Models of Compact Simple Kantor Triple Systems Defined on a Class of Structurable Algebras of Skew-Dimension One

On compact realifications of exceptional simple Kantor triple systems

“…In this paper we continue the work on compact simple Kantor triple systems of [5] and [20,21,22] giving, by a unified formula (Theorem 1), the classification of exceptional compact simple Kantor triple systems defined on the realification of the 2 × 2-matrix algebra determined by Jordan algebra J = H 3 (A C ) of hermitian 3 × 3-matrices over a complex composition algebra A C corresponding to realifications of complex exceptional simple Lie algebras (Theorem 2). In addition, we give an explicit formula for the quadratic canonical trace form for these Kantor triple systems (Corollary 1).…”

“…The results presented here are a continuation of [20] where models of exceptional compact simple Kantor triple systems defined on the 2 × 2-matrix algebra determined by Jordan algebra J = H 3 (A) of hermitian 3 × 3-matrices over a real composition algebra A have been given. Related results are those of [10] where a construction of exceptional simple 5-graded Lie algebras U = ⊕ 2 l=−2 U l and an explicit realization of the subspaces U l have been given by different methods.…”

“…By [15] (Theorem 3.14 and §4.1), in order to classify all compact simple KTS's we have to find one such model for each 5-grading of each real simple Lie algebra. Moreover, by [16] (Theorem 3.3, Table I), all 5-gradings ⊕ 2 l=−2 K l of realifications of complex exceptional simple Lie algebras are such that (dim (20,5), (20,1), (16,8), (32, 10), (32, 1), (35, 7), (56, 1), (64, 14), (14, 1), (8, 7), (4, 1)}.…”

“…We give now a close related structure to the one of the previous chapter which leads to models of compact KTS's such that the corresponding Kantor algebra is the real exceptional simple Lie algebra G C 2 R and moreover the real split G 2 . The approach is closer related to the models of compact KTS's defined in [20] and the presentation of [14], by defining the KTS's on a structurable algebra of skew-dimension one (over R or C), i.e. KTS's defined on a 2 × 2-matrix algebra, than the presentation of (complex) KTS's defined on symmetric tensors of [18].…”

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

“…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. They are related to GJTSs of second order (or (−1, 1)-FKTSs) as introduced and studied in [31,32] and further studied in [3,4,30,[39][40][41][42]45] (see also references therein). Their importance lies with constructions of five graded Lie algebras…”