Split Twisted Inner Derivation Triple Systems

Abstract

The class of twisted inner derivation triple systems (TIDTS) contains the one of Lie triple systems, the one of Jordan triple system, as well as some other kinds of triple systems. TIDTS were introduced and studied, in the finite dimensional setup, by Nora Hopkins in [5]. In order to begin an approach to a structure theory of infinite-dimensional TIDTS, we introduce an study split TIDTS by focussing on those of level 1. We show that any of such TIDTS T with a symmetric root system is of the form with 𝒰 a subspace of the 0-root space T 0 and any I j , a well described ideal of T, satisfying {I j , T, I k } = {T, I j , I k } = {I j , I k , T} = {I j , T, I k }′ = {T, I j , I k }′ = {I j , I k , T}′ = 0 if j ≠ k. Under certain conditions, the simplicity of T is characterized, and it is shown that T is the direct sum of the family of its minimal ideals, each one being a simple split TIDTS.