Talk:Lie theory

Tangential text

The following text was removed as tangential to the topic:

The hope that Lie Theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.

In the longer term, it has not been the direct application of continuous symmetry to geometric questions that has made Lie theory a central chapter of contemporary mathematics. The fact that there is a good structure theory for Lie groups and their representations has made them integral to large parts of abstract algebra. Some major areas of application have been found, for example in automorphic representations and in mathematical physics, and the subject has become a busy crossroads.

Improvements to the Lie theory article can be suggested here.Rgdboer (talk) 01:49, 27 October 2014 (UTC)

Over-reach

The power of Lie theory to describe the Euclidean group E(3) is evidenced in Screw theory#Homography where dual quaternions are employed. In contrast, quaternion analysis shows how homographies with the real quaternions ℍ suffice to generate E(3). It appears that Lie theory would absorb transformation geometry. The generation of a homography group does not employ the exponential function and stands apart from Lie theory. Some authors have claimed that the Möbius group, an important homography group, is just the identity component of SO(3,1) ! — Rgdboer (talk) 03:02, 10 May 2019 (UTC)