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A Peirce decomposition for generalized Jordan triple systems of second order. (English) Zbl 1039.17032

Generalized Jordan triple systems (GJTSs) arise naturally as coordinates for graded Lie algebras of order \(k\). A GJTS is of first-order iff it is a Jordan triple system (JTS). Similarly, GJTSs of second order can be characterized in terms of identities.
In the present paper the authors prove that every tripotent \(e\) of a GJTS of second-order defines a decomposition of the space of the triple into a direct sum of eight components. This decomposition is a generalization of the Peirce decomposition for JTSs. The relation between components are studied in the case when \(e\) is a left unit (several examples of the Peirce decomposition in GJTSs are provided in Section 2).
Since Peirce decomposition is one of the main tools in Loos’ study of the structure of JTSs and Jordan pairs, it could be also useful for GJTSs of second order as well. Moreover, the connection between Jordan systems and symmetric domains was used by Loos to study the behavior of geodesics, rank and boundary in symmetric domains by means of Peirce decompositions. The same type of questions could be considered for the bisymmetric domains associated to GJTSs of second order.

MSC:

17C27 Idempotents, Peirce decompositions
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