Vol. 30, No. 1, 1969

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Algebras formed by the Zorn vector matrix

Tae-il Suh

Vol. 30 (1969), No. 1, 255–258

In the Zorn vector matrix algebra the three dimensional vector algebra is replaced by a finite dimensional Lie algebra L over a field of characteristic not 2 equipped with an associative symmetric bilinear form (a,b) and havin g the property: [a[bc]] = (a,c)b(a,b)c,a,b,c L. We determine all the alternative algebras A obtained in this way: If the bilinear form (a,b) on L is nondegenerate then A is the split Cayley algebra or a quaternion algebra. For a degenerate form (a,b), A is a direct sum of its radical and a subalgebra which is either a quaternion or two dimensional separable algebra. As an immediate consequence of the first result we have shown that if the bilinear form on the Lie algebra L is nondegenerate then L is simple with dimension three or one.

Mathematical Subject Classification
Primary: 17.30
Received: 1 November 1968
Published: 1 July 1969
Tae-il Suh