An operad is never free as a pre-Lie algebra
Emily Burgunder, B\'er\'enice Delcroix-Oger, Dominique Manchon
TL;DR
This paper proves that operads, which have a natural pre-Lie structure, are not free as pre-Lie algebras, by constructing a specific linear operation that distinguishes them.
Contribution
It establishes that operads cannot be free pre-Lie algebras by explicitly constructing a linear operation that vanishes in any operad but not in the free pre-Lie algebra.
Findings
Operads have a natural pre-Lie structure.
A specific linear operation distinguishes operads from free pre-Lie algebras.
Operads are proven not to be free as pre-Lie algebras.
Abstract
An operad is naturally endowed with a pre-Lie structure. We prove that as a pre-Lie algebra an operad is not free. The proof holds on defining a non-vanishing linear operation in the pre-Lie algebra which is zero in any operad.
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Taxonomy
TopicsAdvanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology · Algebraic structures and combinatorial models