## Gradings on Simple Lie AlgebrasGradings are ubiquitous in the theory of Lie algebras, from the root space decomposition of a complex semisimple Lie algebra relative to a Cartan subalgebra to the beautiful Dempwolff decomposition of $E_8$ as a direct sum of thirty-one Cartan subalgebras. This monograph is a self-contained exposition of the classification of gradings by arbitrary groups on classical simple Lie algebras over algebraically closed fields of characteristic not equal to 2 as well as on some nonclassical simple Lie algebras in positive characteristic. Other important algebras also enter the stage: matrix algebras, the octonions, and the Albert algebra. Most of the presented results are recent and have not yet appeared in book form. This work can be used as a textbook for graduate students or as a reference for researchers in Lie theory and neighboring areas. |

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### Contents

Introduction | 1 |

Gradings on Algebras | 9 |

Associative Algebras | 27 |

Classical Lie Algebras | 63 |

Composition Algebras and Type G2 | 123 |

Jordan Algebras and Type F4 | 163 |

Other Simple Lie Algebras in Characteristic Zero | 207 |

Lie Algebras of Cartan Type in Prime Characteristic | 271 |

Appendix A Affine Group Schemes | 299 |

Appendix B Irreducible Root Systems | 321 |

331 | |

### Common terms and phrases

abelian group abelian group gradings affine group schemes Albert algebra algebra g algebra of type algebraic group algebraically closed field anti-automorphism associative algebra assume Aut(A Aut(g Aut(R Aut(T automorphism group basis bicharacter bilinear form Cartan grading Cartan subalgebra Cayley algebra charIF coarsening commutative composition algebra conjugate Corollary corresponding decomposition deﬁned Deﬁnition denoted Der(A dimension equivalence Example ﬁeld field F ﬁne gradings finite finite-dimensional ﬁrst follows G-graded algebra given group G group schemes hence homogeneous components homogeneous elements homomorphism Hopf algebra Hurwitz algebra ideal identiﬁed involution Jordan algebra Lemma Let G linear map matrix algebra maximal quasitorus module morphism multiplication nondegenerate obtain Okubo algebra orthogonal permutation PROOF Proposition quotient restriction root system satisﬁes scalar semisimple simple Lie algebra span subgroup subgroupscheme subspace symmetric composition algebra Theorem unique universal group vector space Weyl group