Gradings on Simple Lie Algebras

Home , Okubo algebra, Simple Lie algebra

Mathematical Surveys and Monographs Volume 189

Alberto Elduque Mikhail Kochetov

American Mathematical Society Atlantic Association for Research in the Mathematical Sciences http://dx.doi.org/10.1090/surv/189

Gradings on Simple Lie Algebras

Mathematical Surveys and Monographs Volume 189

Gradings on Simple Lie Algebras

Alberto Elduque Mikhail Kochetov

American Mathematical Society Providence, RI

Atlantic Association for Research in the Mathematical Sciences Halifax, Nova Scotia, Canada Editorial Committee of Mathematical Surveys and Monographs Ralph L. Cohen, Chair Robert Guralnick Benjamin Sudakov MichaelA.Singer Michael I. Weinstein

Editorial Board of the Atlantic Association for Research in the Mathematical Sciences Jeannette Janssen, Director David Langstroth, Managing Editor Yuri Bahturin Theodore Kolokolnikov Robert Dawson Lin Wang

2010 Mathematics Subject Classiﬁcation. Primary 17B70; Secondary 17B60, 16W50, 17A75, 17C50.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-189

Library of Congress Cataloging-in-Publication Data Elduque, Alberto. Gradings on simple Lie algebras / Alberto Elduque, Mikhail Kochetov. pages cm. — (Mathematical surveys and monographs ; volume 189) Includes bibliographical references and index. ISBN 978-0-8218-9846-8 (alk. paper) 1. Lie algebras 2. Rings (Algebra) 3. Jordan algebras. I. Kochetov, Mikhail, 1977– II. Title. QA252.3.E43 2013 512.482—dc23 2013007217

Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 181716151413 To Pili, a mathematician, and to Eva, a mathematician to be. (A.E.) To the memory of my parents. (M.K.)

Contents

List of Figures ix Preface xi Conventions and Dependence among Chapters xiii Introduction 1 Chapter 1. Gradings on Algebras 9 1.1. General gradings and group gradings 9 1.2. The universal group of a grading 15 1.3. Fine gradings 16 1.4. Duality between gradings and actions 19 1.5. Exercises 25 Chapter 2. Associative Algebras 27 2.1. Graded simple algebras with minimality condition 28 2.2. Graded division algebras over algebraically closed ﬁelds 33 2.3. Classiﬁcation of gradings on matrix algebras 38 2.4. Anti-automorphisms and involutions of graded matrix algebras 49 2.5. Exercises 60 Chapter 3. Classical Lie Algebras 63 3.1. Classical Lie algebras and their automorphism group schemes 64 3.2. ϕ-Gradings on matrix algebras 85 3.3. Type A 105 3.4. Type B 116 3.5. Type C 118 3.6. Type D 119 3.7. Exercises 121

Chapter 4. Composition Algebras and Type G2 123 4.1. Hurwitz algebras 123 4.2. Gradings on Cayley algebras 130 F F 4.3. Gradings on psl3( ), char = 3 137 4.4. Derivations of Cayley algebras and simple Lie algebras of type G2 140 4.5. Gradings on the simple Lie algebras of type G2 146 4.6. Symmetric composition algebras 149 4.7. Exercises 160

Chapter 5. Jordan Algebras and Type F4 163 5.1. The Albert algebra 164

vii viii CONTENTS

5.2. Construction of fine gradings on the Albert algebra 169 5.3. Weyl groups of fine gradings 178 5.4. Classification of gradings on the Albert algebra 184 5.5. Gradings on the simple Lie algebra of type F4 190 5.6. Gradings on simple special Jordan algebras 197 5.7. Exercises 206

Chapter 6. Other Simple Lie Algebras in Characteristic Zero 207 6.1. Fine gradings on the simple Lie algebra of type D4 207 6.2. Freudenthal’s Magic Square 224 6.3. Some nice gradings on the exceptional simple Lie algebras 239 6.4. Fine gradings on the simple Lie algebra of type E6 244 6.5. Fine gradings and gradings by root systems 259 6.6. Summary of known fine gradings for types E6, E7 and E8 265 6.7. Exercises 269 Chapter 7. Lie Algebras of Cartan Type in Prime Characteristic 271 7.1. Restricted Lie algebras 271 7.2. Construction of Cartan type Lie algebras 273 7.3. Automorphism group schemes 276 7.4. Gradings 287 7.5. Exercises 297 Appendix A. Affine Group Schemes 299 A.1. Affine group schemes and commutative Hopf algebras 299 A.2. Morphisms of group schemes 305 A.3. Linear representations 307 A.4. Affine algebraic groups 310 A.5. Infinitesimal theory 314 Appendix B. Irreducible Root Systems 321 Bibliography 323 Index of Notation 331 Index 333 List of Figures

