Extension of the product of a post-Lie algebra and application to the SISO feedback transformation group

TL;DR

This paper explores the algebraic structures underlying the affine SISO feedback transformation group, extending post-Lie algebra concepts to better understand its properties and applications.

Contribution

It introduces an extension of the magmatic product for post-Lie algebras and applies it to the enveloping algebra of the SISO feedback transformation group.

Findings

Construction of the extension of the magmatic product for post-Lie algebras

Description of free post-Lie algebras related to SISO feedback group

Application to the enveloping algebra of g SISO

Abstract

We describe the both post-and pre-Lie algebra g SISO associated to the affine SISO feedback transformation group. We show that it is a member of a family of post-Lie algebras associated to representations of a particular solvable Lie algebra. We first construct the extension of the magmatic product of a post-Lie algebra to its enveloping algebra, which allows to describe free post-Lie algebras and is widely used to obtain the enveloping of g SISO and its dual.