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Mathematical Surveys

and Monographs

Volume 189

Gradings on SimpleLie Algebras

Alberto ElduqueMikhail Kochetov

American Mathematical Society

Atlantic Association for Researchin the Mathematical Sciences

Gradings on Simple Lie Algebras

http://dx.doi.org/10.1090/surv/189

Mathematical Surveys

and Monographs

Volume 189

Gradings on Simple Lie Algebras

Alberto Elduque Mikhail Kochetov

American Mathematical SocietyProvidence, RI

Atlantic Association for Researchin the Mathematical SciencesHalifax, Nova Scotia, Canada

Editorial Committee of Mathematical Surveys and Monographs

Ralph L. Cohen, Chair

Robert GuralnickMichael A. Singer

Benjamin SudakovMichael I. Weinstein

Editorial Board of the Atlantic Association forResearch in the Mathematical Sciences

Jeannette Janssen, DirectorDavid Langstroth, Managing Editor

Yuri Bahturin Theodore KolokolnikovRobert Dawson Lin Wang

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

For additional information and updates on this book, visitwww.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 2013512′.482—dc23

2013007217

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10 9 8 7 6 5 4 3 2 1 18 17 16 15 14 13

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 91.1. General gradings and group gradings 91.2. The universal group of a grading 151.3. Fine gradings 161.4. Duality between gradings and actions 191.5. Exercises 25

Chapter 2. Associative Algebras 272.1. Graded simple algebras with minimality condition 282.2. Graded division algebras over algebraically closed fields 332.3. Classification of gradings on matrix algebras 382.4. Anti-automorphisms and involutions of graded matrix algebras 492.5. Exercises 60

Chapter 3. Classical Lie Algebras 633.1. Classical Lie algebras and their automorphism group schemes 643.2. ϕ-Gradings on matrix algebras 853.3. Type A 1053.4. Type B 1163.5. Type C 1183.6. Type D 1193.7. Exercises 121

Chapter 4. Composition Algebras and Type G2 1234.1. Hurwitz algebras 1234.2. Gradings on Cayley algebras 1304.3. Gradings on psl3(F), charF = 3 1374.4. Derivations of Cayley algebras and simple Lie algebras of type G2 1404.5. Gradings on the simple Lie algebras of type G2 1464.6. Symmetric composition algebras 1494.7. Exercises 160

Chapter 5. Jordan Algebras and Type F4 1635.1. The Albert algebra 164

vii

viii CONTENTS

5.2. Construction of fine gradings on the Albert algebra 1695.3. Weyl groups of fine gradings 1785.4. Classification of gradings on the Albert algebra 1845.5. Gradings on the simple Lie algebra of type F4 1905.6. Gradings on simple special Jordan algebras 1975.7. Exercises 206

Chapter 6. Other Simple Lie Algebras in Characteristic Zero 2076.1. Fine gradings on the simple Lie algebra of type D4 2076.2. Freudenthal’s Magic Square 2246.3. Some nice gradings on the exceptional simple Lie algebras 2396.4. Fine gradings on the simple Lie algebra of type E6 2446.5. Fine gradings and gradings by root systems 2596.6. Summary of known fine gradings for types E6, E7 and E8 2656.7. Exercises 269

Chapter 7. Lie Algebras of Cartan Type in Prime Characteristic 2717.1. Restricted Lie algebras 2717.2. Construction of Cartan type Lie algebras 2737.3. Automorphism group schemes 2767.4. Gradings 2877.5. Exercises 297

Appendix A. Affine Group Schemes 299A.1. Affine group schemes and commutative Hopf algebras 299A.2. Morphisms of group schemes 305A.3. Linear representations 307A.4. Affine algebraic groups 310A.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) where F is an algebraically closedfield, charF �= 2. 44

2.2 Gradings, up to equivalence, on M3(F) where F is an algebraically closedfield, charF �= 3. 46

4.1 Multiplication table of the split Cayley algebra 129

4.2 Gradings on the Cayley algebra over an algebraically closed field ofcharacteristic different from 2 134

4.3 Gradings, up to equivalence, on the simple Lie algebra of type G2 over analgebraically 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

ix

Preface

The aim of this book is to introduce the reader to the theory of gradings on Liealgebras, with a focus on the classification of gradings on simple finite-dimensionalLie algebras over algebraically closed fields. The classic example of such a gradingis the Cartan decomposition with respect to a Cartan subalgebra in characteristiczero, which is a grading by a free abelian group. Since the 1960’s, there has beenmuch work on gradings by other groups, starting with finite cyclic groups, andapplications 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 ratherto present a self-contained exposition of the classification of gradings on classicalsimple Lie algebras in characteristic different from 2 and on some non-classicalsimple Lie algebras in prime characteristic greater than 3. Other important algebrasalso enter the stage: matrix algebras, the octonions and the simple exceptionalJordan algebra. Most of the classification results presented here are recent andhave not yet appeared in book form.

