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On the Lie enveloping algebra of a pre-Lie algebra

Published online by Cambridge University Press:  28 May 2008

J.-M. Oudom  and
D. Guin
J.-M. Oudom
Affiliation:
oudom@math.univ-montp2.frI3M, Université Montpellier II, case 051, Place Eugène Bataillon, 34 095 Montpellier, France
D. Guin
Affiliation:
dguin@math.univ-montp2.frI3M, Université Montpellier II, case 051, Place Eugène Bataillon, 34 095 Montpellier, France
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Abstract

We construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. It turns S(L) into a Hopf algebra which is isomorphic to the enveloping algebra of LLie. Then we prove that in the case of rooted trees our construction gives the Grossman-Larson Hopf algebra, which is known to be the dual of the Connes-Kreimer Hopf algebra. We also show that symmetric brace algebras and pre-Lie algebras are the same. Finally, we give a similar interpretation of the Hopf algebra of planar rooted trees.

Keywords

brace algebraHopf algebrapre-Lie algebrarooted tree
Type
Research Article
Information
Journal of K-Theory , Volume 2 , Issue 1 , August 2008 , pp. 147 - 167
Copyright
Copyright © ISOPP 2008

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