On Dynkin and Klyachko Idempotents in Graded Bialgebras

Let X = x1 xn be an infinite alphabet. The tensor algebra on X has canonically the structure of a Lie algebra and its Lie subalgebra generated by the elements of X identifies with the free Lie algebra on X. The structures of the tensor algebra and of the free Lie algebra are closely related to certain permutation statistics, as emphasized, e.g., in [8, 17]. In this setting, the Dynkin and Klyachko idempotents are fundamental tools. For example, they reduce the construction of basis of the free Lie algebra to the study and counting of given words, such as Lyndon words. The purpose of the present article is to show that these constructions, which could be thought of as intrinsically related to the combinatorics of the tensor bialgebra, generalize in fact to all graded connected cocommutative bialgebras. Recall that, by the Cartier–Milnor–Moore theorem, these bialgebras are, up to isomorphism, the enveloping algebras of graded connected Lie algebras. We establish in particular a Baker-like identity and describe explicitly the kernels of the natural generalizations to these bialgebras of the Dynkin and Klyachko operators (Theorem 16, Corollary 7, and Theorem 6).