On the Cartan–Jacobson Theorem

✍ Scribed by Pablo Alberca Bjerregaard; Alberto Elduque; Cándido Martı́n González; Francisco José Navarro Márquez

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 108 KB
• Volume: 250
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.2001.9084

✦ Synopsis

The well-known Cartan-Jacobson theorem claims that the Lie algebra of derivations of a Cayley algebra is central simple if the characteristic is not 2 or 3. In this paper we have studied these two cases, with the following results: if the characteristic is 2, the theorem is also true, but, if the characteristic is 3, the derivation algebra is not simple. We have also proved that in this last case, there is a unique nonzero proper seven-dimensional ideal, which is a central simple Lie algebra of type A 2 , and the quotient of the derivation algebra modulo this ideal turns out to be isomorphic, as a Lie algebra, to the ideal itself. The original motivation of this work was a series of computer-aided calculations which proved the simplicity of derivation algebras of Cayley algebras in the case of characteristic not 3. These computations also proved the existence of a unique nonzero proper ideal