Introduction to co-split Lie algebras

TL;DR

This paper introduces co-split Lie algebras, a new mathematical structure linking Lie algebras and coalgebras, offering fresh insights into their interrelations and classical forms.

Contribution

It defines the concept of co-split Lie algebras and explores their connection to the Killing form and adjoint representation, providing a novel framework.

Findings

New compatibility condition between Lie algebras and coalgebras

Reveals a new perspective on the Killing form and adjoint representation

Establishes foundational properties of co-split Lie algebras

Abstract

In this work, we introduce a new concept which is obtained by defining a new compatibility condition between Lie algebras and Lie coalgebras. With this terminology, we describe the interrelation between the Killing form and the adjoint representation in a new perspective.