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:
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