PostLie algebra structures on the Lie algebra sl(2,C)

TL;DR

This paper classifies all PostLie algebra structures on the complex Lie algebra sl(2,C), linking algebraic structures with matrix classification techniques, and providing a comprehensive understanding of these enriched algebraic forms.

Contribution

It offers a complete classification of PostLie algebra structures on sl(2,C), reducing the problem to matrix equations and utilizing orthogonal group actions.

Findings

Classification of PostLie structures on sl(2,C) achieved

Reduction of classification to matrix equations

Use of orthogonal group actions in solution

Abstract

The PostLie algebra is an enriched structure of the Lie algebra that has recently arisen from operadic study. It is closely related to pre-Lie algebra, Rota-Baxter algebra, dendriform trialgebra, modified classical Yang-Baxter equations and integrable systems. We give a complete classification of PostLie algebra structures on the Lie algebra sl(2,C) up to isomorphism. We first reduce the classification problem to solving an equation of 3 x 3 matrices. To solve the latter problem, we make use of the classification of complex symmetric matrices up to the congruent action of orthogonal groups.