Cofree Com-PreLie algebras
Lo\"ic Foissy (LMPA)
TL;DR
This paper explores cofree Com-PreLie bialgebras, providing examples, graphical descriptions, and duality properties, including connections to the Connes-Kreimer Hopf algebra of rooted trees.
Contribution
It introduces new examples of cofree Com-PreLie bialgebras, describes their structure graphically, and establishes duality with enveloping algebras of preLie algebras.
Findings
Examples of cofree Com-PreLie bialgebras including homogeneous cases
Graphical description of free unitary Com-PreLie algebras
Duals are enveloping algebras of 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.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Homotopy and Cohomology in Algebraic Topology