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# Lie Groups - Assignment Example

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In the paper “Lie Groups” the author discusses the work of Norwegian mathematician Sophus Lie in the mid-nineteenth century, who has provided means of solving all kinds of differential equations. He found that symmetries of differential equations could be found and used systematically. …

## Extract of sample "Lie Groups"

Download file to see previous pages During this period he gave lectures in mathematics to the students union. He displayed a fine talent in explaining the geometry of the heavens. To provide a means of living, while he decided on his future, he took to lecturing in mathematics. His readings of works in geometry evoked an interest in the subject and he did some original research on the real representation of imaginary quantities in projective geometry. This experience led to the decision to devote his efforts to research in geometry and life as a mathematics researcher. Five years later in 1873, he took the momentous decision of creating a theory of continuous transformation groups, which meant a deviation from his original choice of working in geometry. This research was to occupy the rest of his creative work in mathematics. This creative work of Sophus Lie was to provide mathematics with the Lie theory, Lie groups and symmetries in the solutions of ordinary differential equations (ODEs). (2)
Lie theory has evolved considerably since the days of Sophus Lie, but its basic shape stabilized for the most part upon the publication in 1946 of the pioneering book of Claude Chevalley. A short book by Cartan, which is equally pioneering, had paved the way. If we are willing to ignore a number of details, the subject has two ingredients and three correspondences. The two ingredients are Lie groups and Lie algebras.
A Lie group is a group, wherein it is possible to continuously parameterize its elements by real numbers, like rotation group, which can be parameterized by Euler angles. The significance of Lie groups is in their application to mathematical analysis, physics, and geometry. (3)
There has been sufficient study into the specific classes of infinite-dimensional Lie groups, examples of which are, groups of operators, gauge. ...Download file to see next pagesRead More
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