Geodesic
Lie representations and an algebra containing Solomon's.
Journal of Algebraic Combinatorics, Tome 16 (2002) no. 3, pp. 301-314.

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Summary: We introduce and study a Hopf algebra containing the descent algebra as a sub-Hopf-algebra. It has the main algebraic properties of the descent algebra, and more: it is a sub-Hopf-algebra of the direct sum of the symmetric group algebras; it is closed under the corresponding inner product; it is cocommutative, so it is an enveloping algebra; it contains all Lie idempotents of the symmetric group algebras. Moreover, its primitive elements are exactly the Lie elements which lie in the symmetric group algebras.
Keywords: descent algebra, Hopf algebra, Lie idempotent, symmetric group algebras, quasi-symmetric functions, Lie elements 
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     author = {Patras, Fr\'ed\'eric and Reutenauer, Christophe},
     title = {Lie representations and an algebra containing {Solomon's.}},
     journal = {Journal of Algebraic Combinatorics},
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Patras, Frédéric; Reutenauer, Christophe. Lie representations and an algebra containing Solomon's.. Journal of Algebraic Combinatorics, Tome 16 (2002) no. 3, pp. 301-314. http://geodesic.mathdoc.fr/item/JAC_2002__16_3_a0/