Algebras Formed by the Zorn Vector Matrix

Document Type

Article

Publication Date

1-1-1969

Description

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 having the property: [a[bc]] = (a, c)b — (a, b)c, a, b, c∈ L. We determine all the alternative algebras U obtained in this way: If the bilinear form (a, b) on L is nondegenerate then U is the split Cayley algebra or a quaternion algebra. For a degenerate form (a, b), U is a direct sum of its radical and a subalgebra which is either a quaternion ortwo 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.

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