##
**The geometry of Jordan and Lie structures.**
*(English)*
Zbl 1014.17024

Lecture Notes in Mathematics. 1754. Berlin: Springer. xvi, 269 p. (2000).

In the last ten years the connection between geometry and Jordan algebras has been extended to a broader class of symmetric spaces. The objective of the book under review are general symmetric spaces, but also the potential applications of Jordan theory to harmonic analysis, from a geometric point of view. The method seems to be new. The present book tries to avoid the traditional approach of presentation by starting with geometric concepts (symmetric spaces, Lie groups) and adding additional geometric features (polarizations, almost complex structures). Thus, by applying the condition \(J^2 = \text{id}\) instead of \(J^2 = -\text{id}\) in the definition of complex structure, one arrives at paracomplex structures or polarizations, distinguished in “straight” and “twisted” case. Their formal definition is given in terms of the curvature tensor \(R\) and the invariant almost complex structure \(J\) of the complexified space.

It is a main result of the book that every irreducible symmetric space of classical type admits a twisted complexification and that there are one, two or three inequivalent ones (mostly there exists only one). A surprising conclusion is that a simple Lie group is classical if and only if it admits a twisted complexification. These concepts lead to another type of classification of irreducible symmetric spaces. The conformal group (in the irreducible case) is characterized by a theorem that is a generalization of both, the classical Liouville theorem and the fundamental theorem of projective geometry. A Jordan version of the “Campbell-Hausdorff formula” is given globally for the symmetric space by means of algebraic conditions, the so called Jordan coordinates.

The text is divided into two parts. The objective of the first one (Chapters I–V) is to develop an intrinsic theory of the Jordan-Lie functor and to describe its global geometric content. The theory presented in this part is easy and also general. The reader needs only ordinary tensor calculus and some basic theory of symmetric spaces, given in Chapter I. Prehomogeneous symmetric spaces are defined in Chapter II and their connection with the Lie triple algebra is described. In this chapter the theory of Jordan algebras is treated independently of the other Jordan structures. In Chapter III an interesting introduction and overview of the geometric significance of Jordan triple systems and Jordan pairs is given. The main result states that twisted complexifications correspond bijectively to Jordan triple systems related to the curvature tensor. Chapter IV presents an exhaustive collection of examples of “classical” symmetric spaces with twist: the general linear groups, Grassmannians and spaces of Lagrangian type, such as the orthogonal, symplectic and unitary groups. Chapter V deals with pseudo-Riemannian and pseudo-Hermitian symmetric spaces.

The second part (Chapters VI–XI) is devoted to the global theory of Jordan triple systems and the corresponding conformal group. In this part, the integrability of invariant almost complex structures and polarizations on symmetric spaces is used in an essential way. The material presented in this part is more complicated and advanced. Chapter VI gives “integrated versions” of Jordan pairs and Jordan triple systems. Chapter VII uses the integrability only on the Lie algebra and leads to a geometric version of the Kantor-Koecher-Tits construction. Chapter VIII develops this construction on groups. Chapters IX and XI contain supplementary information on geometric and algebraic aspects of the theory and Chapter X gives the main results on symmetric spaces. Finally, Chapter XII gives the classification and basic structural information on Jordan algebras, Jordan-triple systems, Jordan-pairs and the corresponding symmetric spaces, in a useful form of tables.

Many results in the present book are recent results of the author. It is a contribution to the understanding of the geometric interplay between Lie and Jordan structures. The book is self contained and the reader needs some basic knowledge of Lie groups and symmetric spaces. This work is a slightly extended version of the author’s Habilitationsschrift (Clausthal 1999).

It is a main result of the book that every irreducible symmetric space of classical type admits a twisted complexification and that there are one, two or three inequivalent ones (mostly there exists only one). A surprising conclusion is that a simple Lie group is classical if and only if it admits a twisted complexification. These concepts lead to another type of classification of irreducible symmetric spaces. The conformal group (in the irreducible case) is characterized by a theorem that is a generalization of both, the classical Liouville theorem and the fundamental theorem of projective geometry. A Jordan version of the “Campbell-Hausdorff formula” is given globally for the symmetric space by means of algebraic conditions, the so called Jordan coordinates.

The text is divided into two parts. The objective of the first one (Chapters I–V) is to develop an intrinsic theory of the Jordan-Lie functor and to describe its global geometric content. The theory presented in this part is easy and also general. The reader needs only ordinary tensor calculus and some basic theory of symmetric spaces, given in Chapter I. Prehomogeneous symmetric spaces are defined in Chapter II and their connection with the Lie triple algebra is described. In this chapter the theory of Jordan algebras is treated independently of the other Jordan structures. In Chapter III an interesting introduction and overview of the geometric significance of Jordan triple systems and Jordan pairs is given. The main result states that twisted complexifications correspond bijectively to Jordan triple systems related to the curvature tensor. Chapter IV presents an exhaustive collection of examples of “classical” symmetric spaces with twist: the general linear groups, Grassmannians and spaces of Lagrangian type, such as the orthogonal, symplectic and unitary groups. Chapter V deals with pseudo-Riemannian and pseudo-Hermitian symmetric spaces.

The second part (Chapters VI–XI) is devoted to the global theory of Jordan triple systems and the corresponding conformal group. In this part, the integrability of invariant almost complex structures and polarizations on symmetric spaces is used in an essential way. The material presented in this part is more complicated and advanced. Chapter VI gives “integrated versions” of Jordan pairs and Jordan triple systems. Chapter VII uses the integrability only on the Lie algebra and leads to a geometric version of the Kantor-Koecher-Tits construction. Chapter VIII develops this construction on groups. Chapters IX and XI contain supplementary information on geometric and algebraic aspects of the theory and Chapter X gives the main results on symmetric spaces. Finally, Chapter XII gives the classification and basic structural information on Jordan algebras, Jordan-triple systems, Jordan-pairs and the corresponding symmetric spaces, in a useful form of tables.

Many results in the present book are recent results of the author. It is a contribution to the understanding of the geometric interplay between Lie and Jordan structures. The book is self contained and the reader needs some basic knowledge of Lie groups and symmetric spaces. This work is a slightly extended version of the author’s Habilitationsschrift (Clausthal 1999).

### MSC:

17C36 | Associated manifolds of Jordan algebras |

53C35 | Differential geometry of symmetric spaces |

17C37 | Associated geometries of Jordan algebras |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

17-02 | Research exposition (monographs, survey articles) pertaining to nonassociative rings and algebras |

22E15 | General properties and structure of real Lie groups |

17A40 | Ternary compositions |

17C30 | Associated groups, automorphisms of Jordan algebras |