The Historical Researches Of The Early Developments Of Lie Groups

Having important applications in mathematics and physics, Lie group is of the profound significance. Especially for the applications of the representation theory of Lie group in analysis, differential geometry, topology as well as in quantum mechanics. The theory of Lie group originates from Lie's ideas that extend the Galois theory for algebraic equations to the differential equations. Form its beginning, the theory of Lie group was inextricably linked with the developments of algebra, analysis and geometry. With the classification of semisimple Lie algebra accomplished by Killing and Cartan and the global Lie group concept introduced by Weyl, the continuous transformation group established by Lie has blossomed into the modern theory of Lie group in 1920s. By the mid-1950s, with various influence in theoretical and applied mathematics, the theory of complex and real Lie groups shown out its the key position in modern mathematics.Basing on studying original article and relative conference,beginning with a thorough investigation of the motivation, method, scheme and result, this dissertation focuses on the comparison research of the works of Lie, Killing and Cartan's. Through careful analysis of the origination and the introduction of Lie group and the classification of semisimple Lie algebras, some monolithic historical viewpoint which covered this period are obtained. The following several main aspects are discussed:1. By the systematic summary of the prehistory of contact transformations, the review from the whole mathematical viewpoint of the background of the establishing of Lie group in several aspects, i.e. geometry, differential equations and group theory.2. Taking the role of the contact transformation in the establish of Lie group as the clue, focusing on Lie's motivation, i.e. extending the Galois theory for algebraic equations to the differential equations, the establishing of the Lie group is discussed, the important significance of the contact transformations is illustrated.3. Analysising the motivation, the scheme and the characterize of Lie's research of continue transformation group, this dissertation points out that the determination of the type and the structure of the transformation group is not the fundamental goal but only the midcourse of Lie's research. Furthermore, it is pointed out that Lie's trivial classification of the transformation group.4. Comparing the interpretation of Killing's research with Lie, although they faced the same problems, Killing had turned to the research of the Lie algebra with his different fundamental principle.5. Examining Cartan's dissertation and analysising Cartan's comprehension of work of Lie's and Killing's, this dissertation reveals the significance of Cartan's research in the early development of Lie group. Based on this, the historical reason of Cartan's semi-simplicity criterion and solvability criterion is explained.6. Ascertaining the indispensable factors for the modern Lie group development, it is concluded three important milestones from the early Lie group to the modern Lie group, as well as their important.7. Integrating the comprehensions from advanced standpoint with it from reductive viewpoint, with sufficient applications and arguments in the compare research, this dissertation proposed a global historical viewpoint in mathematical historical research.