Cấu trúc Đại số Lie Đơn giản trên Trường Đặc trưng Dương: Lý thuyết và Ứng dụng

com de Gruyter Expositions in Mathematics 38 Editors O. Kegel, Albert-Ludwigs-Universität, Freiburg V. Maslov, Academy of Sciences, Moscow W. Neumann, Columbia University, New York R., Rice University, Houston www.com de Gruyter Expositions in Mathematics 1 The Analytical and Topological Theory of Semigroups, K.) 2 Combinatorial Homotopy and 4-Dimensional Complexes, H.

Baues 3 The Stefan Problem, A. Meirmanov 4 Finite Soluble Groups, K. Hawkes 5 The Riemann Zeta-Function, A. Voronin 6 Contact Geometry and Linear Differential Equations, V.

Sternin 7 Infinite Dimensional Lie Superalgebras, Yu. Zaicev 8 Nilpotent Groups and their Automorphisms, E. Khukhro 9 Invariant Distances and Metrics in Complex Analysis, M. Pflug 10 The Link Invariants of the Chern-Simons Field Theory, E.

Guadagnini 11 Global Affine Differential Geometry of Hypersurfaces, A. Zhao 12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and Theta Functions, K. Weintraub 13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. Plamenevsky 14 Subgroup Lattices of Groups, R.

Schmidt 15 Orthogonal Decompositions and Integral Lattices, A. Tiep 16 The Adjunction Theory of Complex Projective Varieties, M. Sommese 17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. Bruno 18 Unitary Representation Theory of Exponential Lie Groups, H.

Ludwig 19 Blow-up in Quasilinear Parabolic Equations, A. Mikhailov 20 Semigroups in Algebra, Geometry and Analysis, K.) 21 Compact Projective Planes, H. Stroppel 22 An Introduction to Lorentz Surfaces, T. Weinstein 23 Lectures in Real Geometry, F.) 24 Evolution Equations and Lagrangian Coordinates, A.

Shmarev 25 Character Theory of Finite Groups, B. Huppert 26 Positivity in Lie Theory: Open Problems, J.) 27 Algebra in the Stone-Čech Compactification, N. Strauss 28 Holomorphy and Convexity in Lie Theory, K. Neeb 29 Monoids, Acts and Categories, M.

Mikhalev 30 Relative Homological Algebra, Edgar E. Jenda 31 Nonlinear Wave Equations Perturbed by Viscous Terms, Viktor P. Mosolov 32 Conformal Geometry of Discrete Groups and Manifolds, Boris N. Apanasov 33 Compositions of Quadratic Forms, Daniel B.

Shapiro 34 Extension of Holomorphic Functions, Marek Jarnicki, Peter Pflug 35 Loops in Group Theory and Lie Theory, Péter T. Nagy, Karl Strambach 36 Automatic Sequences, Friedrich von Haeseler 37 Error Calculus for Finance and Physics, Nicolas Bouleau www.com Simple Lie Algebras over Fields of Positive Characteristic I. Structure Theory by Helmut Strade ≥ Walter de Gruyter · Berlin · New York www.com Author Helmut Strade Fachbereich Mathematik Schwerpunkt Algebra und Zahlentheorie Universität Hamburg Bundesstrasse 55 20146 Hamburg, Germany E-mail: strade@math.de Mathematics Subject Classification 2000: 17-02; 17B50, 17B20, 17B05 Key words: simple Lie algebras, classification, Lie algebras of characteristic p ⬎ 2, divided power algebras, Cartan prolongation, recognition theorems 앪 앝 Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability. Library of Congress Cataloging-in-Publication Data Strade, Helmut, 1942⫺ Simple Lie algebras over fields of positive characteristic / by Hel- mut Strade.

cm ⫺ (De Gruyter expositions in mathematics ; 38) Includes bibliographical references and index. 1 : acid-free paper) 1.55⫺dc22 2004043901 ISBN 3-11-014211-2 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at ⬍http://dnb. 쑔 Copyright 2004 by Walter de Gruyter GmbH & Co. KG, 10785 Berlin, Germany.

