Alberto Elduque

Gradings on simple Lie algebras

Introduction Gradings on algebras Associative algebras Classical Lie algebras Composition algebras and type

The Magic Square and Symmetric Compositions

G_2

Gradings on the exceptional Lie algebras

Jordan algebras and type

F_4

F_4

and

Other simple Lie algebras in characteristic zero Lie algebras of Cartan type in prime characteristic Affine group schemes Irreducible root systems Bibliography Index of Notation Index

G_2

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.

revisited

All gradings by abelian groups are classified on the following al- gebras over an algebraically closed field F: the simple Lie algebra of type G2 (charF 6 2,3), the exceptional simple Jordan algebra (charF 6 2), and the simple Lie algebra of type F4 (charF 6 2).

On maximal subalgebras of central simple Malcev algebras

Let F be a field of characteristic different from two and A a central sim- ple non-Lie Malcev algebra over F. Our purpose is to determine the maximal subalgebras of A (with respect to inclusion). Maximal subalgebras of simple Lie algebras over an algebraically closed field have been studied by Dynkin, [3, 41, and maximal subalgebras of central simple associative, alternative and special Jordan algebras have been studied by Racine [ 121. Racine’s results on octonion algebras will be relevant in our investigation. 1.

On triality and automorphisms and derivations of composition algebras

Abstract The Principle of Triality for orthogonal transformations on eight-dimensional spaces is used to obtain different expressions for the automorphisms, derivations and orthogonal transformations of Cayley–Dickson algebras and symmetric composition algebras over fields of arbitrary characteristic.

Mutations of Alternative Algebras

The primary concern in this chapter is to develop a structure theory for mutations of alternative algebras which is parallel to the structure theory for mutations of associative algebras. The framework which organizes this chapter is therefore very similar to that of Chapter II. Since a simple alternative algebra is either associative or a Cayley-Dickson algebra over its center (Theorem I.3.17), this chapter is in large devoted to the structure of mutations of Cayley-Dickson algebras.