Lie Theory

Lie theory (/ˈliː/LEE) is an area of mathematics, developed initially by Sophus Lie. Early expressions of Lie theory are found in books composed by Lie with Friedrich Engel and Georg Scheffers from 1888 to 1896.

In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations.

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.