Reference : Eulerian idempotent, pre-Lie logarithm and combinatorics of trees
E-prints/Working papers : Already available on another site
Physical, chemical, mathematical & earth Sciences : Mathematics
http://hdl.handle.net/10993/31774
Eulerian idempotent, pre-Lie logarithm and combinatorics of trees
English
Bandiera, Ruggero [Università degli Studi di Roma "La Sapienza" > Mathematics]
Schätz, Florian mailto [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >]
2017
1
52
No
[en] Eulerian idempotent ; pre-Lie algebras ; combinatorics
[en] The aim of this paper is to bring together the three objects in the title. Recall that, given a Lie algebra g, the Eulerian idempotent is a canonical projection from the enveloping algebra U(g) to
g. The Baker-Campbell-Hausdorff product and the Magnus expansion can both be expressed in terms of the Eulerian idempotent, which makes it interesting to establish explicit formulas for the latter. We show how to reduce the computation of the Eulerian idempotent to the computation of a logarithm in a certain pre-Lie algebra of planar, binary, rooted trees. The problem of finding formulas for the pre-Lie logarithm, which is interesting in its own right – being related to operad theory, numerical analysis and renormalization – is addressed using techniques inspired by umbral calculus. As a consequence of our analysis, we find formulas both for the Eulerian idempotent and the pre-Lie logarithm in terms of the combinatorics of trees.
http://hdl.handle.net/10993/31774
https://arxiv.org/abs/1702.08907

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