The loop of formal power series with noncommutative coefficients under substitution

TL;DR

This paper explores the algebraic structure formed by formal power series with noncommutative coefficients under substitution, introducing related Lie and Sabinin algebras and providing examples of specific Lie algebras.

Contribution

It initiates the study of the loop structure of such power series and describes associated Lie and Sabinin algebras, including examples satisfying certain identities.

Findings

Identification of the loop structure with substitution product

Description of related Lie and Sabinin algebras

Examples of Lie algebras satisfying identities of degrees 5 and 6

Abstract

The set of formal power series with coefficients in an associative but noncommutative algebra becomes a loop with the substitution product. We initiate the study of this loop by describing certain Lie and Sabinin algebras related to it. Some examples of Lie algebras satisfying the standard identities of degrees $5$ and $6$ appear naturally.