A rigidity theorem for pre-Lie algebras

Livernet, Muriel

doi:10.1016/j.jpaa.2005.10.014

Cited by 58 publications

(81 citation statements)

References 16 publications

(27 reference statements)

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“…This implies that every connected bidendriform bialgebra is freely generated by its primitive elements, so is isomorphic to a H D for a well chosen D (Theorem 39). A similar result is proved in [13] for preLie algebras.…”

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Bidendriform bialgebras, trees, and free quasi-symmetric functions

Foissy

2007

Journal of Pure and Applied Algebra

128

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We introduce bidendriform bialgebras, which are bialgebras such that both product and coproduct can be split into two parts satisfying good compatibilities. For example, the Malvenuto-Reutenauer Hopf algebra and the non-commutative Connes-Kreimer Hopf algebras of planar decorated rooted trees are bidendriform bialgebras. We prove that all connected bidendriform bialgebras are generated by their primitive elements as a dendriform algebra (bidendriform Milnor-Moore theorem) and then is isomorphic to a Connes-Kreimer Hopf algebra. As a corollary, the Hopf algebra of Malvenuto-Reutenauer is isomorphic to the Connes-Kreimer Hopf algebra of planar rooted trees decorated by a certain set. We deduce that the Lie algebra of its primitive elements is free in characteristic zero (G. Duchamp, F. Hivert and J.-Y. Thibon conjecture).Résumé nous introduisons les bigèbres bidendriformes, qui sont des bigèbres dont le produit et le coproduit peuventêtre scindés en deux avec de bonnes compatibilités. Par exemple, l'algèbre de Hopf de Malvenuto-Reutenauer et les algèbres de Hopf non-commutative de Connes-Kreimer sur les arbres plans enracinés décorés sont des bigèbres bidendriformes. Nous montrons que toute bigèbre bidendriforme connexe est engendrée par seséléments primitifs comme algèbre dendriforme (version bidendriforme du théorème de Milnor-Moore) et qu'elle est alors isomorpheà une algèbre de Hopf de Connes-Kreimer. En conséquence, l'algèbre de Hopf de Malvenuto-Reutenauer est isomorpheà l'algèbre de Connes-Kreimer des arbres plans enracinés décorés par un certain ensemble. On en déduit que l'algèbre de Lie de seséléments primitifs est libre en caractéristique zéro (conjecture de G. Duchamp, F. Hivert et J.-Y. Thibon).

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confidence: 77%

“…So we already have (4)-(6), and (12) + (13), (14) + (15). It is then enough to prove (13). We have: Remark.…”

Section: Bidendriform Structure On Fqsymmentioning

confidence: 96%

Bidendriform bialgebras, trees, and free quasi-symmetric functions

Foissy

2007

Journal of Pure and Applied Algebra

128

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“…NAP -algebras. The operation * used as an auxiliary operation in the proof of the main theorem satisfies the following relation: (x * y) * z = (x * z) * y for any labelled trees x, y, z. Algebras with one binary operation satisfying this identity have already occurred in the literature in the work of M. Livernet [4], and were called NAP -algebras (for NonAssociative Perm). She showed that the labelled trees do describe the associated operad and that the product is precisely given by the operation * on trees described in the proof of Theorem 3.1.…”

Section: Comparison With Associative Jordan and Nap-algebrasmentioning

confidence: 99%

The symmetric operation in a free pre-Lie algebra is magmatic

Bergeron¹,

Loday²

2011

Proc. Amer. Math. Soc.

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Abstract. A pre-Lie product is a binary operation whose associator is symmetric in the last two variables. As a consequence its antisymmetrization is a Lie bracket. In this paper we study the symmetrization of the pre-Lie product. We show that it does not satisfy any other universal relation than commutativity. This means that the map from the free commutative-magmatic algebra to the free pre-Lie algebra induced by the symmetrization of the pre-Lie product is injective. This result is in contrast with the associative case, where the symmetrization gives rise to the notion of a Jordan algebra. We first give a selfcontained proof. Then we give a proof which uses the properties of dendriform and duplicial algebras.

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“…The aim of the present text is to prove that r is a free pre-Lie algebra. We use for this the notion of non-associative permutative algebra [14] and a manipulation of formal series. More precisely, we introduce in the second section of this text a nonassociative permutative product on r , and we show that r is free.…”

Section: Introductionmentioning

confidence: 99%

Free Brace Algebras are Free Pre-Lie Algebras

Foissy

2010

Communications in Algebra

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A rigidity theorem for pre-Lie algebras

Cited by 58 publications

References 16 publications

Bidendriform bialgebras, trees, and free quasi-symmetric functions

Bidendriform bialgebras, trees, and free quasi-symmetric functions

The symmetric operation in a free pre-Lie algebra is magmatic

Free Brace Algebras are Free Pre-Lie Algebras

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