Página principal > Tesis > Gradings on simple exceptional Jordan systems and structurable algebras |

Aranda Orna, Diego

Elduque Palomo, Alberto Carlos (dir.)

Universidad de Zaragoza,
2017

(Matemáticas)

Gradings on these structures can be used to construct gradings on the simple exceptional Lie algebras of types E6 , E7 and E8, via the Kantor construction.

In this work, by grading we usually mean group grading. The base ﬁeld is assumed to be algebraically closed of characteristic diﬀerent from 2. Simple Lie algebras over the ﬁeld of complex numbers C were classiﬁed

by W. Killing and E. Cartan in 1894, and Cartan decompositions (i.e., gradings by the associated root system) were used in the proof. After much work on some particular gradings, a systematic study of gradings was initiated by Patera and Zassenhaus in the context of Lie algebras, as a generalization of Cartan decompositions. Since then, gradings have been studied by many authors, and are an important tool for a better understanding of algebras and other structures. It is well-known that, if the base ﬁeld is algebraically closed of characteristic 0, ﬁne gradings by abelian groups are in bijective correspondence with maximal abelian diagonalizable subgroups of the automorphism group and, in that correspondence, equivalence classes of gradings correspond to conjugation classes of subgroups. Note that the same correspondence holds for any characteristic if we consider the automorphism group-scheme instead of the automorphism group. Therefore, the problem of classiﬁcation of ﬁne gradings by abelian groups on an algebra (and other algebraic structures) can be reformulated as an equivalent problem in group theory.

In Chapter 1 we recall the basic deﬁnitions of the algebraic structures that we study in this work. We also recall some well-known results of classiﬁcations of ﬁne gradings on Cayley and Albert algebras, which constitute an extremely important tool to construct gradings in other structures, as shown in further sections. This chapter has no original results of the author. The basic deﬁnitions of gradings (by abelian groups) on Jordan systems are given in Chapter 2, and hereinafter some general results are proven. We recall the deﬁnition of the exceptional simple Jordan pairs and triple systems, i.e., the bi-Cayley and Albert pairs and triple systems. Some results related to the orbits and automorphism groups of the bi-Cayley pair and triple system are given too. In Chapters 3 and 4 we obtain classiﬁcations of the equivalence classes of ﬁne gradings (by abelian groups) on the exceptional simple Jordan pairs and triple systems (over an algebraically closed ﬁeld of characteristic not 2). The associated Weyl group is determined for each grading. We also study the induced ﬁne gradings on e6 and e7 via the TKK-construction. Finally, in Chapter 5 we give a construction of a Z^3_4-grading on the Brown algebra. We then compute the Weyl group of this grading and study how this grading can be used to construct some very special ﬁne gradings on the Lie algebras of types E6, E7 and E8.