Vol. 103, No. 1, 1982

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Compatible Peirce decompositions of Jordan triple systems

Kevin Mor McCrimmon

Vol. 103 (1982), No. 1, 57–102

Jordan triple systems and pairs do not in general possess unit elements, so that certain standard Jordan algebra methods for studying derivations, extensions, and bimodules do not carry over to triples. Unit elements usually arise as a maximal sum of orthogonal idempotents. In Jordan triple systems such orthogonal sums of tripotents are not enough: in order to “cover” the space one must allow families of tripotents which are orthogonal or collinear. We show that well behaved triples and pairs do possess covering systems of mixed tripotents, and that for many purposes such nonorthogonal families serve as an effective substitute for a unit element. In particular, they can be used to reduce the cohomology of a direct sum to the cohomology of the summands.

Mathematical Subject Classification 2000
Primary: 17C10
Secondary: 17C46
Received: 7 March 1980
Revised: 18 June 1981
Published: 1 November 1982
Kevin Mor McCrimmon