By Thomas Hawkins
Written by means of the recipient of the 1997 MAA Chauvenet Prize for mathematical exposition, this booklet tells how the idea of Lie teams emerged from a desirable go fertilization of many lines of nineteenth and early twentieth century geometry, research, mathematical physics, algebra and topology. The reader will meet a number of mathematicians from the interval and turn into familiar with the key mathematical faculties. the 1st half describes the geometrical and analytical issues that initiated the idea by the hands of the Norwegian mathematician, Sophus Lie. the most determine within the moment half is Weierstrass'student Wilhelm Killing, whose curiosity within the foundations of non-Euclidean geometry resulted in his discovery of just about the entire significant recommendations and theorems at the constitution and type of semisimple Lie algebras. The scene then shifts to the Paris mathematical neighborhood and Elie Cartans paintings at the illustration of Lie algebras. the ultimate half describes the influential, unifying contributions of Hermann Weyl and their context: Hilberts Göttingen, normal relativity and the Frobenius-Schur idea of characters. The publication is written with the conviction that mathematical figuring out is deepened through familiarity with underlying motivations and the fewer formal, extra intuitive demeanour of unique belief. The human facet of the tale is evoked via wide use of correspondence among mathematicians. The publication should still end up enlightening to a large variety of readers, together with potential scholars of Lie concept, mathematicians, physicists and historians and philosophers of science.
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Extra resources for Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926
Jordan's memoir was the first publication in which continuous groups of transformations had been considered, but as my account in Sections 1-3 31 By setting up a correspondence between the lines of t: and the spheres of 9l, Lie's sphere mapping also sets up a correspondence between all of line geometry and a sphere geometry. Hermann pointed out that this can be interpreted as reflecting the accidental isomorphism of types A3 and D3 [1976:38]. 4. The Sphere Mapping 31 suggests, it does not appear to have been a significant motivating factor behind the initial consideration of continuous groups by Lie and Klein.
The analogy is established by the mapping IJt : 9t -. £", by means of which the transformations of 11)* go over into those of 11). Motivated by this analogy, on the one hand, and by the analogy between his line-geometric proof and the proof of Dupin's theorem on the other, Klein asked, Does a similar analogy exist between all of line geometry and the metrical geometry of four-dimensional space? , all T E PGL(6, IC) that take the "line manifold M~2) into itself. Likewise, "the metrical geometry of a space of four dimensions" is defined, by analogy with three-dimensional metrical geometry, as follows.
20 Chapter 1. The Geometrical Origins of Lie's Theory occupied with other problems (the sphere mapping and its implicationsSection 4) Lie himself had neither the time nor the temperament and talent for systematic exposition. It was not until the end of his life that Lie, with the assistance of his student G. Scheffers, presented some of his results in [18961. Thus the development of ideas related to the group concept might have ended with the publication of [18711. That it did not was due to two reasons.