Resumen de Gradings on a family of simple structurable algebras

Alejandra Sarina Córdova Martínez

• Gradings by abelian groups on finite-dimensional simple Lie algebras are ubiquitous, starting with the root space decomposition with respect to a Cartan subalgebra in a split simple Lie algebra.

  The main goal of this thesis is the classi fication of gradings (by groups) on one of the families of simple structurable algebras: the tensor product of a Cayley algebra and a Hurwitz algebra with the involution being the tensor product of the standard involutions.

  In the process of obtaining the gradings on the tensor product we found that the problem could be reduced to the problem of finding gradings on the cartesian product. Of course this is not a simple algebra, but a semisimple one, and not much work has been done on gradings on such algebras. With the classi fication of gradings on semisimple algebras (direct products of simple finite-dimensional algebras) at hand, we could fi nally complete the sought classi fication of gradings on the structurable algebras.