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A ℤ 4 3 -grading on a 56 -dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type E

Diego Aranda-Orna;  Alberto Elduque;  Mikhail Kochetov — 2014

Commentationes Mathematicae Universitatis Carolinae

We describe two constructions of a certain ℤ 4 3 -grading on the so-called Brown algebra (a simple structurable algebra of dimension 56 and skew-dimension 1 ) over an algebraically closed field of characteristic different from 2 . 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 E 6 , E 7 and E 8 .

Weyl Groups of Fine Gradings on Simple Lie Algebras of Types A, B, C and D

Elduque, Alberto;  Kochetov, Mikhail — 2012

Serdica Mathematical Journal

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...

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