On split twisted inner derivation triple systems with no restrictions on their 0-root spaces

Abstract

The aim of this paper is to study the structure of arbitrary split twisted inner derivation triple systems. We obtain a sufficient condition for the decomposition of arbitrary twisted inner derivation triple system T{\mathscr{T}} which is of the form T=U+∑[θ]∈ΛT/∼I[θ]{\mathscr{T}}=U+{\sum }_{\left[\theta ]\in {\Lambda }^{{\mathscr{T}}}\text{/} \sim }{I}_{\left[\theta ]} with UU a subspace of T0{{\mathscr{T}}}_{0} and any I[θ]{I}_{\left[\theta ]} a well-described ideal of T{\mathscr{T}}, satisfying {I[θ],T,I[η]}\left\{{I}_{\left[\theta ]},{\mathscr{T}},{I}_{\left[\eta ]}\right\} = {I[θ],I[η],T}\left\{{I}_{\left[\theta ]},{I}_{\left[\eta ]},{\mathscr{T}}\right\} = {T,I[θ],I[η]}\left\{{\mathscr{T}},{I}_{\left[\theta ]},{I}_{\left[\eta ]}\right\} = {I[θ],T,I[η]}′\left\{{I}_{\left[\theta ]},{\mathscr{T}},{I}_{\left[\eta ]}\right\}^{\prime} = {I[θ],I[η],T}′\left\{{I}_{\left[\theta ]},{I}_{\left[\eta ]},{\mathscr{T}}\right\}^{\prime} = {T,I[θ],I[η]}′=0\left\{{\mathscr{T}},{I}_{\left[\theta ]},{I}_{\left[\eta ]}\right\}^{\prime} =0 if [θ]≠[η]\left[\theta ]\ne \left[\eta ]. In particular, a necessary and sufficient condition for the simplicity of the triple system is given.

Keywords

• split
• twisted inner derivation triple system
• root system
• root space
• 17b75
• 17a60
• 17b22
• 17b65