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

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. This paper gives a complete classification of PostLie algebra structures on the Lie algebra sl(2,ℂ) up to isomorphism. The classification problem is first reduced to solving an equation of 3 × 3 matrices. Then the latter problem is solved by making use of the classification of complex symmetric matrices up to the congruent action of orthogonal groups.