Eulerian idempotent and Kashiwara-Vergne conjecture. (English) Zbl 1236.17039
Summary: 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 \(x+y-\log (e^y e^x)=(1-e^{- \text{ad}\, x}) F(x,y)+(e^{\text{ad}\,y}-1)G(x,y).\) 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.
MSC:
| 17B99 | Lie algebras and Lie superalgebras |
| 16T05 | Hopf algebras and their applications |
| 20C30 | Representations of finite symmetric groups |
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
Keywords:
Kashiwara-Vergne conjecture; Baker-Campbell-Hausdorff series; Eulerian idempotent; Dynkin idempotent; Hopf algebrasReferences:
| [1] | Alekseev, A.; Meinrenken, E., On the Kashiwara-Vergne conjecture, Invent. Math., 164, 615-634 (2006) · Zbl 1096.22007 · doi:10.1007/s00222-005-0486-4 |
| [2] | Alekseev, A.; Petracci, E., Uniqueness in the Kashiwara-Vergne conjecture, J. Lie Theory 16, 3, 531-538 (2006) · Zbl 1111.22017 |
| [3] | Cohn, P. M., Integral modules, Lie Rings and Free Groups (1951) |
| [4] | Kashiwara, M.; Vergne, M., The Campbell-Hausdorff formula and invariant hyperfunctions, Invent. Math., 47, 249-272 (1978) · Zbl 0404.22012 · doi:10.1007/BF01579213 |
| [5] | Kontsevich, M., Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66, 157-216 (2003) · Zbl 1058.53065 · doi:10.1023/B:MATH.0000027508.00421.bf |
| [6] | Loday, J.-L., Série de Hausdorff, idempotents eulériens et algèbres de Hopf, Expo. Math., 12, 165-178 (1994) · Zbl 0807.17003 |
| [7] | Patras, F.; Reutenauer, C., On Dynkin and Klyachko idempotents in graded bialgebras, Adv. in Applied Math., 28, 560-579 (2002) · Zbl 1024.16021 · doi:10.1006/aama.2001.0795 |
| [8] | Reutenauer, C., Free Lie Algebras (1993) · Zbl 0798.17001 |
| [9] | Rouvière, F., Démonstration de la conjecture de Kashiwara-Vergne pour l’algèbre \(\text{sl}(2)\), C. R. Acad. Sci. Paris Sér. I Math., 292, 657-660 (1981) · Zbl 0467.22011 |
| [10] | Torossian, C., Sur la conjecture combinatoire de Kashiwara-Vergne, J. Lie Theory, 12, 597-616 (2002) · Zbl 1012.17002 |
| [11] | Vergne, M., Le centre de l’algèbre enveloppante et la formule de Campbell-Hausdorff, C. R. Acad. Sci. Paris, Série I Math., 329, 767-772 (1999) · Zbl 0989.17007 · doi:10.1016/S0764-4442(99)90004-6 |
| [12] | Wigner, D., An identity in the free Lie algebra, Proc. Amer. Math. Soc., 639-640 (1989) · Zbl 0682.17007 · doi:10.1090/S0002-9939-1989-0969322-5 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
