Chevalley Groups over Commutative Rings I. Elementary Calculations

Groups graded by root systems and property (T).

We establish property (T) for a large class of groups graded by root systems, including elementary Chevalley groups and Steinberg groups of rank at least 2 over finitely generated commutative rings with 1. We also construct a group with property (T) which surjects onto all finite simple groups of Lie type and rank at least two.

Chevalley Groups over Commutative Rings

Twelve Bridges from a Reductive Group to Its Langlands Dual

Introduction. These notes are based on a series of lectures given by the author at the Central China Normal University in Wuhan (July 2007). The aim of the lectures was to provide an introduction to the Langlands philosophy. According to a theorem of Chevalley, connected reductive split algebraic groups over a fixed field A are in bijection with certain combinatorial objects called root systems...

A2-proof of Structure Theorems for Chevalley Groups of Type

A new geometric proof is given for the standard description of subgroups in the Chevalley group G = G(F4, R) of type F4 over a commutative ring R that are normalized by the elementary subgroup E(F4, R). There are two major approaches to the proof of such results. Localization proofs (Quillen, Suslin, Bak) are based on a reduction in the dimension. The first proofs of this type for exceptional g...

The generalized total graph of modules respect to proper submodules over commutative rings.

Let $M$ be a module over a commutative ring $R$ and let $N$ be a proper submodule of $M$. The total graph of $M$ over $R$ with respect to $N$, denoted by $T(Gamma_{N}(M))$, have been introduced and studied in [2]. In this paper, A generalization of the total graph $T(Gamma_{N}(M))$, denoted by $T(Gamma_{N,I}(M))$ is presented, where $I$ is an ideal of $R$. It is the graph with all elements of $...