Lie theory

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

historical development

The starting point for Lie's work was the theory of ordinary differential equations . Similar to the model of Galois theory for the solution of algebraic equations, Lie hoped to unite the field of ordinary differential equations by investigating symmetry properties. So he introduced constant transformation groups , which today bear the name Lie group . Elements of this transformation group were continuous, also called continuously, symmetry operations, which convert ordinary differential equations into one another 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 even “infinitesimal” amounts, in contrast to discrete symmetry operations such as reflections . In order to examine and apply continuous transformation groups, he linearized the transformations and examined the infinitesimal generators . The connection properties of the Lie group can be expressed by commutators of the generators; the commutator algebra of generators is now called Lie algebra .

literature

• Fritz Schwarz: Algorithmic Lie Theory for Solving Ordinary Differential Equations . Chapman & Hall, 2007, ISBN 978-1-58488-889-5 .