Twisted descent algebras and the Solomon–Tits algebra

The notion of descent algebra of a bialgebra is lifted to the Barratt–Joyal setting of twisted bialgebras. The new twisted descent algebras share many properties with their classical counterparts. For example, there are twisted analogs of classical Lie idempotents and of the peak algebra. Moreover, the universal twisted descent algebra is equipped with two products and a coproduct, and there is a fundamental rule linking all three. This algebra is shown to be naturally related to the geometry of the Coxeter complex of type A.