Matrix groups. An introduction to Lie group theory.

*(English)*Zbl 1009.22001
Springer Undergraduate Mathematics Series. London: Springer. xi, 330 p. (2002).

This book is an introduction to Lie group theory with focus on the matrix case. Chapter one presents several standard matrix groups: \(Gl_n(K)\), \(Sl_n(K)\), \(O(n)\), \(SO(n)\), \(U(n)\), \(SU(n)\), the Lorentz groups and symplectic groups. Topological aspects are mentioned. Chapter two deals with the exponential of matrices and the related one-parameter subgroups. The Lie algebra and associated entities is the theme of chapter three. The next two chapters treat various topics on algebras and examples (Clifford algebras, spinor groups, quaternionic groups, automorphism groups of algebras). Chapter six details the Lorentz group. A next part introduces to abstract Lie groups and the differential geometric perspective, homogeneous spaces, the connectivity of matrix groups. A final part introduces to compact connected Lie groups, tori, semi-simple decompositions, the adjoint representation. Chapter twelve presents root systems, Weyl groups and Dynkin diagrams. Exercises are included and hints for the solution to some of them are located at the end. One finds also a bibliography (29 entries) and an index.

This book can be recommended to students, making Lie group theory more accessible to them.

This book can be recommended to students, making Lie group theory more accessible to them.

Reviewer: A.Akutowicz (Berlin)

##### MSC:

22-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to topological groups |

57-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes |

22E46 | Semisimple Lie groups and their representations |

17-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to nonassociative rings and algebras |

15A66 | Clifford algebras, spinors |