Knit products of graded Lie algebras and groups

TL;DR

This paper introduces the concept of 'knit products' for graded Lie algebras and groups, exploring their algebraic structure and how homomorphisms interact with these products, linking to the Zappa-Szép product.

Contribution

It defines the knit product for graded Lie algebras and groups, and investigates the behavior of homomorphisms within this framework, connecting to existing algebraic structures.

Findings

Knit product generalizes the double-sided semidirect product for graded Lie algebras.

The knit product of groups aligns with the Zappa-Szép product.

Homomorphisms preserve the knit product structure.

Abstract

If a graded Lie algebra is the direct sum of two graded sub Lie algebras, its bracket can be written in a form that mimics a "double sided semidirect product". It is called the {\it knit product} of the two subalgebras then. The integrated version of this is called a {\it knit product} of groups --- it coincides with the {\it Zappa-Sz\'ep product}. The behavior of homomorphisms with respect to knit products is investigated.