A rigidity theorem for preLie algebras

Details A rigidity theorem for preLie algebras A rigidity theorem for preLie algebras

2005-04-14

arxiv.org/abs/math/0504296v2

Mathematics

Quantum Algebra

20 pages; uses XY-pic. Finale version to appear in JPPA. refernce added and remark 3.8 (formula on heap-ordered trees) added

Scientific paper

In this paper we prove a ``Leray theorem'' for preLie algebras. We define a notion of ''Hopf'' preLie algebra: it is a preLie algebra together with a non associative permutative coproduct D and a compatibility relation between the preLie product and the coproduct D. A nonassociative permutative algebra is a vector space together with a product satisfying the relation (ab)c=(ac)b. A non associative permutative coalgebra is the dual notion. We prove that any such graded connected "Hopf" preLie algebra is a free preLie algebra. It uses the description of preLie algebras in term of rooted trees developped by Chapoton and the author. We interpret also this theorem by way of cogroups in the category of preLie algebras.

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