The Hopf Algebras of Symmetric Functions and Quasi-Symmetric Functions in Non-Commutative Variables Are Free and Co-Free

Abstract

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and co-free. In the process of looking for bases which generate the space we define orders on the set partitions and set compositions which allow us to define bases which have simple and natural rules for the product of basis elements.

From left modules to algebras over an operad: application to combinatorial Hopf algebras

“…It is also the Hopf algebra N CQSym of Bergeron et Zabrocky in [4] and we recover that it is a free algebra.…”

“…We prove that Hopf algebra structures on the faces of the permutohedra given e.g. by Chapoton in [5], Bergeron and Zabrocky in [4] and Patras and Schocker in [22], arise from the operad CTD of commutative tridendriform algebras defined by Loday in [14]. We deduce also some freeness results from our theory.…”

A noncommutative Bohnenblust–Spitzer identity for Rota–Baxter algebras solves Bogoliubov’s recursion

“…Let us recall also Hivert's notion of quasi-symmetric functions over a set of noncommutative variables from [4], [37]. Let f be a surjective map from OEn to OEk, and let X be a countable set of variables, as above.…”

“…It is related to various fundamental objects such as the Coxeter complex of type A n or the corresponding Solomon-Tits and twisted descent algebras. We refer to [42], [4], [37] for further details on the subject. We also introduce, for further use, the notation M n f for the image of M f under the map sending x i to 0 for i > n and x i to itself else.…”

“…Let us simply mention at this stage that these notions will provide the right framework to extend to noncommutative RB algebras and noncommutative symmetric functions the classical results of Rota and Smith relating commutative RB algebras and symmetric functions [46]. Recall that the algebra NCQSym of quasi-symmetric functions in noncommutative variables introduced in the previous section is naturally provided with a Hopf algebra structure [4]. On the elementary quasi-symmetric functions M OEn , the coproduct acts as on a sequence of divided powers…”