Real Spinorial Groups

A Short Mathematical Introduction

verfasst von: Prof. Dr. Sebastià Xambó-Descamps

Verlag: Springer International Publishing

Buchreihe : SpringerBriefs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book explores the Lipschitz spinorial groups (versor, pinor, spinor and rotor groups) of a real non-degenerate orthogonal geometry (or orthogonal geometry, for short) and how they relate to the group of isometries of that geometry.
After a concise mathematical introduction, it offers an axiomatic presentation of the geometric algebra of an orthogonal geometry. Once it has established the language of geometric algebra (linear grading of the algebra; geometric, exterior and interior products; involutions), it defines the spinorial groups, demonstrates their relation to the isometry groups, and illustrates their suppleness (geometric covariance) with a variety of examples. Lastly, the book provides pointers to major applications, an extensive bibliography and an alphabetic index.
Combining the characteristics of a self-contained research monograph and a state-of-the-art survey, this book is a valuable foundation reference resource on applications for both undergraduate and graduate students.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Mathematical Background

Abstract

In this chapter we collect mathematical notions and results that are needed later on. After reviewing some general basic notions and facts about groups in Sect. 1.1, and on linear and multilinear algebra in Sect. 1.2, we study metrics on a real vector space, Sect. 1.3, and finally, in Sect. 1.4, the last, we recall what we need about algebras.

Sebastià Xambó-Descamps

Chapter 2. Grassmann Algebra

Abstract

After presenting the exterior algebra in Sect. 2.1, the remaining three sections are devoted to the main aspects of this algebra that depend on the metric, namely the contraction operator, Sect. 2.2, the extension of the metric to the whole algebra, Sect. 2.3, and the inner product, Sect. 2.4. The point of view here is to call Grassmann algebra to the structure formed by the exterior algebra enriched with the metric, the inner product, and the parity and reverse involutions.

Sebastià Xambó-Descamps

Chapter 3. Geometric Algebra

Abstract

The aim of this chapter is to introduce the geometric algebra of a quadratic space (E, q) by following an axiomatic approach. The root idea is to explore how to minimally enrich the structure (E, q) so that vectors can be multiplied with the usual rules of an algebra, and that non-isotropic vectors can be inverted.

Sebastià Xambó-Descamps

Chapter 4. Orthogonal Geometry with GA

Abstract

The aim of this chapter is to explore the remarkable way by which the isometry group O_q = O_q(E) of an orthogonal space (E, q) and its most significant subgroups are accounted for by the formalism of the geometric algebra \((\mathcal {G},E)\).

Sebastià Xambó-Descamps

Chapter 5. Zooming in on Rotor Groups

Abstract

This chapter is devoted to a closer study of the rotor group \(\mathcal {R}=\mathcal {R}_{r,s}\), and as a byproduct of its primacy (in the sense given to this expression in the last chapter), also of the other spinorial and orthogonal groups.

Sebastià Xambó-Descamps

Chapter 6. Postfaces

Abstract

This chapter is a collection of notes. Most of them are pointed out at suitable spots in the text of the preceding chapters, but an effort has been aimed to ensure that it may be read on its own as a closing chapter. The numbered sections, 6.1–6.5, correspond to Chaps. 1–5, respectively, while this prelude (6.0) corresponds to the Preface.

Sebastià Xambó-Descamps

Backmatter

Titel: Real Spinorial Groups
verfasst von: Prof. Dr. Sebastià Xambó-Descamps
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-00404-0
Print ISBN: 978-3-030-00403-3
DOI: https://doi.org/10.1007/978-3-030-00404-0