Hostname: page-component-7bb8b95d7b-lvwk9 Total loading time: 0 Render date: 2024-09-13T05:19:34.543Z Has data issue: false hasContentIssue false

4 - The classical groups

Published online by Cambridge University Press:  05 June 2015

Summary

The classical groups

Hermann Weyl [243] coined the phrase ‘classical groups’ to describe certain subgroups of general linear groups, and other groups derived from these subgroups, that preserve particular sesquilinear forms. We discuss the collection of all classical groups in this section, but only study reversibility in a selection of them.

Let V denote a finite-dimensional vector space over a commutative field F. The general linear group of V is the group of all invertible linear transformations of V. It is denoted GL(V). Let σ denote a field automorphism of F that is an involution. A sesquilinear form on V, relative to σ, is a map B from V × V to F such that for all vectors u, v, and w in V, and scalars λ and μ in F,

(i) B(λu + μv, w) = λB(u, w) + μB(v, w)

(ii) B(w, λu + μv) = σ(λ)B(w, u) + σ(μ)B(w, v).

If σ is the identity map, then B is said to be bilinear. The sesquilinear form B is nondegenerate if each of the conditions B(u, v) = 0 for all vectors v in V, or B(v, u) = 0 for all vectors v in V, imply that u = 0. A nondegenerate sesquilinear form B is reflexive if the equation B(u, v) = 0 implies that B(v, u) = 0.

A reflexive, nondegenerate sesquilinear form V is said to be

1. alternating, if B is bilinear, and B(v, v) = 0 for each vector v in V

2. Hermitian, if σ is of order 2, and is such that B(u, v) = σ(B(v, u)) for each pair of vectors u and v

3. symmetric, if B is bilinear, and B(u, v) = B(v, u) for each pair of vectors u and v.

A famous theorem of Birkhoff and von Neumann [225, Theorem 7.1] says that each reflexive, nondegenerate sesquilinear form is either alternating, Hermitian, or symmetric. We define the isometry group of B, Isom(B), to be the subgroup of GL(V) consisting of those maps that preserve B. That is

Isom(B) = {g ∈ GL(V) : B(g(u), g(v)) = B(u, v) for each pair u and v in V}.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2015

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

• The classical groups
• Book: Reversibility in Dynamics and Group Theory
• Online publication: 05 June 2015
• Chapter DOI: https://doi.org/10.1017/CBO9781139998321.005
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

• The classical groups
• Book: Reversibility in Dynamics and Group Theory
• Online publication: 05 June 2015
• Chapter DOI: https://doi.org/10.1017/CBO9781139998321.005
Available formats
×

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

• The classical groups
• Book: Reversibility in Dynamics and Group Theory
• Online publication: 05 June 2015
• Chapter DOI: https://doi.org/10.1017/CBO9781139998321.005
Available formats
×