2.1 Gradings, up to equivalence, on M2(F)whereF is an algebraically closed ﬁeld, char F =2. 44

2.2 Gradings, up to equivalence, on M3(F)whereF is an algebraically closed ﬁeld, char F =3. 46

4.1 Multiplication table of the split Cayley algebra 129 4.2 Gradings on the Cayley algebra over an algebraically closed ﬁeld of characteristic diﬀerent from 2 134

4.3 Gradings, up to equivalence, on the simple Lie algebra of type G2 over an algebraically closed ﬁeld 149 4.4 Multiplication table of the split Okubo algebra 151

6.1 Dynkin diagram of D4 211 6.2 Fine gradings on the E-series. 268

Preface

The aim of this book is to introduce the reader to the theory of gradings on Lie algebras, with a focus on the classification of gradings on simple finite-dimensional Lie algebras over algebraically closed fields. The classic example of such a grading is the Cartan decomposition with respect to a Cartan subalgebra in characteristic zero, which is a grading by a free abelian group. Since the 1960’s, there has been much work on gradings by other groups, starting with finite cyclic groups, and applications of such gradings to the theory of Lie algebras and their representations. We do not attempt to give a comprehensive survey of these results but rather to present a self-contained exposition of the classification of gradings on classical simple Lie algebras in characteristic different from 2 and on some non-classical simple Lie algebras in prime characteristic greater than 3. Other important algebras also enter the stage: matrix algebras, the octonions and the simple exceptional Jordan algebra. Most of the classification results presented here are recent and have not yet appeared in book form. This work started with the notes of two courses that the authors gave for the Atlantic Algebra Centre at Memorial University of Newfoundland: “Introduction to affine group schemes” (M. Kochetov, November–December 2008) and “Compo- sition algebras and their gradings” (A. Elduque, May 2009). Affine group schemes are an important tool for the study of gradings on finite-dimensional algebras in arbitrary characteristic, as we explain in Chapter 1. We give a brief exposition of the background on affine group schemes in Appendix A, with references to the literature on this subject. A reader who is interested exclusively in the case of characteristic zero will only need affine algebraic groups (in the “na¨ıve” sense) to follow this book. Apart from this, we assume that the reader is familiar with linear algebra and with the basics on groups and algebras. The book is intended for specialists in Lie theory but may also serve as a textbook for graduate students (in conjunction with an introductory textbook on Lie algebras). In every chapter, at the beginning, we give a brief description of its main results and references to original works; at the end, we give a list of exercises on the covered material. This book would not have been written without the constant support, advice and encouragement of Yuri Bahturin, who himself greatly contributed to the study of gradings by arbitrary groups. It was his enthusiasm that convinced the authors to join efforts in the task of collecting, understanding, unifying and expanding the knowledge about gradings on simple Lie algebras. The second author would also like to use this opportunity to express his gratitude for all the help in his life and career given so generously by Professor Bahturin since becoming his thesis advisor a decade and a half ago.

xi xii PREFACE

The authors have benefited from discussions with many colleagues. Among them, our special thanks are due to Cristina Draper, who explained her results on gradings on exceptional simple Lie algebras long before they were publicly available. The first author acknowledges the support of the former Spanish Ministerio de Ciencia e Innovación—Fondo Europeo de Desarrollo Regional (FEDER)1 and of the Diputación General de Aragón—Fondo Social Europeo (Grupo de Investigación de Algebra).´ He would also like to thank Memorial University for hospitality during his visits to Newfoundland. The second author acknowledges the support of the Natural Sciences and Engi- neering Research Council (NSERC)2 of Canada and the hospitality of the University of Zaragoza during his visits to Spain. Both authors acknowledge the support of the Atlantic Association for Research in the Mathematical Sciences (AARMS) of Canada in the preparation of this book.

