### ${\mathbb{Z}}_{3}$-orthograded quasimonocomposition algebras with one-dimensional null component.

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We prove a series of "going-up" theorems contrasting the structure of semiprime algebras and their subalgebras of invariants under the actions of Lie color algebras.

Attempts to extend our previous work using the octonions to describe fundamental particles lead naturally to the consideration of a particular real, noncompact form of the exceptional Lie group ${E}_{6}$, and of its subgroups. We are therefore led to a description of ${E}_{6}$ in terms of $3\times 3$ octonionic matrices, generalizing previous results in the $2\times 2$ case. Our treatment naturally includes a description of several important subgroups of ${E}_{6}$, notably ${G}_{2}$, ${F}_{4}$, and (the double cover of) $SO(9,1)$. An interpretation of the actions...

Under some conditions we prove that every generalized Jordan triple derivation on a Lie triple system is a generalized derivation. Specially, we conclude that every Jordan triple $\theta $-derivation on a Lie triple system is a $\theta $-derivation.

We investigate the class of finite-dimensional real flexible division algebras. We classify the commutative division algebras, completing an approach by Althoen and Kugler. We solve the isomorphism problem for scalar isotopes of quadratic division algebras, and classify the generalised pseudo-octonion algebras. In view of earlier results by Benkart, Britten and Osborn and Cuenca Mira et al., this reduces the problem of classifying the real flexible division algebras to the normal...

This is an extended version of a talk presented by the second author on the Third Mile High Conference on Nonassociative Mathematics (August 2013, Denver, CO). The purpose of this paper is twofold. First, we would like to review the technique developed in a series of papers for various classes of di-algebras and show how the same ideas work for tri-algebras. Second, we present a general approach to the definition of pre- and post-algebras which turns out to be equivalent to the construction of dendriform...