On the structure of graded Lie triple systems
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We study the structure of an arbitrary graded Lie triple system ${\mathfrak T}$ with restrictions neither on the dimension nor the base field. We show that ${\mathfrak T}$ is of the form ${\mathfrak T}=U + \sum_{j}I_{j}$ with $U$ a linear subspace of the 1-homogeneous component ${\mathfrak T}_1$ and any $I_{j}$ a well described graded ideal of ${\mathfrak T}$, satisfying $[I_j,{\mathfrak T},I_k]=0$ if $j\neq k$. Under mild conditions, the simplicity of ${\mathfrak T}$ is characterized and it is shown that an arbitrary graded Lie triple system ${\mathfrak T}$ is the direct sum of the family of its minimal graded ideals, each one being a simple graded Lie triple system.
Abstract
We study the structure of an arbitrary graded Lie triple system ${\mathfrak T}$ with restrictions neither on the dimension nor the base field. We show that ${\mathfrak T}$ is of the form ${\mathfrak T}=U + \sum_{j}I_{j}$ with $U$ a linear subspace of the 1-homogeneous component ${\mathfrak T}_1$ and any $I_{j}$ a well described graded ideal of ${\mathfrak T}$, satisfying $[I_j,{\mathfrak T},I_k]=0$ if $j\neq k$. Under mild conditions, the simplicity of ${\mathfrak T}$ is characterized and it is shown that an arbitrary graded Lie triple system ${\mathfrak T}$ is the direct sum of the family of its minimal graded ideals, each one being a simple graded Lie triple system.
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