Alberto Elduque and Mikhail Kochetov Zaragoza, Spain February 2013

1MTM2010-18370-C04-02 2Discovery Grant # 341792-07 Conventions and Dependence among Chapters

The symbols Z, Q, R and C will denote, respectively, the integers, rationals, reals and complex numbers. The set of integers modulo m will be denoted by Zm, with individual elements written as numbers with a bar (0,¯ 1,¯ etc.) Unless indicated otherwise, vector spaces, dimensions, linear maps, algebras, tensor products, etc. will be understood over a ground field F. The assumptions on F will vary from section to section and will be stated explicitly. In particular, the characteristic of F will be written as char F. In most cases, we will use italic capitals (U, V , W , etc.) to denote sets and vector spaces, and calligraphic capitals (A, B, C, etc.) to denote algebras. Direct sums of vector spaces will be written as ⊕,and tensor products as ⊗. The trace and determinant of a matrix or an endomorphism will be denoted by tr and det, respectively. An endomorphism whose minimal polynomial has no multiple roots will be called semisimple or (if the ground field is algebraically closed) diagonalizable. Cyclic groups will often be written as Z or Zm. The symmetric group on n symbols will be denoted by Sym(n). Direct and semidirect product of groups will be written as × and , respectively. The stabilizer of an object x under an action of a group G will be denoted by StabG(x), with StabG(x, y) meaning StabG(x)∩StabG(y), etc. In the special case of G acting on itself or its power set by conjugation, we will use CG(x) (centralizer) and NG(x) (normalizer), respectively. Thus, for X ⊂ G,wehave: −1 CG(X):={g ∈ G | gx = xg ∀x ∈ X} and NG(X):={g ∈ G | gXg = X}. The center of G will be denoted by Z(G). The same notation for centralizers, normalizers and center will also be used for Lie algebras. We will use standard notation for classical groups: GLn(F)orGL(V )forthe general linear group and similarly SL for the special linear group, O for the orthogonal group (with respect to a nondegenerate quadratic form), SO for the special orthogonal group, and Sp for the symplectic group (with respect to a nondegenerate symplectic form). If F is finite, its symbol may be replaced by the order: for example, we will write GL3(2) for GL3(F)whereF is the field of two elements. The multiplicative group of F will be denoted by F×. Throughout the book, more notation will be introduced, especially for gradings on various algebras. As a general rule, a grading on an algebra will be denoted by Γ with a subscript indicating the algebra or its type. We have made an effort to collect all such symbols in a separate notation index at the end of the book. The terminology and basic constructions concerning gradings are introduced in Chapter 1. They will be used throughout the book. The chapters depend on each other (in addition to Chapter 1) as follows: Chapter 3 depends on Chapter 2; Chapter 5 depends on Chapter 4 and, to some extent, Chapters 2 and 3; Chapter 6 depends on all preceding chapters. xiii

