Classifying the Absolute Toral Rank Two Case

519 Pages·2017·19.11 MB·English

by  Helmut Strade

Helmut Strade
Simple Lie Algebras over Fields of Positive Characteristic
De Gruyter Expositions in
Mathematics
Edited by
Lev Birbrair, Fortaleza, Brazil
Victor P. Maslov, Moscow, Russia
Walter D. Neumann, New York City, New York, USA
Markus J. Pflaum, Boulder, Colorado, USA
Dierk Schleicher, Bremen, Germany
Volume 42

Für meine Söhne Robert und Jörn
Contents
Introduction
10 Tori in Hamiltonian and Melikian algebras
10.1 Determining absolute toral ranks of Hamiltonian algebras
10.2 More on H(2; (1,2))(2)
[p]
10.3 2-dimensional tori in H(2; 1; Φ(τ))(1)
10.4 Semisimple elements in H(2; 1; Φ(1))
[p]
10.5 Melikian algebras
10.6 Semisimple Lie algebras of absolute toral rank 1 and 2
10.7 Weights
11 1-sections
11.1 Lie algebras of absolute toral rank 1
11.2 1-sections
11.3 Representations of dimension < p2
11.4 More on H(2; 1)(2)
11.5 Low dimensional representations of H(2; 1)(2)
12 Sandwich elements and rigid tori
12.1 Deriving identities
12.2 Sandwich elements
12.3 Rigid roots
12.4 Rigid tori
12.5 Trigonalizability
13 Towards graded algebras
13.1 The pentagon
13.2 An upper bound
13.3 Filtrations
13.4 More on Hamiltonian roots
13.5 Switching tori
13.6 Good triples
13.7 On the existence of good tori and good triples
14 The toral rank 2 case
14.1 No root is exceptional
14.2 S is not of Cartan type
14.3 Graded counterexamples
Notation
Bibliography
Index
Introduction
This three volume monograph on “Simple Lie Algebras over Fields of Positive
Characteristic” presents major methods on modular Lie algebras, all the
examples of simple Lie algebras over algebraically closed fields of characteristic
p ≥ 5 and the complete proof of the Classification Theorem mentioned in the
introduction of Volume 1. The first volume contains the methods, examples and
a first classification result. It turned out during the work on the reproduction of
the classification proof that one has to pay for a reasonable completeness by
extending the text considerably. So the whole work is now planned as a three
volume monograph. This second volume contains the proof of the Classification
Theorem for simple Lie algebras of absolute toral rank 2.
We have already mentioned details outlining the proof of the Classification
Theorem in the introduction of the first volume. Therefore we will just recall
very briefly some strategy in order to place the content of this volume into the
whole picture. Already in the early work on simple Lie algebras over the
complex numbers people determined, as a general procedure, 1-sections
with respect to a toral CSA H, described their
representations in the spaces and determined 2-sections. The
breakthrough paper [B-W88] made this procedure work for modular Lie algebras
as well (if the characteristic is bigger than 7). It turned out, however, that the
many more examples and the richness of their structures made things much more
involved. Imagine that in the classical case only ⊕ H ∩ ker α occurs as the
1-section L(α), while in the modular case the classical algebra , the smallest
Witt algebra W(1; 1) and the smallest Hamiltonian algebra H(2; 1)(2) have
absolute toral rank 1 (by Corollaries 7.5.2 and 7.5.9). Hence each such algebra is
a 1-section of itself. There are nonsplit radical extensions of these Cartan type
Lie algebras, and it is a priori not clear which of these can occur as 1-sections of
simple Lie algebras. Moreover, the representation theories of such extensions are
very rich, and therefore the very details of these theories can hardly be
described. Less information is, fortunately, sufficient for the Classification
Theory. Namely, it is sufficient and possible to describe semisimple quotients
L[α] := L(α)/ rad L(α) with respect to certain tori T (we have to decide which tori
we take into consideration though), and it is also possible to describe the T-
semisimple quotients of 2-sections in terms of simple Lie algebras of absolute
toral rank 1 and 2 by Block’s Theorem Corollary 3.3.6. If one knows the simple
Lie algebras of absolute toral rank not bigger than 2 one is able to describe all
such semisimple quotients of 2-sections in these terms. In this second volume we
will prove the following
Theorem. Every simple Lie algebra over an algebraically closed field of
characteristic p > 3 having absolute toral rank 2 is exactly one of the following:
(a) classical of type A , B or G ;
2 2 2
(b) the restricted Lie algebras W(2; 1), S(3; 1)(1), H(4; 1)(1), K(3; 1); the
naturally graded Lie algebras W(1; 2), H(2; (1, 2))(2); H(2; 1; Φ(τ))(1), H(2;
1; Φ(1));
(c) the Melikian algebra (1,1).
Since we classified the simple Lie algebras of absolute toral rank 1 in Chapter 9
of Volume 1, the result of this second volume will provide sufficient information
on the 2-sections of simple Lie algebras with respect to adequate tori.
The proof of the Classification Theorem for simple Lie algebras of absolute
toral rank 2 is completely different from what we have done in Chapter 9 of
Volume 1 and what one has to do in Volume 3 for the general case. In fact, when
writing this text I have changed some of the original items, so that even more the
description in the introduction of the first volume does not correctly describe the
present procedure. Let me say a few words about the sources for the proofs of
this volume and the citation policy. The breakthrough paper [B-W88] gave the
general procedure and provided many ideas for the solution of the absolute toral
rank 2 case. In the present exposition I stressed the point of using sandwich
elements and graded algebras in combination with the Block–Weisfeiler
description of these. The major contribution of sandwich element methods is due
to A. A. PREMET. Most of the other material can be found in the papers [P-S 97]–
[P-S 01]. I will not quote these results in detail. If the reader is interested in the
original sources he should look into [B-W88], [Pre 85]–[Pre 94] and [P-S 97]–
[P-S 01].
Chapter 10, the first chapter of this volume, is somewhat different from the
rest. In that chapter we determine which of the Cartan type and Melikian
algebras have absolute toral rank 2, determine automorphism groups of these
algebras, describe orbits of toral elements in the minimal p-envelope under the
automorphism group, compute the centralizers of toral elements and estimate the
number of weights on restricted modules. In doing this we decover a lot of
details of the structure of these algebras. This already indicates a weakness of the
theory: at present one needs really much information on the algebra structures to
apply some sophisticated arguments.
The notations in this volume and all references to Chapter 1–9 refer to the
first volume. As a general assumption, F always denotes an algebraically closed
field of characteristic p > 3 (while in the first volume we also included the case p
= 3), and all algebras are regarded to be algebras over F.