© 2018

Real Spinorial Groups

A Short Mathematical Introduction


  • Offers an axiomatic presentation of the geometric algebra of an orthogonal geometry

  • Illustrates topics with a variety of examples and applications

  • Relates Lipschitz spinorial groups and how they connect with the group of isometries in a non-degenerate orthogonal geometry

  • Surveys major areas of application, particularly mathematics, physics and engineering


Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Table of contents

  1. Front Matter
    Pages i-x
  2. Sebastià Xambó-Descamps
    Pages 1-21
  3. Sebastià Xambó-Descamps
    Pages 23-40
  4. Sebastià Xambó-Descamps
    Pages 41-61
  5. Sebastià Xambó-Descamps
    Pages 63-76
  6. Sebastià Xambó-Descamps
    Pages 77-106
  7. Sebastià Xambó-Descamps
    Pages 107-135
  8. Back Matter
    Pages 137-151

About this book


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.


orthogonal geometry geometric algebra orthogonal groups spinorial groups geometric covariance

Authors and affiliations

  1. 1.Universitat Politècnica de CatalunyaBarcelonaSpain

About the authors

Sebastià Xambó-Descamps is an Emeritus Full Professor of Information and Coding Theory at the Department of Mathematics of the Universitat Politècnica de Catalunya (UPC). He holds a Ph.D. in Mathematics from the University of Barcelona, and an M.Sc. degree in Mathematics from Brandeis University, USA. Has been Full Professor at the Department of Algebra of the Universidad Complutense of Madrid, President of the Catalan Mathematical Society, Dean of the Faculty of Mathematics and Statistics of the UPC and President of the Spanish Conference of Deans of Mathematics. Led various R+D+I projects, including the development of the Wiris mathematical platform. Among other productions, authored “Block Error-Correcting Codes--A Computational Primer”, co-authored "The Enumerative Theory of Conics after Halphen,” edited “Enumerative Geometry--Sitges 1987,” and  co-edited “Cosmology, Quantum Vacuum and Zeta Functions”. More recently has co-authored the SpringerBrief "A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry".

Bibliographic information


“This … textbook gives an excellent introduction to real spinor groups very suitable for senior students, graduate and postgraduate students, mathematicians, physicists, computer graphics developers and mathematically interested engineers who seek information on this important and still rapidly developing subject.” (Eckhard M. S. Hitzer, Mathematical Reviews, February, 2020)