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Index of Notation

A(+) (II) , 3, 197 ΓA (G, H, h, β, κ, γ, μ0, g0), 107 A(−),2 (I) ΓA (T,k), 109 Ad, 317 Γ(II)(T,q,s,τ), 110 ad, 317 A ΓBF (G, κ, γ), 199 AlgF, 299 Γ (m, ), 199 Aut(U), 301, 309 BF Γ (G, κ, γ), 117 Aut (A), 14 B G Γ+(G, κ, γ), 204 Aut(Γ), 14 B Γ (q, s), 117 Aut Ξ(κ, γ), 46 B Γ+(q, s), 204 Aut Φ, 65 B 1 2 ΓC,ΓC, 136 AutX (O), X ∈{S, H, K}, 277 1 2 ΓC(G, γ), ΓC(G, H), 136 βσ,35 ΓC (G, T, β, κ, γ, g0), 118 + ΓC (G, T, β, κ, γ, g0), 205 K CD( ,β,γ), 127 ΓC (T,q,s,τ), 119 CD(Q,α), 126 Γ+(T,q,s,τ), 205 C C η, 209 ΓD(G, T, β, κ, γ, g0), 120 C + Cl( ,n), 167 Γ (G, T, β, κ, γ, g0), 204 C D s, 129 ΓD(T,q,s,τ), 120 Γ+ (T,q,s,τ), 205 deg ,9 D Γ Γ1 , Γ2 , Γ3 , Γ4 , 196 Der(A), 4 F4 F4 F4 F4 Γ1 (G, γ), Γ2 (G, H, γ), Γ3 (G, H, g), Diag(Γ), 14 F4 F4 F4 Γ4 (G, H, δ), 196 Diag(Γ), 23 F4 Γ1 ,Γ2 , 146 Dx,y, 225 G2 G2 1 2 dx,y, 140 Γ (G, γ), Γ (G, H), 146 G2 G2 Γ1 , 265 gr H3(Q) EndR (V ), 28 ΓK, 265 F[G], 300 ΓK¯ , 266 D FσT ,34 ΓM ( ,k), ΓM (T,k), 44 ΓM (D,q,s,τ), 88 D G(A), 302 ΓM (G, ,κ,γ), ΓM (G, T, β, κ, γ), 40 Γ1 ,Γ2 , 266 Ga, 300 M3(F) M3(F) α Γ, 16 Γ(I) (G, T, β, κ, γ), 202 1 M + ΓA(G, γ), 188 (II) 1 Γ + (G, H, h, β, κ, γ, μ0, g0), 203 ΓA, 170 M 2 ΓO, 266 ΓA(G, H, γ), 188 2 ΓO(G, P, γ), 291 ΓA, 171 O 3 Γ (s), 292 ΓA(G, H, g), 188 Γ1 ,Γ2 , 265 3 Q Q ΓA, 172 ΓS (G, P, γ, g0), 294 Γ4 (G, H, δ), 189 A ΓS (s), 294 Γ4 , 173 A ΓW (G, P, γ), 292 (I) ΓA (G, T, β, κ, γ), 105 ΓW (s), 292

331 332 INDEX OF NOTATION

1 ΓX(D), 266 tJ, 225 2 2 T (ν), 214 ΓX(K),ΓX(Q), 266 3 ΓX(Q), 267 U(Γ), 15 ˜ J ΓX(D), 267 (U( ),ι), 200 GLn, 300 G , 300 V, 319 m [g] [g] G , 313 V, V ,10 red D Grp, 300 V (G, ,κ,γ), 33 S S g( , ), 230 W (Γ), 15 H(A,ϕ), 3 W (m; n), 274 H A ∗ ( , ), 197 X(G), 305 H(m; n), 275 Ξ(γ), 291 G Hom (V,W), 10 Ξ(κ, γ), 39 Homgr(V,W), 10 gr (m;n) HomR (V,W), 28 Z , 273

Int(L), 66 IDer(L), 4 IDer(C), 225 IDer(J), 225

J(V,b), 197

K(A,ϕ), 3 K(m; n), 276

Lie(G), 315 L(J), 229 L¯(J), 229

M(D,k), 41 M(D,k)ab,43 M(D,q,s,τ), ΓM (T,q,s,τ), 88 M(G, D,κ,γ), 33 M(G, D,κ,γ,δ,g0), 59 M(G, D,κ,γ,μ,g0), 57 M D (G, ,κ,γ,μ0, g0), 87 ModG,10 G G R Mod , ModR ,28 μn, 306

ωS, ωH , ωK , 275 O(m; n), 273

Prim(A), 302

Set, 299 sgn(ϕ), 58 sgn(Si), 56 Σ(τ), 93 σx,y, 191 S(m; n), 275 Spin(C,n), 168 Stab(Γ), 14 StabG, 308 Supp Γ, 9