All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher. Typesetting using the authors’ TEX files: I. Printing and binding: Hubert & Co.

Cover design: Thomas Bonnie, Hamburg.com Für meine liebe Renate, die mit bewundernswerter Geduld die Enstehung dieses Buches begleitet hat.com Contents Introduction 1 1 Toral subalgebras in p-envelopes 17 1.2 The absolute toral rank 23 1.4 Absolute toral ranks of parametrized families 39 1.5 Toral switching 46 2 Lie algebras of special derivations 58 2.1 Divided power mappings 59 2.2 Subalgebras defined by flags 73 2.3 Transitive embeddings of Lie algebras 79 2.4 Automorphisms and derivations 89 2.5 Filtrations and gradations 91 2.6 Minimal embeddings of filtered and associated graded Lie algebras 99 2.9 The constructions can be made basis free 119 3 Derivation simple algebras and modules 133 3.4 Derivation semisimple associative algebras 163 3.6 Conjugacy classes of tori 176 4 Simple Lie algebras 180 4.1 Classical Lie algebras 180 4.2 Lie algebras of Cartan type 184 4.4 Simple Lie algebras in characteristic 3 209 www.com viii Contents 5 Recognition theorems 217 5.2 From local to global Lie algebras 228 5.4 Generating Melikian algebras 258 5.5 The Weak Recognition Theorem 262 5.6 The Recognition Theorem 269 5.7 Wilson’s Theorem 272 6 The isomorphism problem 283 6.2 The compatibility property 295 6.4 Orbits of Hamiltonian forms 314 6.7 Melikian algebras 349 7 Structure of simple Lie algebras 357 7.6 W (1; n) 420 8 Pairings of induced modules 432 8.3 Trigonalizability 461 9 Toral rank 1 Lie algebras 484 9.1 Solvable maximal subalgebras 484 9.2 Cartan subalgebras of toral rank 1 496 Notation 521 Bibliography 527 Index 539 www.com Introduction The theory of finite dimensional Lie algebras over fields F of positive characteristic p was initiated by E. Witt came up with the following example of a simple Lie algebra of dimension p (for p > 3), afterwards named the Witt algebra W (1; 1). On the vector p−2 space i=−1 F ei define the Lie product  (j − i)ei+j if − 1 ≤ i + j ≤ p − 2, [ei , ej ] := 0 otherwise. This algebra behaves completely different from those algebras we know in charac- teristic  0.

As an example, it contains a unique subalgebra of codimension 1, namely i≥0 F ei. It also has sandwich elements, i. Witt himself never published this example or generalizations of it, which he presumably knew of. At that time he was interested in the search for new finite simple groups.

When he realized that these new structures had only known au- tomorphism groups he apparently lost his interest in these algebras. We have only oral and indirect information of Witt’s work on this field by two publications of H. Chang determined the automorphisms and irreducible representations of W (1; 1) over algebraically closed fields. He also men- tioned that Witt himself gave a realization of W (1; 1) in terms of truncated polynomial rings.

Jacobson proved a Galois type theorem for inseparable field exten- sions by substituting the algebra of derivations for the automorphism group of a field extension. More explicitly, he was able to show that the set of intermediate fields of a p field extension F (c1 ,. , cn ) : F with ci ∈ F is in bijection with the set of those Lie subalgebras of Der F F (c1 ,. , cn ), which are F (c1 ,.

, cn )-modules and are closed under the p-power mapping D  → D p. At that early time Jacobson already introduced the term “restrictable” for those Lie algebras, which admit a p-mapping x → x [p] satisfying the equation ad x [p] = (ad x)p for all x. Later he preferred to use the term “restricted Lie algebra” for pairs (L, [p]), when such a p-mapping is fixed. The Lie algebras of linear algebraic groups over F are all equipped with a natural p-mapping, hence they carry canonical restricted Lie algebra structures.