This work started with the notes of two courses that the authors gave for theAtlantic Algebra Centre at Memorial University of Newfoundland: “Introductionto affine group schemes” (M. Kochetov, November–December 2008) and “Compo-sition algebras and their gradings” (A. Elduque, May 2009). Affine group schemesare an important tool for the study of gradings on finite-dimensional algebras inarbitrary characteristic, as we explain in Chapter 1. We give a brief expositionof the background on affine group schemes in Appendix A, with references to theliterature on this subject. A reader who is interested exclusively in the case ofcharacteristic zero will only need affine algebraic groups (in the “naıve” sense) tofollow this book. Apart from this, we assume that the reader is familiar with lin-ear algebra and with the basics on groups and algebras. The book is intended forspecialists 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 tooriginal 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, adviceand encouragement of Yuri Bahturin, who himself greatly contributed to the studyof gradings by arbitrary groups. It was his enthusiasm that convinced the authorsto join efforts in the task of collecting, understanding, unifying and expanding theknowledge about gradings on simple Lie algebras. The second author would alsolike to use this opportunity to express his gratitude for all the help in his life andcareer given so generously by Professor Bahturin since becoming his thesis advisora decade and a half ago.

xi

xii PREFACE

The authors have benefited from discussions with many colleagues. Amongthem, our special thanks are due to Cristina Draper, who explained her results ongradings on exceptional simple Lie algebras long before they were publicly available.

The first author acknowledges the support of the former Spanish Ministerio deCiencia e Innovacion—Fondo Europeo de Desarrollo Regional (FEDER)1 and of theDiputacion General de Aragon—Fondo Social Europeo (Grupo de Investigacion de

Algebra). He would also like to thank Memorial University for hospitality duringhis 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 Universityof Zaragoza during his visits to Spain.

Both authors acknowledge the support of the Atlantic Association for Researchin the Mathematical Sciences (AARMS) of Canada in the preparation of this book.

Alberto Elduque and Mikhail Kochetov

Zaragoza, SpainFebruary 2013

1MTM2010-18370-C04-022Discovery 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 onF will vary from section to section and will be stated explicitly. In particular, thecharacteristic of F will be written as charF. 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 ⊕, andtensor products as ⊗. The trace and determinant of a matrix or an endomorphismwill be denoted by tr and det, respectively. An endomorphism whose minimalpolynomial has no multiple roots will be called semisimple or (if the ground field isalgebraically closed) diagonalizable.

Cyclic groups will often be written as Z or Zm. The symmetric group onn symbols will be denoted by Sym(n). Direct and semidirect product of groupswill be written as × and �, respectively. The stabilizer of an object x underan action of a group G will be denoted by StabG(x), with StabG(x, y) meaningStabG(x)∩StabG(y), etc. In the special case of G acting on itself or its power set byconjugation, we will use CG(x) (centralizer) and NG(x) (normalizer), respectively.Thus, for X ⊂ G, we have:

CG(X) := {g ∈ G | gx = xg ∀x ∈ X} and NG(X) := {g ∈ G | gXg−1 = 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) or GL(V ) for thegeneral linear group and similarly SL for the special linear group, O for the orthog-onal group (with respect to a nondegenerate quadratic form), SO for the specialorthogonal group, and Sp for the symplectic group (with respect to a nondegener-ate symplectic form). If F is finite, its symbol may be replaced by the order: forexample, we will write GL3(2) for GL3(F) where F is the field of two elements. Themultiplicative group of F will be denoted by F×.

Throughout the book, more notation will be introduced, especially for gradingson various algebras. As a general rule, a grading on an algebra will be denoted byΓ with a subscript indicating the algebra or its type. We have made an effort tocollect all such symbols in a separate notation index at the end of the book.