T(C, J), 225 θη, 209 Index

χ-Admissible data, 106 Jordan, 163 δ-Admissible data, 59 degree of, 166 H-admissible grading, 289 exceptional, 163 K-admissible grading, 289 generic minimal polynomial of, 166 S-admissible grading, 289 Lie multiplication algebra of, 229 Affine algebraic group, 312 normalized trace of, 225 connected components of, 314 of a bilinear form, 197 Affine algebraic variety, 312 semisimple, 166 Affine group scheme, 300 special, 163 abelian, 300 unital special universal envelope of, algebraic, 300 200 characters of, 305 Lie, 2 diagonalizable, 309 Malcev, 139 diagonalizable representations of, 309 octonion, 128 dimension of, 300 Okubo, 150 distribution algebra of, 319 para-Hurwitz, 150 finite, 300 Petersson, 150 points of, 300 quaternion, 128 representations of, 307 reduced, 311 smooth, 313 structurable, 266 tangent Lie algebra of, 315 Anti-automorphism of a graded algebra, 49 Algebra Antipode, 302 Albert, 163 Augmentation ideal, 304 alternative, 124 Automorphism group scheme, 301, 309 associator, 125 Cayley, 128 Bialgebra, 302 good basis, 129 finite dual, 304 split, 129 Bicharacter central simple, 70 alternating, 35 Clifford, 167 nondegenerate, 35 composition, 124 Biideal, 303 para-Cayley, 167 para-unit of, 150 Cartan decomposition, 65, 69 related triple, 167 Cartan subalgebra, 65 symmetric, 150 Cartier dual, 303 triality Lie algebra, 190 Chevalley basis, 67 G-graded, 1 Chevalley groups, 70 graded division, 29 Closed imbedding, 305 graded simple, 29 Coaction, 308 Hurwitz, 124 Coalgebra, 301 Cayley–Dickson doubling process, 126 cocommutative, 301 isotropic, 128 Coideal, 301 standard conjugation of, 124 Comodule, 308 trace of, 125 Comorphism, 305

333 334 INDEX

χ-Compatible pair, 81 Gradings Comultiplication, 301 anti-equivalence of, 81 Contact algebra (Cartan type), 276 anti-isomorphism of, 81 Counit, 301 equivalence of, 14 isomorphism of, 14 Dempwolff decomposition, 244 weak isomorphism of, 16 Derivation, 4 ϕ-Gradings, 79 inner, 140 equivalence of, 79 Dual isomorphism of, 79 basis, 52 weak equivalence of, 83 of a module, 52 Group grading, 11 Group-like element, 302 Equivalent gradings, 14 Groups of central type, 35 Fine grading, 18 Hamiltonian algebra (Cartan type), 275 First Tits Construction, 176 Homogeneous Freudenthal’s Magic Square, 228 component, 9 Functor element, 9 change-of-group, 16 map, 10 Lie, 315 Homomorphism representable, 299 of affine algebraic groups, 312 Generalized Pauli matrices, 2 of bialgebras, 303 Graded of coalgebras, 301 algebra, by a (semi)group, 1 of comodules, 308 algebra, general, 11 of graded algebras, 14 bimodule, 28 of graded modules, 28 Density Theorem, 29 of graded spaces, 10 map, 10 of Hopf algebras, 303 module, 28 Hopf algebra, 302 Schur’s Lemma, 29 Hopf ideal, 303 subspace, 10 Hopf subalgebra, 303 vector space, 9 Involution Grading of a graded algebra, 49 automorphism group of, 14 orthogonal, 4 coarsening of, 18 symplectic, 4 diagonal group of, 14 Isomorphic gradings, 14 elementary, 38, 288 fine, 18 Kaplansky’s trick, 127 induced by a homomorphism of groups, Killing form, 65 16 Jordan, 244 Lie algebra, 2 nontrivial, 9 abelian, 5 of Type I and Type II, 80 derived algebra of, 5 on a vector space, 9 metabelian, 5 adapted to a bilinear form, 198 nilpotent, 5 on an algebra, by a (semi)group, 1, 11 radical of, 6 on an algebra, general, 11 semisimple, 5 on Hom(V,W), 10 solvable, 5 on tensor product, 11 adjoint representation of, 5 realization of, 11 Cartan type, 274 refinement of, 18 center of, 5 shift of, 10 classical, 66, 70 stabilizer of, 14 split, 70 support of, 9 direct sum (product), 4 toral, 21 inner automorphisms of, 66 type of, 14 inner derivations of, 4 universal group of, 15 reductive, 218 Weyl group of, 15 representations of, 5 INDEX 335

