The theory of Lie groups is highly developed and of relevance in many parts of contemporary mathematics and theoretical physics. Loosely speaking, a Lie group is a group with the additional structure of a real differentiable manifold, given by local coordinate systems, such that the group operations are smooth functions.
Historically, the study of Lie groups, over the real and complex numbers, arose toward the end of the 19th century, from the analysis of continuous symmetries of differential equations by the mathematician Sophus Lie and others. Around the middle of the 20th century, mathematicians such as Armand Borel and Claude Chevalley found that many of the foundational results concerning Lie groups could be developed completely algebraically, giving rise to the theory of algebraic groups defined over arbitrary fields. This insight opened the way for entirely new directions of investigation. Much of the theory of p-adic Lie groups was developed in the 1960s by mathematicians such as Nicolas Bourbaki, Michel Lazard and Jean-Pierre Serre. Since then the study of p-adic Lie groups and analogues of Lie groups over adele rings has largely been motivated by questions from number theory, e.g. regarding automorphic forms and Galois representations. More recently, p-adic Lie groups have also become a key tool in infinite group theory.
Throughout, let p be a prime. The real numbers ℝ form a completion of the rational numbers ℚ. Similarly, the field of p-adic numbers ℚp is obtained by completing ℚ, albeit with respect to a different, non-archimedean notion of distance.