Eulerian idempotent and Kashiwara-Vergne conjecture

Emily Burgunder[1]

  • [1] Université Montpellier II Institut de Mathématiques et de modélisation de Montpellier UMR CNRS 5149 Place Eugène Bataillon 34095 Montpellier Cedex 5 (France)

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 4, page 1153-1184
  • ISSN: 0373-0956

Abstract

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By using the interplay between the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution ( F , G ) of the first equation of the Kashiwara-Vergne conjecture x + y - log ( e y e x ) = ( 1 - e - ad x ) F ( x , y ) + ( e ad y - 1 ) G ( x , y ) . Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates x and y thanks to the kernel of the Dynkin idempotent.

How to cite

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Burgunder, Emily. "Eulerian idempotent and Kashiwara-Vergne conjecture." Annales de l’institut Fourier 58.4 (2008): 1153-1184. <http://eudml.org/doc/10345>.

@article{Burgunder2008,
abstract = {By using the interplay between the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution $(F,G)$ of the first equation of the Kashiwara-Vergne conjecture\[ x+y-\log (\{\rm e\}^\{y\}\{\rm e\}^\{x\}) = (1-\{\rm e\}^\{-\{\rm ad\}\, x\}) F(x,y)+(\{\rm e\}^\{\{\rm ad\}\, y\}-1) G(x,y) . \]Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates $x$ and $y$ thanks to the kernel of the Dynkin idempotent.},
affiliation = {Université Montpellier II Institut de Mathématiques et de modélisation de Montpellier UMR CNRS 5149 Place Eugène Bataillon 34095 Montpellier Cedex 5 (France)},
author = {Burgunder, Emily},
journal = {Annales de l’institut Fourier},
keywords = {Kashiwara-Vergne conjecture; Baker-Campbell-Hausdorff series; Eulerian idempotent; Dynkin idempotent; Hopf algebras},
language = {eng},
number = {4},
pages = {1153-1184},
publisher = {Association des Annales de l’institut Fourier},
title = {Eulerian idempotent and Kashiwara-Vergne conjecture},
url = {http://eudml.org/doc/10345},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Burgunder, Emily
TI - Eulerian idempotent and Kashiwara-Vergne conjecture
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 4
SP - 1153
EP - 1184
AB - By using the interplay between the Eulerian idempotent and the Dynkin idempotent, we construct explicitly a particular symmetric solution $(F,G)$ of the first equation of the Kashiwara-Vergne conjecture\[ x+y-\log ({\rm e}^{y}{\rm e}^{x}) = (1-{\rm e}^{-{\rm ad}\, x}) F(x,y)+({\rm e}^{{\rm ad}\, y}-1) G(x,y) . \]Then, we explicit all the solutions of the equation in the completion of the free Lie algebra generated by two indeterminates $x$ and $y$ thanks to the kernel of the Dynkin idempotent.
LA - eng
KW - Kashiwara-Vergne conjecture; Baker-Campbell-Hausdorff series; Eulerian idempotent; Dynkin idempotent; Hopf algebras
UR - http://eudml.org/doc/10345
ER -

References

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  8. C. Reutenauer, Free Lie Algebras, (1993), Oxford University Press Zbl0798.17001MR1231799
  9. F. Rouvière, Démonstration de la conjecture de Kashiwara-Vergne pour l’algèbre sl ( 2 ) , C. R. Acad. Sci. Paris Sér. I Math. 292 (1981), 657-660 Zbl0467.22011
  10. C. Torossian, Sur la conjecture combinatoire de Kashiwara-Vergne, J. Lie Theory 12 (2002), 597-616 Zbl1012.17002MR1923789
  11. M. Vergne, Le centre de l’algèbre enveloppante et la formule de Campbell-Hausdorff, C. R. Acad. Sci. Paris, Série I Math. 329 (1999), 767-772 Zbl0989.17007
  12. D. Wigner, An identity in the free Lie algebra, Proc. Amer. Math. Soc. (1989), 639-640 Zbl0682.17007MR969322

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