A new class of nonassociative algebras with involution

Abstract

This article is devoted to introduce a new class of nonassociative algebras with involution including the class of structurable algebras.

On geometric structures associated with triple systems

“…We can also generalize the concept of the GJTS of 2nd order as follows (for examples, see [8], [9], [11], [16] and references therein).…”

“…Note that the concept of a GJTS of 2nd order coincides with that of a (−1, 1)-F-K.t.s. Thus we can construct simple Lie algebras (δ = 1)or superalgebras ( δ = −1) from these triple systems with δ = ±1, by means of the standard embedding method (for example, [2], [3], [8]- [12], [16], [17], [18], [23]). …”

“…Following [16], we exhibit examples of a (-1,-1)-F-K.t.s with a complex structure, is known as an antistructurable algebra.…”

“…Our aim is to use triple systems to investigate the characterization of differential geometry and mathematical physics from the viewpoint of nonassociative algebras that contain a class of Lie algebras or Jordan algebras ( [6], [7], [8], [13], [16], [17], [18]). In this paper, we will study about the Lie superalgebras or Lie algebras associated with triple systems, and the relationship with several vector subspaces associated with these algebras.…”

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

“…If (A, − ) is δ-structurable then A is a Jordan algebra, by [22]. Conversely, by [36,Section 3], any Jordan algebra satisfies (3.8) if V x,y = + V x,y for x, y ∈ A, hence it is structurable.…”