Enriched Lie algebras in topology, I

TL;DR

This paper introduces the concept of enriched Lie algebras as an extension of graded Lie algebras, establishing their role as rational homotopy Lie algebras for connected spaces and laying the groundwork for further study.

Contribution

It defines and explores the properties of complete enriched Lie algebras, linking them to rational homotopy theory and connected topological spaces.

Findings

Enriched Lie algebras generalize graded Lie algebras for topology.

Each complete enriched Lie algebra corresponds to a rational homotopy Lie algebra.

Foundational framework for future research in topology and Lie algebra structures.

Abstract

The complete enriched Lie algebras constitue the natural extension of graded Lie algebras for connected spaces. Each complete enriched Lie algebra is the rational homotopy Lie algebra of a connected space. This text is the first part of a general study of those Lie algebras