|San José State University|
& Tornado Alley
The Discovery of Lie Groups and Algebras|
and Their Properties
Marius Sophus Lie was a Norwegian mathematician who discovered the remarkable properties of transformations of variables. He was born December 17th of 1842. He was not content to be just a Norwegian mathematician. He journeyed to Berlin in 1869 to work with Felix Klein. In 1870 he was in Paris where he worked out his theory of contact transformations. In 1871 he was back in Norway at the University of Kristiana (Oslo) where he became an assistant tutor. In that year he completed his doctorate and in 1872 he was appointed extraordinary professor at the University of Kristiana.
At that time he began his work on continuous transformation groups. That field was soon named after him as Lie groups. These were algebraic groups which were also manifolds. He worked with Ernst Engel for nine years in the development of continuous groups. In 1886 Lie to the place of Felix Klein at the University of Leipzig as the chairman of their mathematics department. He remained in Leipzig until 1898. In 1898 he returned to the University of Kristiana to take a post especially created for him. But he was then in the later stages of tuberculosis and had only about a year left to live. Before he died he published multi-volume compilations of his work.
In the 19th century Lie and other pure mathematicians worked out the strictly mathematical properties of groups of infinitesimal transformations (Lie groups). It wasn't until the 1920's that physicists discovered the applications of Lie groups to physics that research on Lie groups burgeoned. The mathematics involved in the analysis is at a very high level of abstraction. For the physicist or applied mathematician it is best to note the results of the analysis and move on to their utilization. Here are some such results;
These Lie groups are divided into the following categories:
|The Simple Lie Groups|
|A||special unitary group||SU(n)||n×n complex matrices|
of determinant 1
|B||Special orthogonal group||SO(2n+1)||(2n+1)×(2n+1) real matrices|
of determinant equal to 1
|C||Special symplectic group||SP(n)||n×n matrices of quaternions|
that preserve inner products
|D||Special orthogonal group||SO(2n)||(2n)×(2n) real matrices |
with determinant equal to 1
In addition to the above infinite categories there are five exceptional simple Lie groups labeled E6, E7, E8, F4, and G2.
(More on these later.)
(To be continued.)
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