cofree Com-PreLie bialgebras, including all the ones such that the preLie product is homogeneous of degree --1. We also give a graphical description of free unitary Com-PreLie algebras, explicit their canonical bialgebra structure and exhibit with the help of a rigidity theorem certain cofree quotients, including the Connes-Kreimer Hopf algebra of rooted trees. We finally prove that the dual of these bialgebras are also enveloping algebras of preLie algebras, combinatorially described.

Paper Structure

This paper contains 19 sections, 31 theorems, 167 equations, 2 tables.

Table of Contents

  1. Reminders on Com-PreLie algebras
  2. Definitions
  3. Linear endomorphism on primitive elements
  4. Extension of the pre-Lie product
  5. Examples on shuffle algebras
  6. Preliminary lemmas
  7. PreLie products of positive degree
  8. PreLie products of degree $-1$
  9. Free Com-PreLie algebras and quotients
  10. Description of free Com-PreLie algebras
  11. Quotients of $\mathit{UCP}(\mathcal{D})$
  12. PreLie structure of $\mathit{UCP}(\mathcal{D})$ and $\mathit{CP}(\mathcal{D})$
  13. Bialgebra structures on free Com-PreLie algebras
  14. Tensor product of Com-PreLie algebras
  15. Coproduct on $\mathit{UCP}(\mathcal{D})$
  16. ...and 4 more sections

Key Result

Lemma 1.2

Theorems & Definitions (83)

  • Definition 1.1
  • Remark 1.1
  • Lemma 1.2
  • proof
  • Remark 1.2
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • proof
  • Lemma 1.5
  • ...and 73 more