Vol. 15, No. 1, 1965

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On anti-commutative algebras and general Lie triple systems

Arthur Argyle Sagle

Vol. 15 (1965), No. 1, 281–291

A general Lie triple system as defined by K. Yamaguti, is considered as an anti-commutative algebra A with a trilinear operation [x,y,z] in which (among other things) the mappings D(x,y) : z [x,y,z] 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(A) generated by the D(x,y)’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 [x,y,z], then A is (1) a Lie algebra; or (2) a Malcev algebra; or (3) an algebra satisfying J(x,y,z)w = J(w,x,yz) + J(w,y,zx) + J(w,z,xy) where J(x,y,z) = xy z + yz x + zx y. Furthermore in all three cases I(A) is the derivation algebra of A and I(A) is completely reducible in A.

Mathematical Subject Classification
Primary: 17.30
Received: 23 January 1964
Published: 1 March 1965
Arthur Argyle Sagle