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**Clifford algebras. Geometric modelling and chain geometries with application in kinematics. With a foreword by Gunter Weiss.**
*(English)*
Zbl 1310.15037

Wiesbaden: Springer Spektrum; Dresden: TU Dresden (Diss. 2014) (ISBN 978-3-658-07617-7/pbk; 978-3-658-07618-4/ebook). xviii, 216 p. (2015).

The book under review is an introduction to Clifford algebras and their modern applications. Both a theoretical background and recent results obtained by the author are included. The main text is divided into three chapters.

Chapter 1 deals with different representations of Euclidean displacements. Moreover, Clifford algebras and their spin and pin groups are described. The homogeneous model and the conformal Clifford algebra model are introduced. Special Cayley-Klein geometries and their isometry groups are presented in these models. A new geometric algebra which allows the description of inversions with respect to quadrics in principal position as pin group is investigated. This new model is constructed for dimension two and three in detail. The generalization to arbitrary dimension is also presented.

In Section 2, the author translates the study of the cross ratio of four dual unit quaternions to the study of the cross ratio of four spin group elements in a special Clifford algebra. Thus, the theory of chain geometry can be applied to every Clifford algebra and its pin or spin group. The author gives the fundamental chain geometric background and defines the cross ratio for Clifford algebras and their pin and spin groups. He also presents a quadric model corresponding to the dual unit quaternions and homogeneous Clifford algebra models whose pin group corresponds to the group of automorphic collineations of Klein’s, Study’s and Lie’s quadrics.

In the final chapter, the Clifford algebra framework is used for unifying different kinematic mappings. A method to map every pin or spin group element to a point on a certain pseudo-algebraic variety in projective space is derived. The author describes kinematic mappings for the Euclidean \(n\)-space (\(n=2,3,4\)). This very good written book has several pictures that express the geometric meaning of the text. The book is intended for graduate students and researchers who are interested in Clifford algebras and their applications.

Chapter 1 deals with different representations of Euclidean displacements. Moreover, Clifford algebras and their spin and pin groups are described. The homogeneous model and the conformal Clifford algebra model are introduced. Special Cayley-Klein geometries and their isometry groups are presented in these models. A new geometric algebra which allows the description of inversions with respect to quadrics in principal position as pin group is investigated. This new model is constructed for dimension two and three in detail. The generalization to arbitrary dimension is also presented.

In Section 2, the author translates the study of the cross ratio of four dual unit quaternions to the study of the cross ratio of four spin group elements in a special Clifford algebra. Thus, the theory of chain geometry can be applied to every Clifford algebra and its pin or spin group. The author gives the fundamental chain geometric background and defines the cross ratio for Clifford algebras and their pin and spin groups. He also presents a quadric model corresponding to the dual unit quaternions and homogeneous Clifford algebra models whose pin group corresponds to the group of automorphic collineations of Klein’s, Study’s and Lie’s quadrics.

In the final chapter, the Clifford algebra framework is used for unifying different kinematic mappings. A method to map every pin or spin group element to a point on a certain pseudo-algebraic variety in projective space is derived. The author describes kinematic mappings for the Euclidean \(n\)-space (\(n=2,3,4\)). This very good written book has several pictures that express the geometric meaning of the text. The book is intended for graduate students and researchers who are interested in Clifford algebras and their applications.

Reviewer: Georgi Hristov Georgiev (Shumen)

### MSC:

15A66 | Clifford algebras, spinors |

15-02 | Research exposition (monographs, survey articles) pertaining to linear algebra |

51N30 | Geometry of classical groups |

53A17 | Differential geometric aspects in kinematics |