The terminology and basic constructions concerning gradings are introducedin Chapter 1. They will be used throughout the book. The chapters depend oneach 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; Chapter6 depends on all preceding chapters.

xiii

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Index of Notation

A(+), 3, 197

A(−), 2

Ad, 317

ad, 317

AlgF, 299

Aut(U), 301, 309

AutG(A), 14

Aut(Γ), 14

AutΞ(κ, γ), 46

AutΦ, 65

AutX(O), X ∈ {S,H,K}, 277

βσ, 35

CD(K, β, γ), 127

CD(Q, α), 126

Cη , 209

Cl(C, n), 167

Cs, 129

degΓ, 9

Der(A), 4

Diag(Γ), 14

Diag(Γ), 23

Dx,y, 225

dx,y, 140

EndgrR(V ), 28

F[G], 300

FσT , 34

G(A), 302

Ga, 300αΓ, 16

Γ1A(G, γ), 188

Γ1A, 170

Γ2A(G,H, γ), 188

Γ2A, 171

Γ3A(G,H, g), 188

Γ3A, 172

Γ4A(G,H, δ), 189

Γ4A, 173

Γ(I)A (G, T, β, κ, γ), 105

Γ(II)A (G,H, h, β, κ, γ, μ0, g0), 107

Γ(I)A (T, k), 109

Γ(II)A (T, q, s, τ), 110

ΓBF (G,κ, γ), 199

ΓBF (m, �), 199

ΓB(G, κ, γ), 117

Γ+B(G, κ, γ), 204

ΓB(q, s), 117

Γ+B(q, s), 204

Γ1C, Γ2

C, 136

Γ1C(G, γ), Γ2

C(G,H), 136

ΓC(G, T, β, κ, γ, g0), 118

Γ+C(G, T, β, κ, γ, g0), 205

ΓC(T, q, s, τ), 119

Γ+C(T, q, s, τ), 205

ΓD(G, T, β, κ, γ, g0), 120

Γ+D(G, T, β, κ, γ, g0), 204

ΓD(T, q, s, τ), 120

Γ+D(T, q, s, τ), 205

Γ1F4

,Γ2F4

,Γ3F4

,Γ4F4

, 196

Γ1F4

(G, γ), Γ2F4

(G,H, γ), Γ3F4

(G,H, g),

Γ4F4

(G,H, δ), 196

Γ1G2

, Γ2G2

, 146

Γ1G2

(G, γ), Γ2G2

(G,H), 146

Γ1H3(Q)

, 265

ΓK, 265

ΓK, 266

ΓM (D, k), ΓM (T, k), 44

ΓM (D, q, s, τ), 88

ΓM (G,D, κ, γ), ΓM (G, T, β, κ, γ), 40

Γ1M3(F)

, Γ2M3(F)

, 266

Γ(I)

M+(G, T, β, κ, γ), 202

Γ(II)

M+(G,H, h, β, κ, γ, μ0, g0), 203

ΓO, 266

ΓO(G,P, γ), 291

ΓO(s), 292

Γ1Q, Γ2

Q, 265

ΓS(G,P, γ, g0), 294

ΓS(s), 294

ΓW (G,P, γ), 292

ΓW (s), 292

331

332 INDEX OF NOTATION

Γ1X(D)

, 266

Γ2X(K)

, Γ2X(Q)

, 266

Γ3X(Q)

, 267

ΓX(D), 267GLn, 300

Gm, 300Gred, 313Grp, 300

g(S, S′), 230

H(A, ϕ), 3

H(A, ∗), 197H(m;n), 275

HomG(V,W ), 10Homgr(V,W ), 10

HomgrR(V,W ), 28

Int(L), 66

IDer(L), 4IDer(C), 225

IDer(J), 225

J(V, b), 197

K(A, ϕ), 3K(m;n), 276

Lie(G), 315L(J), 229L(J), 229

M(D, k), 41M(D, k)ab, 43

M(D, q, s, τ), ΓM (T, q, s, τ), 88M(G,D, κ, γ), 33M(G,D, κ, γ, δ, g0), 59

M(G,D, κ, γ, μ, g0), 57

M(G,D, κ, γ, μ0, g0), 87

ModG, 10

R ModG, ModGR , 28

μn, 306

ωS , ωH , ωK , 275

O(m;n), 273

Prim(A), 302

Set, 299sgn(ϕ), 58

sgn(Si), 56Σ(τ), 93σx,y, 191

S(m;n), 275Spin(C, n), 168Stab(Γ), 14

StabG, 308Supp Γ, 9

T(C, J), 225θη, 209

tJ, 225

T (ν), 214

U(Γ), 15(U(J), ι), 200

V, 319[g]V, V [g], 10V (G,D, κ, γ), 33

W (Γ), 15W (m;n), 274

X(G), 305Ξ(γ), 291Ξ(κ, γ), 39

Z(m;n), 273

Index

χ-Admissible data, 106δ-Admissible data, 59H-admissible grading, 289K-admissible grading, 289S-admissible grading, 289Affine algebraic group, 312

connected components of, 314Affine algebraic variety, 312Affine group scheme, 300

abelian, 300algebraic, 300characters of, 305diagonalizable, 309diagonalizable representations of, 309dimension of, 300distribution algebra of, 319finite, 300points of, 300representations of, 307smooth, 313tangent Lie algebra of, 315

