ScholarLens Back to search
2012•Oxford University Press eBooksRequires access

Casimir operators for orthogonal groups

Adam M. Bincer

Open publisher page 0 citations

Abstract

Casimir operators for orthogonal groups are defined. Rank for semisimple groups is defined and shown to equal m for SO(2m) and SO(2m+1). It is shown that there are m independent Casimirs and a set of them is presented in the form of polynomials in the generators of degree 2k, 1 ≤ k≤ m. For SO(2m) the Casimir of degree 2m must be replaced in the integrity basis by a Casimir of degree m defined using the invariant ε tensor. This special Casimir plays a crucial role in distinguishing conjugate representations that are inequivalent. Biographical notes on Pfaff are given.

About this research paper

What this paper is about

Casimir operators for orthogonal groups are defined. Rank for semisimple groups is defined and shown to equal m for SO(2m) and SO(2m+1). It is shown that there are m independent Casimirs and a set of them is presented in the form of polynomials in the generators of degree 2k, 1 ≤ k≤ m. For SO(2m) the Casimir of degree 2m must be replaced in the integrity basis by a Casimir of degree m defined using the invariant ε tensor. This special Casimir plays a crucial role in distinguishing conjugate representations that are inequivalent. Biographical notes on Pfaff are given.

Why it matters

A significance statement is not available in the OpenAlex record.

Key contribution

A contribution statement is not available in the OpenAlex record.

Method / approach

Method details are not available in the OpenAlex metadata.

Main findings

Findings are not separately available in the OpenAlex metadata.

Limitations

Limitations are not available in the OpenAlex metadata.

Applications

Application details are not available in the OpenAlex metadata.

Available abstract

Casimir operators for orthogonal groups are defined. Rank for semisimple groups is defined and shown to equal m for SO(2m) and SO(2m+1). It is shown that there are m independent Casimirs and a set of them is presented in the form of polynomials in the generators of degree 2k, 1 ≤ k≤ m. For SO(2m) the Casimir of degree 2m must be replaced in the integrity basis by a Casimir of degree m defined using the invariant ε tensor. This special Casimir plays a crucial role in distinguishing conjugate representations that are inequivalent. Biographical notes on Pfaff are given.

Key concepts: Casimir effect, Casimir element, Mathematics, Pure mathematics, Invariant (physics), Rank (graph theory), Degree (music), Algebra over a field

Related papers

Related paper 1Related paper 2Related paper 3
Back to paper searchBrowse research topicsOriginal source
Casimir operators for orthogonal groups — Research Paper | ScholarLens