restricted, 272 Standard realization of a division grading, root graded, 259 39 coordinate algebra, 260 Subbialgebra, 303 grading subalgebra, 259 Subcoalgebra, 301 semidirect sum (product), 4 Subcomodule, 308 symmetric pair, 248 Subgroupscheme, 304 inverse image of, 306 MAD subgroups, 21 normal, 307 Module Support, 9 graded, 28 Symplectic triple system, 251 graded irreducible, 29 graded simple, 29 Theorem over graded division algebra, 29 abelian gradings on matrix algebras Morphism of affine group schemes anti-automorphisms, 56, 57 differential of, 316 fine gradings up to equivalence, 44 Morphism of group schemes, 305 image of, 306 gradings up to isomorphism, 40 kernel of, 306 involutions, 59 Multiplicative orthogonal decomposition, Weyl groups of fine gradings, 47 160 automorphism group schemes of A,77 Multiset, 39 automorphism group schemes of B, C, multiplicity of an element, 39 and D,75 classification of symmetric composition Natural map, 299 algebras, 153 density (graded version), 29 Primitive element, 302 division gradings on matrix algebras, 37 fine gradings induce root gradings, 264 Quadratic form, 124 generalized Hurwitz, 127 multiplicative, 124 graded division algebras, 34 nonsingular, 124 graded simple associative algebras polar form, 124 anti-automorphisms, 53 Quasitorus, 20 isomorphisms, 32 maximal, 21 structure, 30 saturated, 20 gradings on A1 Quotient map, 305 fine gradings up to equivalence, 109 up to isomorphism, 106 Representable functor, 299 gradings on A , r ≥ 2 Representing object, 299 r fine gradings up to equivalence, 84, 111 Restricted enveloping algebra, 273 up to isomorphism, 81, 107 Restricted Lie algebra, 272 Weyl groups of fine gradings, Type I, toral rank of, 272 113 toral subalgebras, or tori, 272 Weyl groups of fine gradings, Type II, Root lattice, 65 114 Root system, 65 ≥ automorphism group of, 65 gradings on Br, r 2 base of, 65 fine gradings up to equivalence, 79, 117 Cartan matrix of, 66 up to isomorphism, 79, 117 diagram automorphisms of, 66 Weyl groups of fine gradings, 117 ≥ Dynkin diagram of, 66 gradings on Cr, r 2 irreducible, 66 fine gradings up to equivalence, 79, 119 Weyl group of, 65 up to isomorphism, 79, 118 Weyl groups of fine gradings, 119 Semigroup grading, 11 gradings on D4 Sequence of divided powers, 319 fine gradings up to equivalence, 224 Sesquilinear form, 53 gradings on Dr, r =3orr ≥ 5 balanced, 54 fine gradings up to equivalence, 79, 121 Special algebra (Cartan type), 275 up to isomorphism, 79, 120 Spin group, 168 Weyl groups of fine gradings, 121 natural and spin representations, 168 gradings on E6 336 INDEX

fine gradings of inner type, up to Yoneda’s Lemma, 299 equivalence, 248 fine gradings of outer type, up to Zariski topology, 311 equivalence, 259 gradings on F4, 196 gradings on G2, 146 gradings on Albert algebra fine gradings up to equivalence, 184 up to isomorphism, 189 Weyl groups of fine gradings, 179, 181–183 gradings on Cartan type Lie algebras, 289 by groups without p-torsion, 288 fine gradings on S(m;1)(2) up to equivalence, 296 fine gradings on W (m;1)upto equivalence, 293 gradings on S(m;1)(2) up to isomorphism, 294 gradings on W (m;1)upto isomorphism, 293 gradings on Cayley algebras, 131 up to equivalence, 133 up to isomorphism, 136 Weyl groups of fine gradings, 135 gradings on Jordan algebras (+) Mn(F) , 203 fine gradings on H(Mn(F),t), n even, up to equivalence, 205 fine gradings on H(Mn(F),t), n odd, up to equivalence, 204 fine gradings on H(Mn(F),ts), up to equivalence, 206 gradings on H(Mn(F),t), n even, up to isomorphism, 205 gradings on H(Mn(F),t), n odd, up to isomorphism, 204 gradings on H(Mn(F),ts), up to isomorphism, 205 of bilinear forms, 198 gradings on O(m;1), 291 gradings on Okubo algebras, 159 Poincaré–Birkhoff–Witt, 2, 273 transfer of gradings, 24 Tits construction, 224 Tits–Kantor–Koecher Lie algebra, 229 Twisted group algebra, 34, 177

