N. A. Vavilov

Chevalley Groups over Commutative Rings: I. Elementary Calculations

This is the first in a series of papers dedicated to the structure of Chevalley groups over commutative rings. The goal of this series is to systematically develop methods of calculations in Chevalley groups over rings, based on the use of their minimal modules. As an application, we give new direct proofs for normality of the elementary subgroup, description of normal subgroups and similar results due to E. Abe, G. Taddei, L. N. Vaserstein, and others, as well as some generalizations. In this first part we outline the whole project, reproduce construction of Chevalley groups and their elementary subgroups, recall familiar facts about the elementary calculations in these groups, and fix a specific choice of the structure constants.

K1 of Chevalley groups are nilpotent

Abstract Let Φ be a reduced irreducible root system and R be a commutative ring. Further, let G ( Φ , R ) be a Chevalley group of type Φ over R and E ( Φ , R ) be its elementary subgroup. We prove that if the rank of Φ is at least 2 and the Bass-Serre dimension of R is finite, then the quotient G ( Φ , R )/ E ( Φ , R ) is nilpotent by abelian. In particular, when G ( Φ , R ) is simply connected the quotient K 1 ( Φ , R )= G ( Φ , R )/ E ( Φ , R ) is nilpotent. This result was previously established by Bak for the series A 1 and by Hazrat for C 1 and D 1 . As in the above papers we use the localisation-completion method of Bak, with some technical simplifications.

Normality for elementary subgroup functors

We define a notion of group functor G on categories of graded modules, which unifies previous concepts of a group functor G possessing a notion of elementary subfunctor E . We show under a general condition which is easily checked in practice that the elementary subgroup E ( M ) of G ( M ) is normal for all quasi-weak Noetherian objects M in the source category of G . This result includes all previous ones on Chevalley and classical groups G of rank ≥ 2 over a commutative or module finite ring M (since such rings are quasi-weak Noetherian) and settles positively unanswered cases of normality for these group functors.

An a 3 -Proof of Structure Theorems for Chevalley Groups of Types E 6 and E 7 .

In the nineties the author, A. Stepanov and E. Plotkin developed a geometric approach towards calculations in exceptional groups at the level of K1, decomposition of unipotents. However, it relied on the presence of large classical embeddings, such as A5 ≤ E6 or A7 ≤ E7. Recently the author, M. Gavrilovich and S. Nikolenko devised a sharper geometric method which only uses embeddings A2 ≤ E6, E7, F4. Here we show that one can make a further step and simultaneously stabilize two columns of a root unipotent by an element of type A3 ≤ E6, E7. This opens the way to applications of this method at the level of K2.

Net subgroups of Chevalley groups

We introduce and study net subgroups of Chevalley groups of normal and certain twisted types. Another subgroup of Chevalley groups related to a net was discussed in Ref. Zh. Mat. 1976; 10A151; 1977, 10A301; 1978, 6A476.

Visual Basic Representations

We depict the weight diagrams (alias, crystal graphs) of basic and adjoint representations of complex simple Lie algebras/algebraic groups and describe some of their uses.

Weight elements of Chevalley groups

The paper is devoted to a detailed study of some remarkable semisimple elements of (extended) Chevalley groups that are diagonalizable over the ground field — the weight elements. These are the conjugates of certain semisimple elements hω(e) of extended Chevalley groups G = G(Φ ,K ), where ω is a weight of the dual root system Φ ∨ and e ∈ K ∗ . In the adjoint case the hω(e)s were defined by Chevalley himself and in the simply connected case they were constructed by Berman and Moody. The conjugates of hω(e) are called weight elements of type ω .V arious constructions of weight elements are discussed in the paper, in particular, their action in irreducible rational representations and weight elements induced on a regularly embedded Chevalley subgroup by the conjugation action of a larger Chevalley group. It is proved that for a given x ∈ G all elements x(e )= xhω(e)x −1 , e ∈ K ∗ ,a part maybe from a finite number of them, lie in the same Bruhat coset BwB ,w herew is an involution of the Weyl group W = W (Φ). The elements hω(e) are particularly important when ω = � i is a microweight of Φ ∨ . The main result of the paper is a calculation of the factors of the Bruhat decomposition of microweight elements x(e )f or the case whereω = � i. It turns out that all nontrivial x(e)s lie in the same Bruhat coset BwB ,w herew is a product of reflections in pairwise strictly orthogonal roots γ1 ,...,γ r+s. Moreover, if among these roots r are long and s are short, then r +2 s does not exceed the width of the unipotent radical of the ith maximal parabolic subgroup in G. A version of this result was first announced in a paper by the author in Soviet Mathematics: Doklady in 1988. From a technical viewpoint, this amounts to the determination of Borel orbits of a Levi factor of a parabolic subgroup with Abelian unipotent radical and generalizes some results of Richardson, Rohrle, and Steinberg. These results are instrumental in the description of overgroups of a split maximal torus and in the recent papers by the author and V. Nesterov on the geometry of tori.

Do It Yourself: the Structure Constants for Lie Algebras of Types El

Two algorithms for computing the structure table of Lie algebras of type El with respect to a Chevalley base are compared: the usual inductive algorithm and an algorithm based on the use of the Frenkel-Kac cocycle. It turns out that the Frenkel–Kac algorithm is several dozen times faster, but under the “natural” choice of a bilinear form and a sign function it has no success in a positive Chevalley base. We show how one can modify the sign function to obtain a proper choice of the structure constants. Cohen, Griess, and Lisser obtained a similar result by varying the bilinear form. We recall the hyperbolic realization of the root systems of type El, which dramatically simplifies calculations as compared with the usual Euclidean realization. We give Mathematica definitions, which realize root systems and implement the inductive and Frenkel–Kac algorithms. Using these definitions, one can compute the whole structure table for E8 in a quarter of an hour with a home computer. At the end of the paper, we reproduce tables of roots ordered in accordance with HeightLex and the resulting tables of structure constants. Bibliography: 43 titles.

Normalizer of the Chevalley group of type

We consider the simply connected Chevalley group

Can one see the signs of structure constants

G(E_7,R)