Yaglom I.M. Felix Klein and Sophus Lie: Evolution of the Idea of Symmetry in the Nineteenth Century

Translated by S. Sossinsky. — Edited by H. Grant and A. Shenitzer. — Boston; Basel: Birkhäuser, 1988. — 251 p.While sufficient attention is paid in the book to the origins of the concept of symmetry in the "new" European mathematics, the further development of the relevant ideas is not exhaustively treated here. The ideas derived from Klein and Lie had a profound impact on all of twentieth-century mathematics; in the present book the narrative is largely restricted to the nineteenth century. We deal neither with the rather extensive nineteenth-century research in the natural sciences related to the rise of scientific crystallography nor with the massive twentieth-century applications of symmetry in theoretical physics in general and in the theory of elementary particles in particular. But then this simple introductory book does not claim to be in any sense exhaustive. Our purpose is to kindle the reader's interest and to prompt him or her to explore the relevant literature.Foreword.
The Precursors: Evariste Galois and Camille Jordan.
Jordan’s Pupils.
Nineteenth-Century Geometry: Projective Geometry.
Nineteenth-Century Geometry: Non-Euclidean Geometries.
Nineteenth-Century Geometry: Multidimensional Spaces; Vectors and (Hyper) Complex Numbers.
Sophus Lie and Continuous Groups.
Felix Klein and His Erlangen Program.
Biographical Sketches.Book in Russian: /file/298769/