AlgebraAlbert, 163alternative, 124associator, 125Cayley, 128

good basis, 129split, 129

central simple, 70

Clifford, 167composition, 124

para-Cayley, 167para-unit of, 150related triple, 167symmetric, 150triality Lie algebra, 190

G-graded, 1graded division, 29graded simple, 29Hurwitz, 124

Cayley–Dickson doubling process, 126isotropic, 128standard conjugation of, 124trace of, 125

Jordan, 163

degree of, 166

exceptional, 163

generic minimal polynomial of, 166

Lie multiplication algebra of, 229

normalized trace of, 225

of a bilinear form, 197

semisimple, 166

special, 163

unital special universal envelope of,200

Lie, 2

Malcev, 139

octonion, 128

Okubo, 150

para-Hurwitz, 150

Petersson, 150

quaternion, 128

reduced, 311

structurable, 266

Anti-automorphism of a graded algebra, 49

Antipode, 302

Augmentation ideal, 304

Automorphism group scheme, 301, 309

Bialgebra, 302

finite dual, 304

Bicharacter

alternating, 35

nondegenerate, 35

Biideal, 303

Cartan decomposition, 65, 69

Cartan subalgebra, 65

Cartier dual, 303

Chevalley basis, 67

Chevalley groups, 70

Closed imbedding, 305

Coaction, 308

Coalgebra, 301

cocommutative, 301

Coideal, 301

Comodule, 308

Comorphism, 305

333

334 INDEX

χ-Compatible pair, 81

Comultiplication, 301

Contact algebra (Cartan type), 276Counit, 301

Dempwolff decomposition, 244

Derivation, 4

inner, 140Dual

basis, 52

of a module, 52

Equivalent gradings, 14

Fine grading, 18First Tits Construction, 176

Freudenthal’s Magic Square, 228

Functorchange-of-group, 16

Lie, 315

representable, 299

Generalized Pauli matrices, 2Graded

algebra, by a (semi)group, 1

algebra, general, 11

bimodule, 28Density Theorem, 29

map, 10

module, 28

Schur’s Lemma, 29subspace, 10

vector space, 9

Gradingautomorphism group of, 14

coarsening of, 18

diagonal group of, 14

elementary, 38, 288fine, 18

induced by a homomorphism of groups,16

Jordan, 244

nontrivial, 9of Type I and Type II, 80

on a vector space, 9

adapted to a bilinear form, 198on an algebra, by a (semi)group, 1, 11

on an algebra, general, 11

on Hom(V,W ), 10

on tensor product, 11realization of, 11

refinement of, 18

shift of, 10

stabilizer of, 14support of, 9

toral, 21

type of, 14universal group of, 15

Weyl group of, 15

Gradings

anti-equivalence of, 81

anti-isomorphism of, 81equivalence of, 14

isomorphism of, 14

weak isomorphism of, 16

ϕ-Gradings, 79equivalence of, 79

isomorphism of, 79

weak equivalence of, 83Group grading, 11

Group-like element, 302

Groups of central type, 35

Hamiltonian algebra (Cartan type), 275Homogeneous

component, 9

element, 9map, 10

Homomorphism

of affine algebraic groups, 312

of bialgebras, 303of coalgebras, 301

of comodules, 308

of graded algebras, 14

of graded modules, 28of graded spaces, 10

of Hopf algebras, 303

Hopf algebra, 302Hopf ideal, 303

Hopf subalgebra, 303

Involution

of a graded algebra, 49orthogonal, 4

symplectic, 4

Isomorphic gradings, 14

Kaplansky’s trick, 127

Killing form, 65

Lie algebra, 2

abelian, 5derived algebra of, 5

metabelian, 5

nilpotent, 5

radical of, 6semisimple, 5

solvable, 5

adjoint representation of, 5Cartan type, 274

center of, 5

classical, 66, 70

split, 70direct sum (product), 4

inner automorphisms of, 66

inner derivations of, 4reductive, 218

representations of, 5

INDEX 335

restricted, 272

root graded, 259

coordinate algebra, 260

grading subalgebra, 259

semidirect sum (product), 4

symmetric pair, 248

MAD subgroups, 21

Module

graded, 28

graded irreducible, 29

graded simple, 29

over graded division algebra, 29

Morphism of affine group schemes

differential of, 316

Morphism of group schemes, 305

image of, 306

kernel of, 306

Multiplicative orthogonal decomposition,160

Multiset, 39

multiplicity of an element, 39

Natural map, 299

Primitive element, 302

Quadratic form, 124

multiplicative, 124

nonsingular, 124

polar form, 124

Quasitorus, 20

maximal, 21

saturated, 20

Quotient map, 305

Representable functor, 299

Representing object, 299

Restricted enveloping algebra, 273

Restricted Lie algebra, 272

toral rank of, 272

toral subalgebras, or tori, 272

Root lattice, 65

