On anti-commutative algebras and general Lie triple systems

Arthur Argyle Sagle

Vol. 15 (1965), No. 1, 281–291

DOI: 10.2140/pjm.1965.15.281

Abstract

A general Lie triple system as defined by K. Yamaguti, is considered as an anti-commutative algebra $A$ with a trilinear operation $\left[x,y,z\right]$ in which (among other things) the mappings $D\left(x,y\right):z\to \left[x,y,z\right]$ are derivations of $A$. It is shown that if the trilinear operation is homogeneous, and $A$ is irreducible as a general L.t.s. or irreducible relative to the Lie algebra $I\left(A\right)$ generated by the $D\left(x,y\right)$’s, then $A$ is a simple algebra. The main result is the following. If $A$ is a simple finite-dimensional anti-commutative algebra over a field of characteristic zero which is a general L.t.s. with a homogeneous trilinear operation $\left[x,y,z\right]$, then $A$ is (1) a Lie algebra; or (2) a Malcev algebra; or (3) an algebra satisfying $J\left(x,y,z\right)w=J\left(w,x,yz\right)+J\left(w,y,zx\right)+J\left(w,z,xy\right)$ where $J\left(x,y,z\right)=xy\cdot z+yz\cdot x+zx\cdot y$. Furthermore in all three cases $I\left(A\right)$ is the derivation algebra of $A$ and $I\left(A\right)$ is completely reducible in $A$.

Mathematical Subject Classification

Primary: 17.30

Milestones

Received: 23 January 1964

Published: 1 March 1965