Vol. 15, No. 2, 1965

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On simple extended Lie algebras over fields of characteristic zero

Arthur Argyle Sagle

Vol. 15 (1965), No. 2, 621–648

In this paper we shall investigate algebras which generalize Lie algebras, Malcev algebras and binary-Lie algebras (every two elements generate a Lie subalgebra). Such an algebra A is called an extented Lie algebra (briefly el-algebra) and is defined by

xy = − yx and J(x,y,xy) = 0

for all x, y in A where J(x,y,z) = xy z + yz x + zx y. We prove the following.

Theorem. Let A be a simple finite dimensional el-algebra over an algebraically closed field of characteristic zero, then A is a simple Lie algebra or the simple seven dimensional Malcev algebra if and only if the trace form, (x,y) = trace RxRy, is a nondegenerate invariant form.

Mathematical Subject Classification
Primary: 17.30
Received: 7 November 1963
Published: 1 June 1965
Arthur Argyle Sagle