Root system, 65

automorphism group of, 65

base of, 65

Cartan matrix of, 66

diagram automorphisms of, 66

Dynkin diagram of, 66

irreducible, 66

Weyl group of, 65

Semigroup grading, 11

Sequence of divided powers, 319

Sesquilinear form, 53

balanced, 54

Special algebra (Cartan type), 275

Spin group, 168

natural and spin representations, 168

Standard realization of a division grading,39

Subbialgebra, 303

Subcoalgebra, 301

Subcomodule, 308

Subgroupscheme, 304

inverse image of, 306

normal, 307

Support, 9

Symplectic triple system, 251

Theorem

abelian gradings on matrix algebras

anti-automorphisms, 56, 57

fine gradings up to equivalence, 44

gradings up to isomorphism, 40

involutions, 59

Weyl groups of fine gradings, 47

automorphism group schemes of A, 77

automorphism group schemes of B, C,

and D, 75

classification of symmetric compositionalgebras, 153

density (graded version), 29

division gradings on matrix algebras, 37

fine gradings induce root gradings, 264

generalized Hurwitz, 127

graded division algebras, 34

graded simple associative algebras

anti-automorphisms, 53

isomorphisms, 32

structure, 30

gradings on A1


up to isomorphism, 106

gradings on Ar, r ≥ 2

fine gradings up to equivalence, 84, 111

up to isomorphism, 81, 107

Weyl groups of fine gradings, Type I,113

Weyl groups of fine gradings, Type II,114

gradings on Br, r ≥ 2




gradings on Cr, r ≥ 2




gradings on D4


gradings on Dr, r = 3 or r ≥ 5




gradings on E6

336 INDEX

fine gradings of inner type, up toequivalence, 248

fine gradings of outer type, up toequivalence, 259

gradings on F4, 196

gradings on G2, 146

gradings on Albert algebra



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 toequivalence, 296

fine gradings on W (m; 1) up toequivalence, 293

gradings on S(m; 1)(2) up toisomorphism, 294

gradings on W (m; 1) up to

isomorphism, 293

gradings on Cayley algebras, 131

up to equivalence, 133



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 toequivalence, 206

gradings on H(Mn(F), t), n even, up toisomorphism, 205

gradings on H(Mn(F), t), n odd, up to

isomorphism, 204

gradings on H(Mn(F), ts), up toisomorphism, 205

of bilinear forms, 198

gradings on O(m; 1), 291

gradings on Okubo algebras, 159

Poincare–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

Yoneda’s Lemma, 299

Zariski topology, 311

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180 Martin W. Liebeck and Gary M. Seitz, Unipotent and Nilpotent Classes in SimpleAlgebraic 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

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171 Leonid Pastur and Mariya Shcherbina, Eigenvalue Distribution of Large RandomMatrices, 2011

170 Kevin Costello, Renormalization and Effective Field Theory, 2011

169 Robert R. Bruner and J. P. C. Greenlees, Connective Real K-Theory of FiniteGroups, 2010

168 Michiel Hazewinkel, Nadiya Gubareni, and V. V. Kirichenko, Algebras, Ringsand Modules, 2010

167 Michael Gekhtman, Michael Shapiro, and Alek Vainshtein, Cluster Algebras andPoisson Geometry, 2010

166 Kyung Bai Lee and Frank Raymond, Seifert Fiberings, 2010

165 Fuensanta Andreu-Vaillo, Jose M. Mazon, Julio D. Rossi, and J. JulianToledo-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, JamesIsenberg, 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 Maz′ya and Jurgen Rossmann, Elliptic Equations in Polyhedral Domains,2010

161 Kanishka Perera, Ravi P. Agarwal, and Donal O’Regan, Morse Theoretic Aspectsof 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 theBerkovich Projective Line, 2010

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

SURV/189

For additional informationand updates on this book, visit

www.ams.org/bookpages/surv-189

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 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.

American Mathematical Society www.ams.org

Atlantic Association for Researchin the Mathematical Sciences www.aarms.math.ca

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gradings on simple lie algebras · 2019-02-12 · [bou98] n.bourbaki,lie groupsand lie...

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