# Nonassociative Solomon's descent algebras

## TL;DR

This paper extends Solomon's descent algebra to a nonassociative setting using Sabinin algebras, introducing new algebraic structures and combinatorial frameworks related to nonassociative symmetric functions.

## Contribution

It establishes a nonassociative analogue of Solomon's descent algebra via Sabinin algebras and develops a combinatorial nonassociative Hopf algebra framework.

## Key findings

- Proves isomorphism between universal enveloping algebras of Sabinin algebras and nonassociative symmetric functions.
- Introduces a nonassociative lifting of the Malvenuto-Reutenauer Hopf algebra.
- Shows the nonassociative algebra is free and cofree with a Mackey type formula.

## Abstract

Descent algebras of graded bialgebras were introduced by F. Patras as a generalization of Solomon's descent algebras for Coxeter groups of type $A$, i.e. symmetric groups. The universal enveloping algebra of the free Lie algebra on a countable number of generators, its descent algebra and Solomon's descent algebra, with its outer product, for symmetric groups are isomorphic to the Hopf algebra of noncommutative symmetric functions, a free associative algebra on a countable number of generators. In this paper we prove a similar result for universal enveloping algebras of relatively free Sabinin algebras, which makes them nonassociative analogues of the Solomon's descent algebra and of the algebra of noncommutative symmetric functions. In the absolutely free case a combinatorial setting is also presented by introducing a nonassociative lifting of the Malvenuto-Reutenauer Hopf algebra of permutations and of the dual of the Hopf algebra of noncommutative quasi-symmetric functions. We prove that this lifting is a free nonassociative algebra as well as a cofree graded coassociative coalgebra which also admits an associative inner product that satisfies a Mackey type formula when restricted to its nonassociative Solomon's descent algebra. Pseudo-compositions of sets instead of pseudo-compositions of numbers is the natural language for the underlying combinatorics.