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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 $E_{8}$ .

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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

A $ℤ_{4}^{3}$ -grading on a $56$ -dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type $E$