Lie theory

The Lie theory in mathematics is a theory that deals with solving differential equations. It was founded by Sophus Lie in the 1870s and the 1880s. The Lie groups and Lie algebra have evolved from the Lie theory out, but are now regarded as separate research areas.

Historical Development

Starting point for Lie's works was the theory of ordinary differential equations. Similar to the model of the Galois theory for the solution of algebraic equations hoped Lie by the study of symmetry properties to unite the field of ordinary differential equations.

So he led a continuous transformation groups which today bear the name Lie group. Elements of this transformation group were continuous ( or continuous ) symmetry operations which transform ordinary differential equations into each other and thus form equivalence classes of differential equations. Such continuous symmetry operations are, for example displacements and rotations by arbitrary and in a certain sense " infinitesimal " amounts, in contrast to discrete symmetry operations such as reflections.

To investigate and apply continuous transformation groups, he linearized the transformations and examined the infinitesimal generators. The association properties of the Lie group can be expressed by commutators of the generators; the commutator algebra of generators is called a Lie algebra today.