The author describes the development of the Lie groups and their ad vancement into the modern mathematics. At the end of the last century Sophus Lie investigated in his voluminous work the local properties of the Lie groups. Not before about 1925 mathematicians, instigated by E. Cartan and H. Weyl, engaged in researching their global structure. Very soon the conception of Lie groups triumphantly advanced within the whole scope of mathematics. The author gives a detailed account of the various applications of the Lie groups, namely the linear representations of the Lie groups, their applications in differential geometry, differential topology, in algebraic geometry and re cently in arithmetic and in the theory of finite groups. Resume L'auteur deerit le developpement des "groupes de Lie" et leur entree dans les mathematiques modernes. Sophus Lie a etudie les proprietes loeales des "groupes de Lie" dans son reuvre volumineux aIa fin du dernier siede. Ce n' etait qu' a partir de 1925 qu' on s'est oeeupe de leur strueture globale aI'instigation de E. Cartan et H. Wey!.

Klappentext

The author describes the development of the Lie groups and their ad vancement into the modern mathematics. At the end of the last century Sophus Lie investigated in his voluminous work the local properties of the Lie groups. Not before about 1925 mathematicians, instigated by E. Cartan and H. Weyl, engaged in researching their global structure. Very soon the conception of Lie groups triumphantly advanced within the whole scope of mathematics. The author gives a detailed account of the various applications of the Lie groups, namely the linear representations of the Lie groups, their applications in differential geometry, differential topology, in algebraic geometry and re cently in arithmetic and in the theory of finite groups. Resume L'auteur deerit le developpement des «groupes de Lie» et leur entree dans les mathematiques modernes. Sophus Lie a etudie les proprietes loeales des «groupes de Lie» dans son reuvre volumineux aIa fin du dernier siede. Ce n' etait qu' a partir de 1925 qu' on s'est oeeupe de leur strueture globale aI'instigation de E. Cartan et H. Wey!.

Inhalt
Die Lieschen Gruppen in der modernen Mathematik.- Das Werk von Sophus Lie.- Lineare Darstellung der Lieschen Gruppen.- Anwendungen in der Differentialgeometrie und der Differentialtopologie.- Anwendungen in der algebraischen Geometrie und der Arithmetik.- Anwendung auf die Theorie der endlichen Gruppen.- Literatur.- Diskussionbeiträge.- Mathematische Probleme der modernen Wellenoptik.- Diskussionsbeiträge.