Zassenhaus [Zas39] generalized the construction of E. Witt in a natural way. Let n  G be a subgroup of order p in the additive group of F and give the direct sum g∈G F ug a Lie algebra structure via [ug , uh ] := (h − g)ug+h for all g, h ∈ G. Such Lie algebras are now called Zassenhaus algebras.

He also proved the first classification result, saying that a simple Lie algebra having a 1-dimensional CSA (= Cartan subalgebra) such that all roots are GF(p)-dependent and all root spaces are 1-dimensional is isomorphic to sl(2) or W (1; 1). Since then a great number of publications on this new theory of modular Lie alge- bras have appeared. We were shown how to construct analogues of the characteristic 0 simple Lie algebras [Jac41], [Jac43], [Che56], [M-S57] (these algebras, including the exceptional ones, are called classical in the modular theory), and in which way classes of non-classical algebras (called Cartan type) arise from infinite dimensional algebras of differential operators over C [K-S66], [K-S69], [Wil69], [Kac74], [Wil76]. In some sense [Wil76] was a cornerstone.

In this paper the previously known finite dimensional simple Lie algebras had been categorized into the classes of classical Lie algebras and “generalized” Cartan type Lie algebras for characteristic p > 3. People began to believe that the list of finite dimensional Lie algebras known so far could pos- sibly be complete, at least for p > 5. There were some indications that characteristic 5 is a borderline case. In fact, additional examples of simple Lie algebras were known in characteristics 2 and 3 (G.

Kostrikin) as early as from 1967. Melikian published a new family of simple Lie algebras in characteristic 5 ([Mel80]), now named Melikian algebras. The present Classifica- tion Theory of Block–Wilson–Strade–Premet indeed proves that the classical, Cartan type, and Melikian algebras exhaust the class of simple Lie algebras for p > 3. It could also well be that the list of known simple Lie algebras in characteristic 3 is close to complete.

But, as an example, Yu. Leites [K-L92] constructed simple Lie algebras in characteristic 2 from superalgebras. This indicates that a greater variety of constructions could yield many more examples in characteristic 2. A more complete history of this search for new simple Lie algebras would have to mention many other mathematicians who prepared the ground well, whose names, unfortunately, will remain in the dark during this short introduction.

Let us briefly describe the known simple Lie algebras for p > 3. The construction of C. Chevalley provides in a finite dimensional simple Lie algebra g over C a basis B of root vectors with respect to a CSA h such that the multiplication coefficients are integers of absolute value < 5. The Z-span gZ of B is a Z-form in g closed under taking Lie brackets.

Therefore, gF := F ⊗Z gZ is a Lie algebra over F with basis 1 ⊗ B and structure constants obtained from those for gZ by reducing modulo p. For p > 3, the Lie algebra gF fails to be simple if and only if the root system  = (g, h) has type Al where l = mp−1 for some m ∈ N. If  has type Amp−1 , then gF ∼= sl(mp) has the one dimensional center of scalar matrices and the Lie algebra www.com Introduction 3 gF /z(gF ) ∼= psl(mp) is simple. The simple Lie algebras over F thus obtained are called classical.

All classical Lie algebras are restricted with p-mapping given by (1 ⊗ eα )[p] = 0 and (1 ⊗ hi )[p] = 1 ⊗ hi for all α ∈  and 1 ≤ i ≤ l. As in characteristic 0, they are parametrized by Dynkin diagrams of types Al , Bl , Cl , Dl , G2 , F4 , E6 , E7 , E8. We stress that, by abuse of characteristic 0 notation, the classical simple Lie algebras over F include the Lie algebras of simple algebraic F -groups of exceptional types. All classical simple Lie algebras are closely related to simple algebraic groups over F.

Šafarevič gave a unified description of a large class of non-classical simple Lie algebras over F. Their construction was mo- tivated by classical work of E. Cartan [Car09] on infinite dimensional, simple, tran- sitive pseudogroups of transformations. To define finite dimensional modular ana- logues of complex Cartan type Lie algebras Kostrikin and Šafarevič replaced algebras of formal power series over C by divided power algebras over F.

Let Nm denote mof non-negative integers.