A structure theory of Freudenthal-Kantor triple systems

On δ-Lie supertriple systems associated with (ε, δ)-Freudenthal-Kantor supertriple systems

“…or an anti-Lie supertriple system (the case ofd = -l) with respect to the triple product defined by 6K{oi,Cj) e ((-irL(d j ,a l )-6L(b i ,c j ) • From this Proposition, we can obtain the 6-Lie supertriple system U(e, 6) ® U(e, 8) associated with U(e,6) and denote it by T(5) (for special cases of degree 0 and 6 = 1, see [8] or [11]). …”

“…These systems are actually special cases of more general S-L'ie triple systems and Preudenthal-Kantor triple systems denned by certain identities among triple products (cf. [8,12,13,21]). …”

“…Definition 2.1. For e = ±1, 5 = ±1, a vector space U(e, 8) = Si®Ui(e, 5) over a field with a triple product (-, -, -) is said to be a (e, <5)-Freudenthal-Kantor supertriple system if [10]). We note that the following identities are equivalent:…”

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

“…We can generalize the concept of the GJTS of second order as follows (see [13,14,17,23,50] and the earlier references therein). For ε = ±1 and δ = ±1, a triple product that satisfies the identities…”

“…Hence within the general framework of ( , δ)-FKTSs ( , δ = ±1) and the standard embedding Lie (super)algebra construction studied in [6,7,[13][14][15]28] (see also references therein) we define δ-structurable algebras as a class of nonassociative algebras with involution which coincides with the class of structurable algebras for δ = 1 as introduced and studied in [1,2]. Structurable algebras are a class of nonassociative algebras with involution that include Jordan algebras (with trivial involution), associative algebras with involution, and alternative algebras with involution.…”