The new construction given by Barton and Sudbery of the Freudenthal-Tits magic square, which includes the exceptional classical simple Lie algebras, will be interpreted and extended by using a pair of symmetric composition algebras, instead of the standard unital composition algebras.
The purpose of these survey notes is to give a presentation of a classical theorem of Nomizu [] that relates the invariant affine connections on reductive homogeneous spaces and nonassociative algebras.
In this paper the structure of the maximal elements of the lattice of subalgebras of central simple non-Lie Malcev algebras is considered. Such maximal subalgebras are studied in two ways: first by using theoretical results concerning Malcev algebras, and second by using the close connection between these simple non-Lie Malcev algebras and the Cayley-Dickson algebras, which have been extensively studied (see [4]).
We describe two constructions of a certain -grading on the so-called Brown algebra (a simple structurable algebra of dimension and skew-dimension ) over an algebraically closed field of characteristic different from . The Weyl group of this grading is computed. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types , and .
2010 Mathematics Subject Classification: Primary 17B70, secondary 17B40, 16W50. Given a grading Γ : L ⨁ = g ∈ G L g on a nonassociative algebra L by an abelian group G, we have two subgroups of Aut(L): the automorphisms that stabilize each component L g (as a subspace) and the automorphisms that permute the components. By the Weyl group of Γ we mean the quotient of the latter subgroup by the former. In the case of a Cartan decomposition of a semisimple complex Lie algebra, this is the...
A Lie algebra L is said to be minimal non supersolvable if all its subalgebras, except L itself, are supersolvable.
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