A Structure Theory of (−1,−1)-Freudenthal Kantor Triple Systems

Abstract

In this paper we discuss the simplicity criteria of (−1, −1)-Freudenthal Kantor triple systems and give examples of such triple systems, from which we can construct some Lie superalgebras. We also show that we can associate a Jordan triple system to any (ε, δ)-Freudenthal Kantor triple system. Further, we introduce the notion of δ-structurable algebras and connect them to (−1, δ)-Freudenthal Kantor triple systems and the corresponding Lie (super)algebra construction.2000 Mathematics subject classification: primary 17A40, 17B60.

Symmetry of Lie algebras associated with (ε, δ)-Freudenthal-Kantor triple system

“…[14]). Let (V, (xyz)) be a (ε, δ)-FKTS with an endomorphism P such that P 2 = −εδ Id and P (xyz) = (P xP yP z).…”

“…Any unitary (ε, ε)-FKTS is special [9,14]. Moreover, (h, g, f ) constructed above are now contained in L(W, W ) by replacing K(x, y) in Remark 2.3 by i K(a i , b i ) = Id.…”

“…the triple product, Also, following [11], we note that if the product of Table 1 in [11] is defined by abc = (cā)b, then the fourth family with q = 0 (i.e. abc − acb + bca) reduces to (cā)b − (cb)a + (ab)c, which is called an anti-structurable algebra of a special case in (−1, −1)-FKTSs [14]. Hence, these concepts are closely related to that of structurable algebras, but the details will be discussed in another paper.…”

Examples of Freudenthal-Kantor Triple Systems

“…It seems that the concept of a triple system (or called a ternary algebra) in nonassociative algebras started from the metasymplectic geometry due to Freudenthal. After a generalization of the concept has been studied by Tits, Koecher, Kantor, Yamaguti, Allison and authors ( [1,2] for many earlier references on the subject). Also it is well known the object of investigation of Jordan and Lie algebras with application to symmetric spaces or domains [3] and to physics [4,5].…”

“…These imply that we are considering to structure of the subspase L 1 of the five graded Lie (super)algebra L (ϵ,δ) = L -2 ⊕ L -1 ⊕ L 0 ⊕ L 1 ⊕ L 2 satisfying [L i , L j ] ⊆ L i+j ,, associated with an (ϵ,δ) Freudenthal-Kantor triple system which contains a class of Jordan triple systems related 3 graded Lie algebra L -1 ⊕ L 0 ⊕ L 1 . For these consideretions without untilizing properties of root systems or Cartan matrices, we would like to refer to the articles of the present author and earlier references quoted therein [1,2,10,11].…”