Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Homotopy and Commutativity Principle

Ravi A. Rao, Sampat Sharma

Abstract

In this article, we prove commutativity principal for linear, symplectic and transvection groups. This principle is a consequence of Quillen-Suslin local global principle and using a non-symmetric application of it as done by A. Bak. The existence of a Local-Global Principle enables us to prove similar results in various groups. We restrict ourselves to the classical symplectic, orthogonal groups (and their relative versions); and to the automorphism groups of a projective module (with a unimodular element), a symplectic module (with ahyperbolic summand), and an orthogonal module (with a hyperbolic symmand). We could show that the symplectic quotients were abelian, but we could only establish that the orthogonal quotients are solvable of length atmost two. We do believe that the orthogonal quotient groups are also abelian; and prove this when the base ring is a regular local ring containing a field.

Homotopy and Commutativity Principle

Abstract

In this article, we prove commutativity principal for linear, symplectic and transvection groups. This principle is a consequence of Quillen-Suslin local global principle and using a non-symmetric application of it as done by A. Bak. The existence of a Local-Global Principle enables us to prove similar results in various groups. We restrict ourselves to the classical symplectic, orthogonal groups (and their relative versions); and to the automorphism groups of a projective module (with a unimodular element), a symplectic module (with ahyperbolic summand), and an orthogonal module (with a hyperbolic symmand). We could show that the symplectic quotients were abelian, but we could only establish that the orthogonal quotients are solvable of length atmost two. We do believe that the orthogonal quotient groups are also abelian; and prove this when the base ring is a regular local ring containing a field.

Paper Structure

This paper contains 4 sections, 44 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Linear and Symplectic group
  3. Transvection groups
  4. Orthogonal groups

Key Result

Theorem 1.1

$($A. Bak$)$ For an almost commutative ring $R$ with identity with centre $C(R)$. The group $\frac{SL_{n}(R)}{E_{n}(R)}$ is nilpotent of class atmost $\delta(C(R)) + 3 - n$, where $\delta(C(R)) < \infty ~and~ n\geq 3$, where $\delta(C(R))$ is the Bass--Serre- dimension of $C(R)$.

Theorems & Definitions (74)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.11
  • ...and 64 more