Universal enveloping algebra, 273 Universal group of a grading, 15

Verschiebung operator, 319

Weakly isomorphic gradings, 16 Weyl group of a grading, 15 Weyl group of a root system, 65 Witt algebra (Cartan type), 274 Selected Published Titles in This Series

189 Alberto Elduque and Mikhail Kochetov, Gradings on Simple Lie Algebras, 2013 188 David Lannes, The Water Waves Problem, 2013 187 Nassif Ghoussoub and Amir Moradifam, Functional Inequalities: New Perspectives and New Applications, 2013 186 Gregory Berkolaiko and Peter Kuchment, Introduction to Quantum Graphs, 2013 185 Patrick Iglesias-Zemmour, Diffeology, 2013 184 Frederick W. Gehring and Kari Hag, The Ubiquitous Quasidisk, 2012 183 Gershon Kresin and Vladimir Maz’ya, Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems, 2012 182 Neil A. Watson, Introduction to Heat Potential Theory, 2012 181 Graham J. Leuschke and Roger Wiegand, Cohen-Macaulay Representations, 2012 180 Martin W. Liebeck and Gary M. Seitz, Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras, 2012 179 Stephen D. Smith, Subgroup complexes, 2011 178 Helmut Brass and Knut Petras, Quadrature Theory, 2011 177 Alexei Myasnikov, Vladimir Shpilrain, and Alexander Ushakov, Non-commutative Cryptography and Complexity of Group-theoretic Problems, 2011 176 Peter E. Kloeden and Martin Rasmussen, Nonautonomous Dynamical Systems, 2011 175 Warwick de Launey and Dane Flannery, Algebraic Design Theory, 2011 174 Lawrence S. Levy and J. Chris Robson, Hereditary Noetherian Prime Rings and Idealizers, 2011 173 Sariel Har-Peled, Geometric Approximation Algorithms, 2011 172 Michael Aschbacher, Richard Lyons, Stephen D. Smith, and Ronald Solomon, The Classification of Finite Simple Groups, 2011 171 Leonid Pastur and Mariya Shcherbina, Eigenvalue Distribution of Large Random Matrices, 2011 170 Kevin Costello, Renormalization and Effective Field Theory, 2011 169 Robert R. Bruner and J. P. C. Greenlees, Connective Real K-Theory of Finite Groups, 2010 168 Michiel Hazewinkel, Nadiya Gubareni, and V. V. Kirichenko, Algebras, Rings and Modules, 2010 167 Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster Algebras and Poisson Geometry, 2010 166 Kyung Bai Lee and Frank Raymond, Seifert Fiberings, 2010 165 Fuensanta Andreu-Vaillo, JoséM.Mazón, Julio D. Rossi, and J. Julián Toledo-Melero, Nonlocal Diffusion Problems, 2010 164 Vladimir I. Bogachev, Differentiable Measures and the Malliavin Calculus, 2010 163 Bennett Chow, Sun-Chin Chu, David Glickenstein, Christine Guenther, James Isenberg, Tom Ivey, Dan Knopf, Peng Lu, Feng Luo, and Lei Ni, The Ricci Flow: Techniques and Applications: Part III: Geometric-Analytic Aspects, 2010 162 Vladimir Mazya and Jürgen Rossmann, Elliptic Equations in Polyhedral Domains, 2010 161 KanishkaPerera,RaviP.Agarwal,andDonalO’Regan, Morse Theoretic Aspects of p-Laplacian Type Operators, 2010 160 Alexander S. Kechris, Global Aspects of Ergodic Group Actions, 2010 159 Matthew Baker and Robert Rumely, Potential Theory and Dynamics on the Berkovich Projective Line, 2010

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/survseries/.

Gradings are ubiquitous in the theory of Lie algebras, from the root space decomposition of a complex semisimple Lie algebra rela- tive to a Cartan subalgebra to the beautiful Dempwolff decomposition of E8 as a direct

sum of thirty-one Cartan subalgebras. This Elduque Photograph courtesy of Eva Elduque Photograph courtesy of Eva 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 nonclas- sical 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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